Deck 4: Polynomial and Rational Functions Applications to Optimization

ملء الشاشة (f)
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سؤال
Find all fixed points of the function. f(x)=13+x1f ( x ) = 13 + \sqrt { x - 1 }

A) x = 17
B) x = 0, x = 170
C) x = 10
D) x = 17, x = 10
E) no fixed points
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سؤال
Suppose that the revenue generated by selling x units of a certain commodity is given by R=15x2+600xR = - \frac { 1 } { 5 } x ^ { 2 } + 600 x . Assume that R is in dollars. What is the maximum revenue possible in this situation?

A) $450,000
B) $440,000
C) $430,000
D) $900,000
E) $480,000
سؤال
Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x 4 + 5

A)  <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x<sup> 4 </sup> + 5</strong> A)   x-intercepts: 5 Y-intercept: - 50,625 B)   x-intercepts: -5 Y-intercept: - 50,625 C)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: - 5 D)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: 5 E)   x-intercept: 0 Y-intercept: 0 <div style=padding-top: 35px>  x-intercepts: 5
Y-intercept: - 50,625
B)  <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x<sup> 4 </sup> + 5</strong> A)   x-intercepts: 5 Y-intercept: - 50,625 B)   x-intercepts: -5 Y-intercept: - 50,625 C)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: - 5 D)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: 5 E)   x-intercept: 0 Y-intercept: 0 <div style=padding-top: 35px>  x-intercepts: -5
Y-intercept: - 50,625
C)  <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x<sup> 4 </sup> + 5</strong> A)   x-intercepts: 5 Y-intercept: - 50,625 B)   x-intercepts: -5 Y-intercept: - 50,625 C)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: - 5 D)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: 5 E)   x-intercept: 0 Y-intercept: 0 <div style=padding-top: 35px>  x-intercepts: 534,534- \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }
Y-intercept: - 5
D)  <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x<sup> 4 </sup> + 5</strong> A)   x-intercepts: 5 Y-intercept: - 50,625 B)   x-intercepts: -5 Y-intercept: - 50,625 C)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: - 5 D)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: 5 E)   x-intercept: 0 Y-intercept: 0 <div style=padding-top: 35px>  x-intercepts: 534,534- \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }
Y-intercept: 5
E)  <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x<sup> 4 </sup> + 5</strong> A)   x-intercepts: 5 Y-intercept: - 50,625 B)   x-intercepts: -5 Y-intercept: - 50,625 C)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: - 5 D)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: 5 E)   x-intercept: 0 Y-intercept: 0 <div style=padding-top: 35px>  x-intercept: 0
Y-intercept: 0
سؤال
Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts. s=14t2t1s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1

A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.  <strong>Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts.  s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1 </strong> A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.   B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.   C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.   D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept:  \pm 2  ; s-intercept: 1.   E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept:  \pm \sqrt { 8 }  ;s-intercept: 2.   <div style=padding-top: 35px>
B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.  <strong>Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts.  s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1 </strong> A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.   B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.   C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.   D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept:  \pm 2  ; s-intercept: 1.   E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept:  \pm \sqrt { 8 }  ;s-intercept: 2.   <div style=padding-top: 35px>
C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.  <strong>Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts.  s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1 </strong> A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.   B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.   C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.   D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept:  \pm 2  ; s-intercept: 1.   E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept:  \pm \sqrt { 8 }  ;s-intercept: 2.   <div style=padding-top: 35px>
D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept: ±2\pm 2 ; s-intercept: 1.  <strong>Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts.  s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1 </strong> A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.   B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.   C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.   D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept:  \pm 2  ; s-intercept: 1.   E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept:  \pm \sqrt { 8 }  ;s-intercept: 2.   <div style=padding-top: 35px>
E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept: ±8\pm \sqrt { 8 } ;s-intercept: 2.  <strong>Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts.  s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1 </strong> A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.   B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.   C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.   D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept:  \pm 2  ; s-intercept: 1.   E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept:  \pm \sqrt { 8 }  ;s-intercept: 2.   <div style=padding-top: 35px>
سؤال
Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3) 3 - 1

A) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3)<sup> 3 </sup> - 1</strong> A)   x-intercept: -1 Y-intercept: -1 B)   x-intercept: -2 Y-intercept: 26 C)   x-intercept: 2 Y-intercept: 26 D)   x-intercept: 2 Y-intercept: - 26 E)   x-intercept: 3 Y-intercept: 27 <div style=padding-top: 35px> x-intercept: -1
Y-intercept: -1
B) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3)<sup> 3 </sup> - 1</strong> A)   x-intercept: -1 Y-intercept: -1 B)   x-intercept: -2 Y-intercept: 26 C)   x-intercept: 2 Y-intercept: 26 D)   x-intercept: 2 Y-intercept: - 26 E)   x-intercept: 3 Y-intercept: 27 <div style=padding-top: 35px> x-intercept: -2
Y-intercept: 26
C) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3)<sup> 3 </sup> - 1</strong> A)   x-intercept: -1 Y-intercept: -1 B)   x-intercept: -2 Y-intercept: 26 C)   x-intercept: 2 Y-intercept: 26 D)   x-intercept: 2 Y-intercept: - 26 E)   x-intercept: 3 Y-intercept: 27 <div style=padding-top: 35px> x-intercept: 2
Y-intercept: 26
D) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3)<sup> 3 </sup> - 1</strong> A)   x-intercept: -1 Y-intercept: -1 B)   x-intercept: -2 Y-intercept: 26 C)   x-intercept: 2 Y-intercept: 26 D)   x-intercept: 2 Y-intercept: - 26 E)   x-intercept: 3 Y-intercept: 27 <div style=padding-top: 35px> x-intercept: 2
Y-intercept: - 26
E) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3)<sup> 3 </sup> - 1</strong> A)   x-intercept: -1 Y-intercept: -1 B)   x-intercept: -2 Y-intercept: 26 C)   x-intercept: 2 Y-intercept: 26 D)   x-intercept: 2 Y-intercept: - 26 E)   x-intercept: 3 Y-intercept: 27 <div style=padding-top: 35px> x-intercept: 3
Y-intercept: 27
سؤال
A triangle is inscribed in a semicircle of diameter 6R. Show that the smallest possible value for the area of the shaded region is 9(π2)R22\frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .  <strong>A triangle is inscribed in a semicircle of diameter 6R. Show that the smallest possible value for the area of the shaded region is  \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  .   Hint: The area of the shaded region is a minimum when the area of the triangle is a maximum. Find the value of x that maximizes the square of the area of the triangle. This will be the same x that maximizes the area of the triangle.</strong> A) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The area of the triangle is equal to  A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } }  . The square of the area of the triangle is equal to  ( A ( x ) ) ^ { 2 } = 9 R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 }  , and the substitution  x ^ { 2 } = t  will transform this expression into the quadratic function  - \frac { 1 } { 4 } t ^ { 2 } + 9 R ^ { 2 } t ( 1 )  Since we want to find the maximum value of t, we will substitute the value  t = 18 R ^ { 2 } = x ^ { 2 }  into the equation. Solving for t gives us the following minimum area of the shaded region:  t = \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  . B) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The area of the triangle is equal to  A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } }  . The square of the area of the triangle is equal to  ( A ( x ) ) ^ { 2 } = 9 R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 }  The substitution  x ^ { 2 } = t  will transform this expression into the quadratic function  - \frac { 1 } { 4 } t ^ { 2 } + 9 R ^ { 2 } t ( 1 )  Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is  t = \frac { - b } { 2 a } = 18 R ^ { 2 }  Substituting the value  t = 18 R ^ { 2 }  into the equation  t = x ^ { 2 }  gives us  x ^ { 2 } = 18 R ^ { 2 }  and consequently  x = 3 R \sqrt { 2 }  (The negative root can be rejected since the side of a triangle can't be negative). With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to  \frac { 9 \pi R ^ { 2 } } { 2 } - \frac { 1 } { 2 } x \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } }  Substituting the value  x = 3 R \sqrt { 2 }  in the equation (2) gives us that the minimum value of the shaded region is equal to  \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  . C) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The square of the area of the triangle is equal to  \frac { 1 } { 4 } 9 R ^ { 2 } x ^ { 2 } + \frac { 1 } { 4 } x ^ { 4 }  . The substitution  x ^ { 2 } = t  will transform this into the quadratic function  \frac { 1 } { 4 } 9 R ^ { 2 } t + \frac { 1 } { 4 } t ^ { 2 }  Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is  t = \frac { - b } { 2 a } = 9 R ^ { 2 }  Substituting the value  t = 9 R ^ { 2 }  into the equation  t = x ^ { 2 }  gives us  x ^ { 2 } = 9 R ^ { 2 }  and consequently  x = 3 R  . With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to  \pi 9 R ^ { 2 } - \frac { 1 } { 2 } x \sqrt { 6 R ^ { 2 } - x ^ { 2 } }  Substituting the value  x = 3 R  into the equation (2), we find that the minimum value of the shaded region is equal to  \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  . D) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The square of the area of the triangle is equal to  A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } }  , which is a quadratic function. The graph of this function will be a parabola opening downward, so we can write the maximum value of this function as:  x ^ { 2 } = \frac { - b } { 2 a } = 18 R ^ { 2 }  We can then write  x ^ { 2 } = 18 R ^ { 2 }  as  x ^ { 2 } = 18 R ^ { 2 }  and calculate the minimum area of the shaded region. Substituting this value into the area equation, we find its minimum area:  \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  . E) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The square of the area of the triangle is equal to  \frac { 1 } { 2 } 9 ^ { 2 } R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 }  . The substitution  x ^ { 2 } = t  will transform this into the quadratic function  \frac { 1 } { 2 } 9 ^ { 2 } R ^ { 2 } t - \frac { 1 } { 4 } t ^ { 2 }  Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is  t = \frac { - b } { 2 a } = 3 R ^ { 2 }  Substituting the value  t = 3 R ^ { 2 }  into the equation  t = x ^ { 2 }  gives us  x ^ { 2 } = 3 R ^ { 2 }  and consequently  x = R \sqrt { 7 }  . With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to  \frac { \pi 9 R ^ { 2 } } { 4 } - \frac { 1 } { 2 } x \sqrt { 9 R ^ { 2 } - x ^ { 2 } }  Substituting the value  x = R \sqrt { 7 }  into the equation (2), we find that the minimum value of the shaded region is equal to  \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  . <div style=padding-top: 35px>  Hint: The area of the shaded region is a minimum when the area of the triangle is a maximum. Find the value of x that maximizes the square of the area of the triangle. This will be the same x that maximizes the area of the triangle.

A) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The area of the triangle is equal to A(x)=x2(6R)2x2A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } } .
The square of the area of the triangle is equal to (A(x))2=9R2x214x4( A ( x ) ) ^ { 2 } = 9 R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 } , and the substitution x2=tx ^ { 2 } = t will transform this expression into the quadratic function 14t2+9R2t(1)- \frac { 1 } { 4 } t ^ { 2 } + 9 R ^ { 2 } t ( 1 )
Since we want to find the maximum value of t, we will substitute the value t=18R2=x2t = 18 R ^ { 2 } = x ^ { 2 } into the equation. Solving for t gives us the following minimum area of the shaded region: t=9(π2)R22t = \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .
B) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The area of the triangle is equal to A(x)=x2(6R)2x2A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } } .
The square of the area of the triangle is equal to (A(x))2=9R2x214x4( A ( x ) ) ^ { 2 } = 9 R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 } The substitution x2=tx ^ { 2 } = t will transform this expression into the quadratic function 14t2+9R2t(1)- \frac { 1 } { 4 } t ^ { 2 } + 9 R ^ { 2 } t ( 1 )
Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is t=b2a=18R2t = \frac { - b } { 2 a } = 18 R ^ { 2 }
Substituting the value t=18R2t = 18 R ^ { 2 } into the equation t=x2t = x ^ { 2 } gives us x2=18R2x ^ { 2 } = 18 R ^ { 2 } and consequently x=3R2x = 3 R \sqrt { 2 } (The negative root can be rejected since the side of a triangle can't be negative). With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to 9πR2212x(6R)2x2\frac { 9 \pi R ^ { 2 } } { 2 } - \frac { 1 } { 2 } x \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } }
Substituting the value x=3R2x = 3 R \sqrt { 2 } in the equation (2) gives us that the minimum value of the shaded region is equal to 9(π2)R22\frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .
C) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The square of the area of the triangle is equal to 149R2x2+14x4\frac { 1 } { 4 } 9 R ^ { 2 } x ^ { 2 } + \frac { 1 } { 4 } x ^ { 4 } .
The substitution x2=tx ^ { 2 } = t will transform this into the quadratic function 149R2t+14t2\frac { 1 } { 4 } 9 R ^ { 2 } t + \frac { 1 } { 4 } t ^ { 2 }
Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is t=b2a=9R2t = \frac { - b } { 2 a } = 9 R ^ { 2 }
Substituting the value t=9R2t = 9 R ^ { 2 } into the equation t=x2t = x ^ { 2 } gives us x2=9R2x ^ { 2 } = 9 R ^ { 2 } and consequently x=3Rx = 3 R . With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to π9R212x6R2x2\pi 9 R ^ { 2 } - \frac { 1 } { 2 } x \sqrt { 6 R ^ { 2 } - x ^ { 2 } }
Substituting the value x=3Rx = 3 R into the equation (2), we find that the minimum value of the shaded region is equal to 9(π2)R22\frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .
D) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle.
The square of the area of the triangle is equal to A(x)=x2(6R)2x2A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } } , which is a quadratic function. The graph of this function will be a parabola opening downward, so we can write the maximum value of this function as: x2=b2a=18R2x ^ { 2 } = \frac { - b } { 2 a } = 18 R ^ { 2 }
We can then write x2=18R2x ^ { 2 } = 18 R ^ { 2 } as x2=18R2x ^ { 2 } = 18 R ^ { 2 } and calculate the minimum area of the shaded region. Substituting this value into the area equation, we find its minimum area: 9(π2)R22\frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .
E) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The square of the area of the triangle is equal to 1292R2x214x4\frac { 1 } { 2 } 9 ^ { 2 } R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 } .
The substitution x2=tx ^ { 2 } = t will transform this into the quadratic function 1292R2t14t2\frac { 1 } { 2 } 9 ^ { 2 } R ^ { 2 } t - \frac { 1 } { 4 } t ^ { 2 }
Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is t=b2a=3R2t = \frac { - b } { 2 a } = 3 R ^ { 2 }
Substituting the value t=3R2t = 3 R ^ { 2 } into the equation t=x2t = x ^ { 2 } gives us x2=3R2x ^ { 2 } = 3 R ^ { 2 } and consequently x=R7x = R \sqrt { 7 } . With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to π9R2412x9R2x2\frac { \pi 9 R ^ { 2 } } { 4 } - \frac { 1 } { 2 } x \sqrt { 9 R ^ { 2 } - x ^ { 2 } }
Substituting the value x=R7x = R \sqrt { 7 } into the equation (2), we find that the minimum value of the shaded region is equal to 9(π2)R22\frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .
سؤال
Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.

A) Estimated slope: 2; Estimated y-intercept: -12.5 <strong>Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.</strong> A) Estimated slope: 2; Estimated y-intercept: -12.5   B) Estimated slope: 2.5; Estimated y-intercept: -13   C) Estimated slope: 2.9; Estimated y-intercept: -15.3   D) Estimated slope: 1.2; Estimated y-intercept: -1.5   E) Estimated slope: 2.5; Estimated y-intercept: -16   <div style=padding-top: 35px>
B) Estimated slope: 2.5; Estimated y-intercept: -13 <strong>Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.</strong> A) Estimated slope: 2; Estimated y-intercept: -12.5   B) Estimated slope: 2.5; Estimated y-intercept: -13   C) Estimated slope: 2.9; Estimated y-intercept: -15.3   D) Estimated slope: 1.2; Estimated y-intercept: -1.5   E) Estimated slope: 2.5; Estimated y-intercept: -16   <div style=padding-top: 35px>
C) Estimated slope: 2.9; Estimated y-intercept: -15.3 <strong>Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.</strong> A) Estimated slope: 2; Estimated y-intercept: -12.5   B) Estimated slope: 2.5; Estimated y-intercept: -13   C) Estimated slope: 2.9; Estimated y-intercept: -15.3   D) Estimated slope: 1.2; Estimated y-intercept: -1.5   E) Estimated slope: 2.5; Estimated y-intercept: -16   <div style=padding-top: 35px>
D) Estimated slope: 1.2; Estimated y-intercept: -1.5 <strong>Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.</strong> A) Estimated slope: 2; Estimated y-intercept: -12.5   B) Estimated slope: 2.5; Estimated y-intercept: -13   C) Estimated slope: 2.9; Estimated y-intercept: -15.3   D) Estimated slope: 1.2; Estimated y-intercept: -1.5   E) Estimated slope: 2.5; Estimated y-intercept: -16   <div style=padding-top: 35px>
E) Estimated slope: 2.5; Estimated y-intercept: -16 <strong>Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.</strong> A) Estimated slope: 2; Estimated y-intercept: -12.5   B) Estimated slope: 2.5; Estimated y-intercept: -13   C) Estimated slope: 2.9; Estimated y-intercept: -15.3   D) Estimated slope: 1.2; Estimated y-intercept: -1.5   E) Estimated slope: 2.5; Estimated y-intercept: -16   <div style=padding-top: 35px>
سؤال
Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2) 2 + 4

A) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2)<sup> 2 </sup> + 4</strong> A)   There is no x-intercept. y-intercept: 8 B)   There is no x-intercept. y-intercept: 8 C)   There is no x-intercept. y-intercept: 7 D)   x-intercept: 0 y-intercept: 4 E)   There is no x-intercept. y-intercept: 7 <div style=padding-top: 35px> There is no x-intercept. y-intercept: 8
B) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2)<sup> 2 </sup> + 4</strong> A)   There is no x-intercept. y-intercept: 8 B)   There is no x-intercept. y-intercept: 8 C)   There is no x-intercept. y-intercept: 7 D)   x-intercept: 0 y-intercept: 4 E)   There is no x-intercept. y-intercept: 7 <div style=padding-top: 35px> There is no x-intercept. y-intercept: 8
C) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2)<sup> 2 </sup> + 4</strong> A)   There is no x-intercept. y-intercept: 8 B)   There is no x-intercept. y-intercept: 8 C)   There is no x-intercept. y-intercept: 7 D)   x-intercept: 0 y-intercept: 4 E)   There is no x-intercept. y-intercept: 7 <div style=padding-top: 35px> There is no x-intercept. y-intercept: 7
D) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2)<sup> 2 </sup> + 4</strong> A)   There is no x-intercept. y-intercept: 8 B)   There is no x-intercept. y-intercept: 8 C)   There is no x-intercept. y-intercept: 7 D)   x-intercept: 0 y-intercept: 4 E)   There is no x-intercept. y-intercept: 7 <div style=padding-top: 35px> x-intercept: 0 y-intercept: 4
E) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2)<sup> 2 </sup> + 4</strong> A)   There is no x-intercept. y-intercept: 8 B)   There is no x-intercept. y-intercept: 8 C)   There is no x-intercept. y-intercept: 7 D)   x-intercept: 0 y-intercept: 4 E)   There is no x-intercept. y-intercept: 7 <div style=padding-top: 35px> There is no x-intercept. y-intercept: 7
سؤال
What is the largest possible area for a rectangle with a perimeter of 40 cm?

A) 400 cm 2
B) 100 cm 2
C) 90 cm 2
D) 130 cm 2
E) 150 cm 2
سؤال
Find the linear function satisfying the given conditions. g(0)=0g ( 0 ) = 0 and g(2)=3g ( 2 ) = \sqrt { 3 }

A) g(x)=3x2g ( x ) = \frac { \sqrt { 3 x } } { 2 }
B) g(x)=32xg ( x ) = \frac { \sqrt { 3 } } { 2 } x
C) g(x)=32x3g ( x ) = \frac { \sqrt { 3 } } { 2 } x - 3
D) g(x)=32x+2g ( x ) = \frac { \sqrt { 3 } } { 2 } x + 2
E) g(x)=233xg ( x ) = \frac { 2 \sqrt { 3 } } { 3 } x
سؤال
For the following figure, express the length AB as a function of x. (Hint: Note the similar triangles.)  <strong>For the following figure, express the length AB as a function of x. (Hint: Note the similar triangles.)  </strong> A)  A B ( x ) = \frac { ( x + 2 ) \sqrt { x ^ { 2 } + 9 } } { x ^ { 2 } }  B)  A B ( x ) = \frac { ( x + 2 ) \sqrt { x ^ { 2 } + 9 } } { x }  C)  A B ( x ) = \frac { ( x + 3 ) \sqrt { x + 4 } } { x }  D)  A B ( x ) = \frac { ( x + 3 ) ^ { 2 } ( x + 2 ) } { x }  E)  A B ( x ) = \frac { ( x + 3 ) \left( x ^ { 2 } + 2 \right) } { x }  <div style=padding-top: 35px>

A) AB(x)=(x+2)x2+9x2A B ( x ) = \frac { ( x + 2 ) \sqrt { x ^ { 2 } + 9 } } { x ^ { 2 } }
B) AB(x)=(x+2)x2+9xA B ( x ) = \frac { ( x + 2 ) \sqrt { x ^ { 2 } + 9 } } { x }
C) AB(x)=(x+3)x+4xA B ( x ) = \frac { ( x + 3 ) \sqrt { x + 4 } } { x }
D) AB(x)=(x+3)2(x+2)xA B ( x ) = \frac { ( x + 3 ) ^ { 2 } ( x + 2 ) } { x }
E) AB(x)=(x+3)(x2+2)xA B ( x ) = \frac { ( x + 3 ) \left( x ^ { 2 } + 2 \right) } { x }
سؤال
Find all fixed points of the function. f(x)=3x+4f ( x ) = - 3 x + 4

A) x = 1
B) x = - 1
C) x = 4
D) x = 3
E) no fixed points
سؤال
A factory owner buys a new machine for $25,000. After eight years, the machine has a salvage value of $1,000. Find a formula for the value of the machine after t years, where 0t80 \leq t \leq 8

A) V (t) = - 3,000x + 25,000
B) V (t) = - 3,000x + 1,000
C) V (t) = 3,000x + 25,000
D) V (t) = 3,000x - 25,000
E) V (t) = - 3,000x - 25,000
سؤال
Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)

A) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)</strong> A)   x-intercepts: - 2, 1, 2 Y-intercept: -4 B)   x-intercepts: - 1, 1, 2 Y-intercept: 2 C)   x-intercepts: - 1, 1, 2 Y-intercept: -2 D)   x-intercepts: - 2, 1, 2 Y-intercept: 4 E)   x-intercepts: - 1, 1 Y-intercepts: - 2.25 <div style=padding-top: 35px> x-intercepts: - 2, 1, 2
Y-intercept: -4
B) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)</strong> A)   x-intercepts: - 2, 1, 2 Y-intercept: -4 B)   x-intercepts: - 1, 1, 2 Y-intercept: 2 C)   x-intercepts: - 1, 1, 2 Y-intercept: -2 D)   x-intercepts: - 2, 1, 2 Y-intercept: 4 E)   x-intercepts: - 1, 1 Y-intercepts: - 2.25 <div style=padding-top: 35px> x-intercepts: - 1, 1, 2
Y-intercept: 2
C) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)</strong> A)   x-intercepts: - 2, 1, 2 Y-intercept: -4 B)   x-intercepts: - 1, 1, 2 Y-intercept: 2 C)   x-intercepts: - 1, 1, 2 Y-intercept: -2 D)   x-intercepts: - 2, 1, 2 Y-intercept: 4 E)   x-intercepts: - 1, 1 Y-intercepts: - 2.25 <div style=padding-top: 35px> x-intercepts: - 1, 1, 2
Y-intercept: -2
D) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)</strong> A)   x-intercepts: - 2, 1, 2 Y-intercept: -4 B)   x-intercepts: - 1, 1, 2 Y-intercept: 2 C)   x-intercepts: - 1, 1, 2 Y-intercept: -2 D)   x-intercepts: - 2, 1, 2 Y-intercept: 4 E)   x-intercepts: - 1, 1 Y-intercepts: - 2.25 <div style=padding-top: 35px> x-intercepts: - 2, 1, 2
Y-intercept: 4
E) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)</strong> A)   x-intercepts: - 2, 1, 2 Y-intercept: -4 B)   x-intercepts: - 1, 1, 2 Y-intercept: 2 C)   x-intercepts: - 1, 1, 2 Y-intercept: -2 D)   x-intercepts: - 2, 1, 2 Y-intercept: 4 E)   x-intercepts: - 1, 1 Y-intercepts: - 2.25 <div style=padding-top: 35px> x-intercepts: - 1, 1
Y-intercepts: - 2.25
سؤال
Two points A and B move along the x-axis. After t sec, their positions are given by the equations A: x=4t+120\quad x = 4 t + 120
B: x=30t36\quad x = 30 t - 36 At what time t do A and B have the same x-coordinate?

A) 7 sec
B) 6 sec
C) 5 sec
سؤال
Find all fixed points of the function. h(x)=x27x9h ( x ) = x ^ { 2 } - 7 x - 9

A) x = - 1, x = - 9
B) x = - 1, x = 9
C) x = - 9, x = 1
D) x = 9
E) no fixed points
سؤال
Determine the input that produces the largest or smallest output (whichever is appropriate). State whether the output is largest or smallest. s = - 24t 2 + 147t + 15

A) t=4916t = \frac { 49 } { 16 } ; largest
B) t=1649t = \frac { 16 } { 49 } ; largest
C) x=49x = 49 ; smallest
D) x=49x = 49 ; largest
E) t=4916t = \frac { 49 } { 16 } ; smallest
سؤال
A piece of wire 7πy7 \pi y inches long is bent into a circle. Express the area of the circle as a function of y.

A) A(x)=49πy2A ( x ) = 49 \pi y ^ { 2 }
B) A(x)=49πy22A ( x ) = \frac { 49 \pi y ^ { 2 } } { 2 }
C) A(x)=25y2A ( x ) = 25 y ^ { 2 }
D) A(x)=25πy24A ( x ) = \frac { 25 \pi y ^ { 2 } } { 4 }
E) A(x)=49πy24A ( x ) = \frac { 49 \pi y ^ { 2 } } { 4 }
سؤال
Suppose that the height of an object shot straight up is given by h =544t - 16t 2 . (Here h is in feet and t is in seconds.) Find the maximum height.

A) 4,724 ft
B) 4,624 ft
C) 1,156 ft
D) 4,290 ft
E) 5,189 ft
سؤال
Among all rectangles having a perimeter of 24m, find the dimensions of the one with the largest area.

A) 9 m by 15 m9 \mathrm {~m} \text { by } 15 \mathrm {~m}
B) 94 m by 154 m\frac { 9 } { 4 } \mathrm {~m} \text { by } \frac { 15 } { 4 } \mathrm {~m}
C) 4 m by 6 m4 \mathrm {~m} \text { by } 6 \mathrm {~m}
D) 6mby6 m6 \mathrm { mby } 6 \mathrm {~m}
E) 92 m by 152 m\frac { 9 } { 2 } \mathrm {~m} \text { by } \frac { 15 } { 2 } \mathrm {~m}
سؤال
Sketch the graph of the rational function. Specify the intercepts and the asymptotes. y=1(x+1)2y = \frac { 1 } { ( x + 1 ) ^ { 2 } }

A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { 1 } { ( x + 1 ) ^ { 2 } } </strong> A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;   <div style=padding-top: 35px>
B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { 1 } { ( x + 1 ) ^ { 2 } } </strong> A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;   <div style=padding-top: 35px>
C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { 1 } { ( x + 1 ) ^ { 2 } } </strong> A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;   <div style=padding-top: 35px>
D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { 1 } { ( x + 1 ) ^ { 2 } } </strong> A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;   <div style=padding-top: 35px>
E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { 1 } { ( x + 1 ) ^ { 2 } } </strong> A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;   <div style=padding-top: 35px>
سؤال
Sketch the graph of the function and specify all x- and y-intercepts.y = x 3 - 9x

A) <strong>Sketch the graph of the function and specify all x- and y-intercepts.y = x<sup> 3 </sup> - 9x</strong> A)   x-intercept: 0 Y-intercept: 0 B)   x-intercept: 0 Y-intercept: 0 C)   x-intercepts: -3, 0, 3 Y-intercept: 0 D)   x-intercepts: -3, 0, 3 Y-intercept: 0 E)   x-intercept: 0 Y-intercept: 0 <div style=padding-top: 35px> x-intercept: 0
Y-intercept: 0
B) <strong>Sketch the graph of the function and specify all x- and y-intercepts.y = x<sup> 3 </sup> - 9x</strong> A)   x-intercept: 0 Y-intercept: 0 B)   x-intercept: 0 Y-intercept: 0 C)   x-intercepts: -3, 0, 3 Y-intercept: 0 D)   x-intercepts: -3, 0, 3 Y-intercept: 0 E)   x-intercept: 0 Y-intercept: 0 <div style=padding-top: 35px> x-intercept: 0
Y-intercept: 0
C) <strong>Sketch the graph of the function and specify all x- and y-intercepts.y = x<sup> 3 </sup> - 9x</strong> A)   x-intercept: 0 Y-intercept: 0 B)   x-intercept: 0 Y-intercept: 0 C)   x-intercepts: -3, 0, 3 Y-intercept: 0 D)   x-intercepts: -3, 0, 3 Y-intercept: 0 E)   x-intercept: 0 Y-intercept: 0 <div style=padding-top: 35px> x-intercepts: -3, 0, 3
Y-intercept: 0
D) <strong>Sketch the graph of the function and specify all x- and y-intercepts.y = x<sup> 3 </sup> - 9x</strong> A)   x-intercept: 0 Y-intercept: 0 B)   x-intercept: 0 Y-intercept: 0 C)   x-intercepts: -3, 0, 3 Y-intercept: 0 D)   x-intercepts: -3, 0, 3 Y-intercept: 0 E)   x-intercept: 0 Y-intercept: 0 <div style=padding-top: 35px> x-intercepts: -3, 0, 3
Y-intercept: 0
E) <strong>Sketch the graph of the function and specify all x- and y-intercepts.y = x<sup> 3 </sup> - 9x</strong> A)   x-intercept: 0 Y-intercept: 0 B)   x-intercept: 0 Y-intercept: 0 C)   x-intercepts: -3, 0, 3 Y-intercept: 0 D)   x-intercepts: -3, 0, 3 Y-intercept: 0 E)   x-intercept: 0 Y-intercept: 0 <div style=padding-top: 35px> x-intercept: 0
Y-intercept: 0
سؤال
Sketch the graph of the rational function. Specify the intercepts and the asymptotes. y=x3x+3y = \frac { x - 3 } { x + 3 }

A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { x - 3 } { x + 3 } </strong> A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;   B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;   D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;   <div style=padding-top: 35px>
B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { x - 3 } { x + 3 } </strong> A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;   B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;   D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;   <div style=padding-top: 35px>
C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { x - 3 } { x + 3 } </strong> A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;   B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;   D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;   <div style=padding-top: 35px>
D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { x - 3 } { x + 3 } </strong> A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;   B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;   D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;   <div style=padding-top: 35px>
E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { x - 3 } { x + 3 } </strong> A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;   B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;   D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;   <div style=padding-top: 35px>
سؤال
Sketch the graph of the function and specify all x- and y-intercepts. y = x 3 (x + 1)

A) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = x<sup> 3 </sup> (x + 1)</strong> A)   x-intercepts: 0, 1 Y-intercept: 0 B)   x-intercepts: 0, - 1 Y-intercept: 0 C)   x-intercepts: 0, 1 Y-intercept: 0 D)   x-intercepts: 0, 1 Y-intercept: 0 E)   x-intercepts: 0, - 1 Y-intercept: 0 <div style=padding-top: 35px> x-intercepts: 0, 1
Y-intercept: 0
B) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = x<sup> 3 </sup> (x + 1)</strong> A)   x-intercepts: 0, 1 Y-intercept: 0 B)   x-intercepts: 0, - 1 Y-intercept: 0 C)   x-intercepts: 0, 1 Y-intercept: 0 D)   x-intercepts: 0, 1 Y-intercept: 0 E)   x-intercepts: 0, - 1 Y-intercept: 0 <div style=padding-top: 35px> x-intercepts: 0, - 1
Y-intercept: 0
C) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = x<sup> 3 </sup> (x + 1)</strong> A)   x-intercepts: 0, 1 Y-intercept: 0 B)   x-intercepts: 0, - 1 Y-intercept: 0 C)   x-intercepts: 0, 1 Y-intercept: 0 D)   x-intercepts: 0, 1 Y-intercept: 0 E)   x-intercepts: 0, - 1 Y-intercept: 0 <div style=padding-top: 35px> x-intercepts: 0, 1
Y-intercept: 0
D) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = x<sup> 3 </sup> (x + 1)</strong> A)   x-intercepts: 0, 1 Y-intercept: 0 B)   x-intercepts: 0, - 1 Y-intercept: 0 C)   x-intercepts: 0, 1 Y-intercept: 0 D)   x-intercepts: 0, 1 Y-intercept: 0 E)   x-intercepts: 0, - 1 Y-intercept: 0 <div style=padding-top: 35px> x-intercepts: 0, 1
Y-intercept: 0
E) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = x<sup> 3 </sup> (x + 1)</strong> A)   x-intercepts: 0, 1 Y-intercept: 0 B)   x-intercepts: 0, - 1 Y-intercept: 0 C)   x-intercepts: 0, 1 Y-intercept: 0 D)   x-intercepts: 0, 1 Y-intercept: 0 E)   x-intercepts: 0, - 1 Y-intercept: 0 <div style=padding-top: 35px> x-intercepts: 0, - 1
Y-intercept: 0
سؤال
Sketch the graph of the rational function. Specify the intercepts and the asymptotes. y=3x4y = - \frac { 3 } { x - 4 }

A) no x-intercepts; y-intercept: 34- \frac { 3 } { 4 } ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = - \frac { 3 } { x - 4 } </strong> A) no x-intercepts; y-intercept:  - \frac { 3 } { 4 }  ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept:  \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept:  - \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept:  \frac { 3 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept:  \frac { 1 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   <div style=padding-top: 35px>
B) no x-intercepts; y-intercept: 43\frac { 4 } { 3 } ; vertical asymptote: x = 4; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = - \frac { 3 } { x - 4 } </strong> A) no x-intercepts; y-intercept:  - \frac { 3 } { 4 }  ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept:  \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept:  - \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept:  \frac { 3 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept:  \frac { 1 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   <div style=padding-top: 35px>
C) no x-intercepts; y-intercept: 43- \frac { 4 } { 3 } ; vertical asymptote: x = 4; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = - \frac { 3 } { x - 4 } </strong> A) no x-intercepts; y-intercept:  - \frac { 3 } { 4 }  ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept:  \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept:  - \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept:  \frac { 3 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept:  \frac { 1 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   <div style=padding-top: 35px>
D) no x-intercepts; y-intercept: 34\frac { 3 } { 4 } ; vertical asymptote: x = 4; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = - \frac { 3 } { x - 4 } </strong> A) no x-intercepts; y-intercept:  - \frac { 3 } { 4 }  ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept:  \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept:  - \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept:  \frac { 3 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept:  \frac { 1 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   <div style=padding-top: 35px>
E) no x-intercepts; y-intercept: 14\frac { 1 } { 4 } ; vertical asymptote: x = 4; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = - \frac { 3 } { x - 4 } </strong> A) no x-intercepts; y-intercept:  - \frac { 3 } { 4 }  ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept:  \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept:  - \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept:  \frac { 3 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept:  \frac { 1 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   <div style=padding-top: 35px>
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Deck 4: Polynomial and Rational Functions Applications to Optimization
1
Find all fixed points of the function. f(x)=13+x1f ( x ) = 13 + \sqrt { x - 1 }

A) x = 17
B) x = 0, x = 170
C) x = 10
D) x = 17, x = 10
E) no fixed points
x = 17
2
Suppose that the revenue generated by selling x units of a certain commodity is given by R=15x2+600xR = - \frac { 1 } { 5 } x ^ { 2 } + 600 x . Assume that R is in dollars. What is the maximum revenue possible in this situation?

A) $450,000
B) $440,000
C) $430,000
D) $900,000
E) $480,000
$450,000
3
Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x 4 + 5

A)  <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x<sup> 4 </sup> + 5</strong> A)   x-intercepts: 5 Y-intercept: - 50,625 B)   x-intercepts: -5 Y-intercept: - 50,625 C)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: - 5 D)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: 5 E)   x-intercept: 0 Y-intercept: 0  x-intercepts: 5
Y-intercept: - 50,625
B)  <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x<sup> 4 </sup> + 5</strong> A)   x-intercepts: 5 Y-intercept: - 50,625 B)   x-intercepts: -5 Y-intercept: - 50,625 C)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: - 5 D)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: 5 E)   x-intercept: 0 Y-intercept: 0  x-intercepts: -5
Y-intercept: - 50,625
C)  <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x<sup> 4 </sup> + 5</strong> A)   x-intercepts: 5 Y-intercept: - 50,625 B)   x-intercepts: -5 Y-intercept: - 50,625 C)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: - 5 D)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: 5 E)   x-intercept: 0 Y-intercept: 0  x-intercepts: 534,534- \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }
Y-intercept: - 5
D)  <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x<sup> 4 </sup> + 5</strong> A)   x-intercepts: 5 Y-intercept: - 50,625 B)   x-intercepts: -5 Y-intercept: - 50,625 C)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: - 5 D)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: 5 E)   x-intercept: 0 Y-intercept: 0  x-intercepts: 534,534- \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }
Y-intercept: 5
E)  <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = - 3x<sup> 4 </sup> + 5</strong> A)   x-intercepts: 5 Y-intercept: - 50,625 B)   x-intercepts: -5 Y-intercept: - 50,625 C)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: - 5 D)   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: 5 E)   x-intercept: 0 Y-intercept: 0  x-intercept: 0
Y-intercept: 0
   x-intercepts:  - \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }  Y-intercept: 5  x-intercepts: 534,534- \sqrt [ 4 ] { \frac { 5 } { 3 } } , \sqrt [ 4 ] { \frac { 5 } { 3 } }
Y-intercept: 5
4
Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts. s=14t2t1s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1

A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.  <strong>Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts.  s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1 </strong> A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.   B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.   C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.   D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept:  \pm 2  ; s-intercept: 1.   E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept:  \pm \sqrt { 8 }  ;s-intercept: 2.
B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.  <strong>Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts.  s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1 </strong> A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.   B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.   C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.   D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept:  \pm 2  ; s-intercept: 1.   E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept:  \pm \sqrt { 8 }  ;s-intercept: 2.
C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.  <strong>Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts.  s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1 </strong> A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.   B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.   C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.   D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept:  \pm 2  ; s-intercept: 1.   E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept:  \pm \sqrt { 8 }  ;s-intercept: 2.
D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept: ±2\pm 2 ; s-intercept: 1.  <strong>Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts.  s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1 </strong> A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.   B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.   C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.   D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept:  \pm 2  ; s-intercept: 1.   E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept:  \pm \sqrt { 8 }  ;s-intercept: 2.
E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept: ±8\pm \sqrt { 8 } ;s-intercept: 2.  <strong>Graph the quadratic function. Specify the vertex, axis of symmetry, maximum or minimum value, and intercepts.  s = - \frac { 1 } { 4 } t ^ { 2 } - t - 1 </strong> A) vertex: (- 2, 0); axis of symmetry: t = - 2; maximum value: 0; t-intercept: - 2; s-intercept: -1.   B) vertex: (- 2, 0); axis of symmetry: t = - 2; minimum value: 0; t-intercept: - 2; s-intercept: 1.   C) vertex: (2, 0); axis of symmetry: t = 2; maximum value: 0; t-intercept: 2; s-intercept: -1.   D) vertex: (0,1); axis of symmetry: t = 0; maximum value: 1; t-intercept:  \pm 2  ; s-intercept: 1.   E) vertex: (0, 2);axis of symmetry: t = 0; maximum value: 2; t-intercept:  \pm \sqrt { 8 }  ;s-intercept: 2.
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5
Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3) 3 - 1

A) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3)<sup> 3 </sup> - 1</strong> A)   x-intercept: -1 Y-intercept: -1 B)   x-intercept: -2 Y-intercept: 26 C)   x-intercept: 2 Y-intercept: 26 D)   x-intercept: 2 Y-intercept: - 26 E)   x-intercept: 3 Y-intercept: 27 x-intercept: -1
Y-intercept: -1
B) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3)<sup> 3 </sup> - 1</strong> A)   x-intercept: -1 Y-intercept: -1 B)   x-intercept: -2 Y-intercept: 26 C)   x-intercept: 2 Y-intercept: 26 D)   x-intercept: 2 Y-intercept: - 26 E)   x-intercept: 3 Y-intercept: 27 x-intercept: -2
Y-intercept: 26
C) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3)<sup> 3 </sup> - 1</strong> A)   x-intercept: -1 Y-intercept: -1 B)   x-intercept: -2 Y-intercept: 26 C)   x-intercept: 2 Y-intercept: 26 D)   x-intercept: 2 Y-intercept: - 26 E)   x-intercept: 3 Y-intercept: 27 x-intercept: 2
Y-intercept: 26
D) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3)<sup> 3 </sup> - 1</strong> A)   x-intercept: -1 Y-intercept: -1 B)   x-intercept: -2 Y-intercept: 26 C)   x-intercept: 2 Y-intercept: 26 D)   x-intercept: 2 Y-intercept: - 26 E)   x-intercept: 3 Y-intercept: 27 x-intercept: 2
Y-intercept: - 26
E) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =- ( x - 3)<sup> 3 </sup> - 1</strong> A)   x-intercept: -1 Y-intercept: -1 B)   x-intercept: -2 Y-intercept: 26 C)   x-intercept: 2 Y-intercept: 26 D)   x-intercept: 2 Y-intercept: - 26 E)   x-intercept: 3 Y-intercept: 27 x-intercept: 3
Y-intercept: 27
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6
A triangle is inscribed in a semicircle of diameter 6R. Show that the smallest possible value for the area of the shaded region is 9(π2)R22\frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .  <strong>A triangle is inscribed in a semicircle of diameter 6R. Show that the smallest possible value for the area of the shaded region is  \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  .   Hint: The area of the shaded region is a minimum when the area of the triangle is a maximum. Find the value of x that maximizes the square of the area of the triangle. This will be the same x that maximizes the area of the triangle.</strong> A) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The area of the triangle is equal to  A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } }  . The square of the area of the triangle is equal to  ( A ( x ) ) ^ { 2 } = 9 R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 }  , and the substitution  x ^ { 2 } = t  will transform this expression into the quadratic function  - \frac { 1 } { 4 } t ^ { 2 } + 9 R ^ { 2 } t ( 1 )  Since we want to find the maximum value of t, we will substitute the value  t = 18 R ^ { 2 } = x ^ { 2 }  into the equation. Solving for t gives us the following minimum area of the shaded region:  t = \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  . B) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The area of the triangle is equal to  A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } }  . The square of the area of the triangle is equal to  ( A ( x ) ) ^ { 2 } = 9 R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 }  The substitution  x ^ { 2 } = t  will transform this expression into the quadratic function  - \frac { 1 } { 4 } t ^ { 2 } + 9 R ^ { 2 } t ( 1 )  Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is  t = \frac { - b } { 2 a } = 18 R ^ { 2 }  Substituting the value  t = 18 R ^ { 2 }  into the equation  t = x ^ { 2 }  gives us  x ^ { 2 } = 18 R ^ { 2 }  and consequently  x = 3 R \sqrt { 2 }  (The negative root can be rejected since the side of a triangle can't be negative). With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to  \frac { 9 \pi R ^ { 2 } } { 2 } - \frac { 1 } { 2 } x \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } }  Substituting the value  x = 3 R \sqrt { 2 }  in the equation (2) gives us that the minimum value of the shaded region is equal to  \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  . C) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The square of the area of the triangle is equal to  \frac { 1 } { 4 } 9 R ^ { 2 } x ^ { 2 } + \frac { 1 } { 4 } x ^ { 4 }  . The substitution  x ^ { 2 } = t  will transform this into the quadratic function  \frac { 1 } { 4 } 9 R ^ { 2 } t + \frac { 1 } { 4 } t ^ { 2 }  Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is  t = \frac { - b } { 2 a } = 9 R ^ { 2 }  Substituting the value  t = 9 R ^ { 2 }  into the equation  t = x ^ { 2 }  gives us  x ^ { 2 } = 9 R ^ { 2 }  and consequently  x = 3 R  . With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to  \pi 9 R ^ { 2 } - \frac { 1 } { 2 } x \sqrt { 6 R ^ { 2 } - x ^ { 2 } }  Substituting the value  x = 3 R  into the equation (2), we find that the minimum value of the shaded region is equal to  \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  . D) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The square of the area of the triangle is equal to  A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } }  , which is a quadratic function. The graph of this function will be a parabola opening downward, so we can write the maximum value of this function as:  x ^ { 2 } = \frac { - b } { 2 a } = 18 R ^ { 2 }  We can then write  x ^ { 2 } = 18 R ^ { 2 }  as  x ^ { 2 } = 18 R ^ { 2 }  and calculate the minimum area of the shaded region. Substituting this value into the area equation, we find its minimum area:  \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  . E) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The square of the area of the triangle is equal to  \frac { 1 } { 2 } 9 ^ { 2 } R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 }  . The substitution  x ^ { 2 } = t  will transform this into the quadratic function  \frac { 1 } { 2 } 9 ^ { 2 } R ^ { 2 } t - \frac { 1 } { 4 } t ^ { 2 }  Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is  t = \frac { - b } { 2 a } = 3 R ^ { 2 }  Substituting the value  t = 3 R ^ { 2 }  into the equation  t = x ^ { 2 }  gives us  x ^ { 2 } = 3 R ^ { 2 }  and consequently  x = R \sqrt { 7 }  . With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to  \frac { \pi 9 R ^ { 2 } } { 4 } - \frac { 1 } { 2 } x \sqrt { 9 R ^ { 2 } - x ^ { 2 } }  Substituting the value  x = R \sqrt { 7 }  into the equation (2), we find that the minimum value of the shaded region is equal to  \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 }  .  Hint: The area of the shaded region is a minimum when the area of the triangle is a maximum. Find the value of x that maximizes the square of the area of the triangle. This will be the same x that maximizes the area of the triangle.

A) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The area of the triangle is equal to A(x)=x2(6R)2x2A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } } .
The square of the area of the triangle is equal to (A(x))2=9R2x214x4( A ( x ) ) ^ { 2 } = 9 R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 } , and the substitution x2=tx ^ { 2 } = t will transform this expression into the quadratic function 14t2+9R2t(1)- \frac { 1 } { 4 } t ^ { 2 } + 9 R ^ { 2 } t ( 1 )
Since we want to find the maximum value of t, we will substitute the value t=18R2=x2t = 18 R ^ { 2 } = x ^ { 2 } into the equation. Solving for t gives us the following minimum area of the shaded region: t=9(π2)R22t = \frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .
B) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The area of the triangle is equal to A(x)=x2(6R)2x2A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } } .
The square of the area of the triangle is equal to (A(x))2=9R2x214x4( A ( x ) ) ^ { 2 } = 9 R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 } The substitution x2=tx ^ { 2 } = t will transform this expression into the quadratic function 14t2+9R2t(1)- \frac { 1 } { 4 } t ^ { 2 } + 9 R ^ { 2 } t ( 1 )
Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is t=b2a=18R2t = \frac { - b } { 2 a } = 18 R ^ { 2 }
Substituting the value t=18R2t = 18 R ^ { 2 } into the equation t=x2t = x ^ { 2 } gives us x2=18R2x ^ { 2 } = 18 R ^ { 2 } and consequently x=3R2x = 3 R \sqrt { 2 } (The negative root can be rejected since the side of a triangle can't be negative). With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to 9πR2212x(6R)2x2\frac { 9 \pi R ^ { 2 } } { 2 } - \frac { 1 } { 2 } x \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } }
Substituting the value x=3R2x = 3 R \sqrt { 2 } in the equation (2) gives us that the minimum value of the shaded region is equal to 9(π2)R22\frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .
C) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The square of the area of the triangle is equal to 149R2x2+14x4\frac { 1 } { 4 } 9 R ^ { 2 } x ^ { 2 } + \frac { 1 } { 4 } x ^ { 4 } .
The substitution x2=tx ^ { 2 } = t will transform this into the quadratic function 149R2t+14t2\frac { 1 } { 4 } 9 R ^ { 2 } t + \frac { 1 } { 4 } t ^ { 2 }
Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is t=b2a=9R2t = \frac { - b } { 2 a } = 9 R ^ { 2 }
Substituting the value t=9R2t = 9 R ^ { 2 } into the equation t=x2t = x ^ { 2 } gives us x2=9R2x ^ { 2 } = 9 R ^ { 2 } and consequently x=3Rx = 3 R . With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to π9R212x6R2x2\pi 9 R ^ { 2 } - \frac { 1 } { 2 } x \sqrt { 6 R ^ { 2 } - x ^ { 2 } }
Substituting the value x=3Rx = 3 R into the equation (2), we find that the minimum value of the shaded region is equal to 9(π2)R22\frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .
D) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle.
The square of the area of the triangle is equal to A(x)=x2(6R)2x2A ( x ) = \frac { x } { 2 } \sqrt { ( 6 R ) ^ { 2 } - x ^ { 2 } } , which is a quadratic function. The graph of this function will be a parabola opening downward, so we can write the maximum value of this function as: x2=b2a=18R2x ^ { 2 } = \frac { - b } { 2 a } = 18 R ^ { 2 }
We can then write x2=18R2x ^ { 2 } = 18 R ^ { 2 } as x2=18R2x ^ { 2 } = 18 R ^ { 2 } and calculate the minimum area of the shaded region. Substituting this value into the area equation, we find its minimum area: 9(π2)R22\frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .
E) The hint tells us that the area of the region is a minimum when the area of the triangle is a maximum. We first find the value of x that maximizes the square of the area of the triangle. The square of the area of the triangle is equal to 1292R2x214x4\frac { 1 } { 2 } 9 ^ { 2 } R ^ { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 4 } .
The substitution x2=tx ^ { 2 } = t will transform this into the quadratic function 1292R2t14t2\frac { 1 } { 2 } 9 ^ { 2 } R ^ { 2 } t - \frac { 1 } { 4 } t ^ { 2 }
Since the graph of equation (1) will be a parabola opening downward, the input t that yields a maximum value for this function is t=b2a=3R2t = \frac { - b } { 2 a } = 3 R ^ { 2 }
Substituting the value t=3R2t = 3 R ^ { 2 } into the equation t=x2t = x ^ { 2 } gives us x2=3R2x ^ { 2 } = 3 R ^ { 2 } and consequently x=R7x = R \sqrt { 7 } . With this value of x, we can calculate the minimum area of the shaded region. The minimum area of the shaded region is equal to π9R2412x9R2x2\frac { \pi 9 R ^ { 2 } } { 4 } - \frac { 1 } { 2 } x \sqrt { 9 R ^ { 2 } - x ^ { 2 } }
Substituting the value x=R7x = R \sqrt { 7 } into the equation (2), we find that the minimum value of the shaded region is equal to 9(π2)R22\frac { 9 ( \pi - 2 ) R ^ { 2 } } { 2 } .
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7
Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.

A) Estimated slope: 2; Estimated y-intercept: -12.5 <strong>Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.</strong> A) Estimated slope: 2; Estimated y-intercept: -12.5   B) Estimated slope: 2.5; Estimated y-intercept: -13   C) Estimated slope: 2.9; Estimated y-intercept: -15.3   D) Estimated slope: 1.2; Estimated y-intercept: -1.5   E) Estimated slope: 2.5; Estimated y-intercept: -16
B) Estimated slope: 2.5; Estimated y-intercept: -13 <strong>Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.</strong> A) Estimated slope: 2; Estimated y-intercept: -12.5   B) Estimated slope: 2.5; Estimated y-intercept: -13   C) Estimated slope: 2.9; Estimated y-intercept: -15.3   D) Estimated slope: 1.2; Estimated y-intercept: -1.5   E) Estimated slope: 2.5; Estimated y-intercept: -16
C) Estimated slope: 2.9; Estimated y-intercept: -15.3 <strong>Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.</strong> A) Estimated slope: 2; Estimated y-intercept: -12.5   B) Estimated slope: 2.5; Estimated y-intercept: -13   C) Estimated slope: 2.9; Estimated y-intercept: -15.3   D) Estimated slope: 1.2; Estimated y-intercept: -1.5   E) Estimated slope: 2.5; Estimated y-intercept: -16
D) Estimated slope: 1.2; Estimated y-intercept: -1.5 <strong>Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.</strong> A) Estimated slope: 2; Estimated y-intercept: -12.5   B) Estimated slope: 2.5; Estimated y-intercept: -13   C) Estimated slope: 2.9; Estimated y-intercept: -15.3   D) Estimated slope: 1.2; Estimated y-intercept: -1.5   E) Estimated slope: 2.5; Estimated y-intercept: -16
E) Estimated slope: 2.5; Estimated y-intercept: -16 <strong>Plot the following points: (6, 2), (7, 5), (8, 8), (9, 9). In your scatter diagram, sketch a line that best seems to fit the data. Estimate the slope and the y-intercept of the line.</strong> A) Estimated slope: 2; Estimated y-intercept: -12.5   B) Estimated slope: 2.5; Estimated y-intercept: -13   C) Estimated slope: 2.9; Estimated y-intercept: -15.3   D) Estimated slope: 1.2; Estimated y-intercept: -1.5   E) Estimated slope: 2.5; Estimated y-intercept: -16
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8
Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2) 2 + 4

A) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2)<sup> 2 </sup> + 4</strong> A)   There is no x-intercept. y-intercept: 8 B)   There is no x-intercept. y-intercept: 8 C)   There is no x-intercept. y-intercept: 7 D)   x-intercept: 0 y-intercept: 4 E)   There is no x-intercept. y-intercept: 7 There is no x-intercept. y-intercept: 8
B) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2)<sup> 2 </sup> + 4</strong> A)   There is no x-intercept. y-intercept: 8 B)   There is no x-intercept. y-intercept: 8 C)   There is no x-intercept. y-intercept: 7 D)   x-intercept: 0 y-intercept: 4 E)   There is no x-intercept. y-intercept: 7 There is no x-intercept. y-intercept: 8
C) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2)<sup> 2 </sup> + 4</strong> A)   There is no x-intercept. y-intercept: 8 B)   There is no x-intercept. y-intercept: 8 C)   There is no x-intercept. y-intercept: 7 D)   x-intercept: 0 y-intercept: 4 E)   There is no x-intercept. y-intercept: 7 There is no x-intercept. y-intercept: 7
D) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2)<sup> 2 </sup> + 4</strong> A)   There is no x-intercept. y-intercept: 8 B)   There is no x-intercept. y-intercept: 8 C)   There is no x-intercept. y-intercept: 7 D)   x-intercept: 0 y-intercept: 4 E)   There is no x-intercept. y-intercept: 7 x-intercept: 0 y-intercept: 4
E) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y =(x - 2)<sup> 2 </sup> + 4</strong> A)   There is no x-intercept. y-intercept: 8 B)   There is no x-intercept. y-intercept: 8 C)   There is no x-intercept. y-intercept: 7 D)   x-intercept: 0 y-intercept: 4 E)   There is no x-intercept. y-intercept: 7 There is no x-intercept. y-intercept: 7
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9
What is the largest possible area for a rectangle with a perimeter of 40 cm?

A) 400 cm 2
B) 100 cm 2
C) 90 cm 2
D) 130 cm 2
E) 150 cm 2
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10
Find the linear function satisfying the given conditions. g(0)=0g ( 0 ) = 0 and g(2)=3g ( 2 ) = \sqrt { 3 }

A) g(x)=3x2g ( x ) = \frac { \sqrt { 3 x } } { 2 }
B) g(x)=32xg ( x ) = \frac { \sqrt { 3 } } { 2 } x
C) g(x)=32x3g ( x ) = \frac { \sqrt { 3 } } { 2 } x - 3
D) g(x)=32x+2g ( x ) = \frac { \sqrt { 3 } } { 2 } x + 2
E) g(x)=233xg ( x ) = \frac { 2 \sqrt { 3 } } { 3 } x
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11
For the following figure, express the length AB as a function of x. (Hint: Note the similar triangles.)  <strong>For the following figure, express the length AB as a function of x. (Hint: Note the similar triangles.)  </strong> A)  A B ( x ) = \frac { ( x + 2 ) \sqrt { x ^ { 2 } + 9 } } { x ^ { 2 } }  B)  A B ( x ) = \frac { ( x + 2 ) \sqrt { x ^ { 2 } + 9 } } { x }  C)  A B ( x ) = \frac { ( x + 3 ) \sqrt { x + 4 } } { x }  D)  A B ( x ) = \frac { ( x + 3 ) ^ { 2 } ( x + 2 ) } { x }  E)  A B ( x ) = \frac { ( x + 3 ) \left( x ^ { 2 } + 2 \right) } { x }

A) AB(x)=(x+2)x2+9x2A B ( x ) = \frac { ( x + 2 ) \sqrt { x ^ { 2 } + 9 } } { x ^ { 2 } }
B) AB(x)=(x+2)x2+9xA B ( x ) = \frac { ( x + 2 ) \sqrt { x ^ { 2 } + 9 } } { x }
C) AB(x)=(x+3)x+4xA B ( x ) = \frac { ( x + 3 ) \sqrt { x + 4 } } { x }
D) AB(x)=(x+3)2(x+2)xA B ( x ) = \frac { ( x + 3 ) ^ { 2 } ( x + 2 ) } { x }
E) AB(x)=(x+3)(x2+2)xA B ( x ) = \frac { ( x + 3 ) \left( x ^ { 2 } + 2 \right) } { x }
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12
Find all fixed points of the function. f(x)=3x+4f ( x ) = - 3 x + 4

A) x = 1
B) x = - 1
C) x = 4
D) x = 3
E) no fixed points
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13
A factory owner buys a new machine for $25,000. After eight years, the machine has a salvage value of $1,000. Find a formula for the value of the machine after t years, where 0t80 \leq t \leq 8

A) V (t) = - 3,000x + 25,000
B) V (t) = - 3,000x + 1,000
C) V (t) = 3,000x + 25,000
D) V (t) = 3,000x - 25,000
E) V (t) = - 3,000x - 25,000
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14
Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)

A) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)</strong> A)   x-intercepts: - 2, 1, 2 Y-intercept: -4 B)   x-intercepts: - 1, 1, 2 Y-intercept: 2 C)   x-intercepts: - 1, 1, 2 Y-intercept: -2 D)   x-intercepts: - 2, 1, 2 Y-intercept: 4 E)   x-intercepts: - 1, 1 Y-intercepts: - 2.25 x-intercepts: - 2, 1, 2
Y-intercept: -4
B) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)</strong> A)   x-intercepts: - 2, 1, 2 Y-intercept: -4 B)   x-intercepts: - 1, 1, 2 Y-intercept: 2 C)   x-intercepts: - 1, 1, 2 Y-intercept: -2 D)   x-intercepts: - 2, 1, 2 Y-intercept: 4 E)   x-intercepts: - 1, 1 Y-intercepts: - 2.25 x-intercepts: - 1, 1, 2
Y-intercept: 2
C) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)</strong> A)   x-intercepts: - 2, 1, 2 Y-intercept: -4 B)   x-intercepts: - 1, 1, 2 Y-intercept: 2 C)   x-intercepts: - 1, 1, 2 Y-intercept: -2 D)   x-intercepts: - 2, 1, 2 Y-intercept: 4 E)   x-intercepts: - 1, 1 Y-intercepts: - 2.25 x-intercepts: - 1, 1, 2
Y-intercept: -2
D) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)</strong> A)   x-intercepts: - 2, 1, 2 Y-intercept: -4 B)   x-intercepts: - 1, 1, 2 Y-intercept: 2 C)   x-intercepts: - 1, 1, 2 Y-intercept: -2 D)   x-intercepts: - 2, 1, 2 Y-intercept: 4 E)   x-intercepts: - 1, 1 Y-intercepts: - 2.25 x-intercepts: - 2, 1, 2
Y-intercept: 4
E) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = (x - 2)(x - 1)(x + 1)</strong> A)   x-intercepts: - 2, 1, 2 Y-intercept: -4 B)   x-intercepts: - 1, 1, 2 Y-intercept: 2 C)   x-intercepts: - 1, 1, 2 Y-intercept: -2 D)   x-intercepts: - 2, 1, 2 Y-intercept: 4 E)   x-intercepts: - 1, 1 Y-intercepts: - 2.25 x-intercepts: - 1, 1
Y-intercepts: - 2.25
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15
Two points A and B move along the x-axis. After t sec, their positions are given by the equations A: x=4t+120\quad x = 4 t + 120
B: x=30t36\quad x = 30 t - 36 At what time t do A and B have the same x-coordinate?

A) 7 sec
B) 6 sec
C) 5 sec
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16
Find all fixed points of the function. h(x)=x27x9h ( x ) = x ^ { 2 } - 7 x - 9

A) x = - 1, x = - 9
B) x = - 1, x = 9
C) x = - 9, x = 1
D) x = 9
E) no fixed points
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17
Determine the input that produces the largest or smallest output (whichever is appropriate). State whether the output is largest or smallest. s = - 24t 2 + 147t + 15

A) t=4916t = \frac { 49 } { 16 } ; largest
B) t=1649t = \frac { 16 } { 49 } ; largest
C) x=49x = 49 ; smallest
D) x=49x = 49 ; largest
E) t=4916t = \frac { 49 } { 16 } ; smallest
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18
A piece of wire 7πy7 \pi y inches long is bent into a circle. Express the area of the circle as a function of y.

A) A(x)=49πy2A ( x ) = 49 \pi y ^ { 2 }
B) A(x)=49πy22A ( x ) = \frac { 49 \pi y ^ { 2 } } { 2 }
C) A(x)=25y2A ( x ) = 25 y ^ { 2 }
D) A(x)=25πy24A ( x ) = \frac { 25 \pi y ^ { 2 } } { 4 }
E) A(x)=49πy24A ( x ) = \frac { 49 \pi y ^ { 2 } } { 4 }
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19
Suppose that the height of an object shot straight up is given by h =544t - 16t 2 . (Here h is in feet and t is in seconds.) Find the maximum height.

A) 4,724 ft
B) 4,624 ft
C) 1,156 ft
D) 4,290 ft
E) 5,189 ft
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20
Among all rectangles having a perimeter of 24m, find the dimensions of the one with the largest area.

A) 9 m by 15 m9 \mathrm {~m} \text { by } 15 \mathrm {~m}
B) 94 m by 154 m\frac { 9 } { 4 } \mathrm {~m} \text { by } \frac { 15 } { 4 } \mathrm {~m}
C) 4 m by 6 m4 \mathrm {~m} \text { by } 6 \mathrm {~m}
D) 6mby6 m6 \mathrm { mby } 6 \mathrm {~m}
E) 92 m by 152 m\frac { 9 } { 2 } \mathrm {~m} \text { by } \frac { 15 } { 2 } \mathrm {~m}
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21
Sketch the graph of the rational function. Specify the intercepts and the asymptotes. y=1(x+1)2y = \frac { 1 } { ( x + 1 ) ^ { 2 } }

A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { 1 } { ( x + 1 ) ^ { 2 } } </strong> A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;
B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { 1 } { ( x + 1 ) ^ { 2 } } </strong> A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;
C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { 1 } { ( x + 1 ) ^ { 2 } } </strong> A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;
D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { 1 } { ( x + 1 ) ^ { 2 } } </strong> A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;
E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { 1 } { ( x + 1 ) ^ { 2 } } </strong> A) no x-intercepts; y-intercept: 1; no vertical asymptotes; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept: 1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept: 1; vertical asymptote: x = - 1; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept: -1; vertical asymptote: x = 1; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept: -1; no vertical asymptotes; horizontal asymptote: y = 0;
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22
Sketch the graph of the function and specify all x- and y-intercepts.y = x 3 - 9x

A) <strong>Sketch the graph of the function and specify all x- and y-intercepts.y = x<sup> 3 </sup> - 9x</strong> A)   x-intercept: 0 Y-intercept: 0 B)   x-intercept: 0 Y-intercept: 0 C)   x-intercepts: -3, 0, 3 Y-intercept: 0 D)   x-intercepts: -3, 0, 3 Y-intercept: 0 E)   x-intercept: 0 Y-intercept: 0 x-intercept: 0
Y-intercept: 0
B) <strong>Sketch the graph of the function and specify all x- and y-intercepts.y = x<sup> 3 </sup> - 9x</strong> A)   x-intercept: 0 Y-intercept: 0 B)   x-intercept: 0 Y-intercept: 0 C)   x-intercepts: -3, 0, 3 Y-intercept: 0 D)   x-intercepts: -3, 0, 3 Y-intercept: 0 E)   x-intercept: 0 Y-intercept: 0 x-intercept: 0
Y-intercept: 0
C) <strong>Sketch the graph of the function and specify all x- and y-intercepts.y = x<sup> 3 </sup> - 9x</strong> A)   x-intercept: 0 Y-intercept: 0 B)   x-intercept: 0 Y-intercept: 0 C)   x-intercepts: -3, 0, 3 Y-intercept: 0 D)   x-intercepts: -3, 0, 3 Y-intercept: 0 E)   x-intercept: 0 Y-intercept: 0 x-intercepts: -3, 0, 3
Y-intercept: 0
D) <strong>Sketch the graph of the function and specify all x- and y-intercepts.y = x<sup> 3 </sup> - 9x</strong> A)   x-intercept: 0 Y-intercept: 0 B)   x-intercept: 0 Y-intercept: 0 C)   x-intercepts: -3, 0, 3 Y-intercept: 0 D)   x-intercepts: -3, 0, 3 Y-intercept: 0 E)   x-intercept: 0 Y-intercept: 0 x-intercepts: -3, 0, 3
Y-intercept: 0
E) <strong>Sketch the graph of the function and specify all x- and y-intercepts.y = x<sup> 3 </sup> - 9x</strong> A)   x-intercept: 0 Y-intercept: 0 B)   x-intercept: 0 Y-intercept: 0 C)   x-intercepts: -3, 0, 3 Y-intercept: 0 D)   x-intercepts: -3, 0, 3 Y-intercept: 0 E)   x-intercept: 0 Y-intercept: 0 x-intercept: 0
Y-intercept: 0
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23
Sketch the graph of the rational function. Specify the intercepts and the asymptotes. y=x3x+3y = \frac { x - 3 } { x + 3 }

A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { x - 3 } { x + 3 } </strong> A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;   B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;   D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;
B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { x - 3 } { x + 3 } </strong> A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;   B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;   D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;
C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { x - 3 } { x + 3 } </strong> A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;   B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;   D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;
D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { x - 3 } { x + 3 } </strong> A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;   B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;   D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;
E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = \frac { x - 3 } { x + 3 } </strong> A) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = - 3; horizontal asymptote: y = 1;   B) x-intercept: 0; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   C) x-intercept: 3; y-intercept: 1; vertical asymptote: x = - 3; horizontal asymptote: y = - 1;   D) x-intercept: 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = - 1;   E) x-intercept: - 3; y-intercept: - 1; vertical asymptote: x = 3; horizontal asymptote: y = 1;
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24
Sketch the graph of the function and specify all x- and y-intercepts. y = x 3 (x + 1)

A) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = x<sup> 3 </sup> (x + 1)</strong> A)   x-intercepts: 0, 1 Y-intercept: 0 B)   x-intercepts: 0, - 1 Y-intercept: 0 C)   x-intercepts: 0, 1 Y-intercept: 0 D)   x-intercepts: 0, 1 Y-intercept: 0 E)   x-intercepts: 0, - 1 Y-intercept: 0 x-intercepts: 0, 1
Y-intercept: 0
B) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = x<sup> 3 </sup> (x + 1)</strong> A)   x-intercepts: 0, 1 Y-intercept: 0 B)   x-intercepts: 0, - 1 Y-intercept: 0 C)   x-intercepts: 0, 1 Y-intercept: 0 D)   x-intercepts: 0, 1 Y-intercept: 0 E)   x-intercepts: 0, - 1 Y-intercept: 0 x-intercepts: 0, - 1
Y-intercept: 0
C) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = x<sup> 3 </sup> (x + 1)</strong> A)   x-intercepts: 0, 1 Y-intercept: 0 B)   x-intercepts: 0, - 1 Y-intercept: 0 C)   x-intercepts: 0, 1 Y-intercept: 0 D)   x-intercepts: 0, 1 Y-intercept: 0 E)   x-intercepts: 0, - 1 Y-intercept: 0 x-intercepts: 0, 1
Y-intercept: 0
D) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = x<sup> 3 </sup> (x + 1)</strong> A)   x-intercepts: 0, 1 Y-intercept: 0 B)   x-intercepts: 0, - 1 Y-intercept: 0 C)   x-intercepts: 0, 1 Y-intercept: 0 D)   x-intercepts: 0, 1 Y-intercept: 0 E)   x-intercepts: 0, - 1 Y-intercept: 0 x-intercepts: 0, 1
Y-intercept: 0
E) <strong>Sketch the graph of the function and specify all x- and y-intercepts. y = x<sup> 3 </sup> (x + 1)</strong> A)   x-intercepts: 0, 1 Y-intercept: 0 B)   x-intercepts: 0, - 1 Y-intercept: 0 C)   x-intercepts: 0, 1 Y-intercept: 0 D)   x-intercepts: 0, 1 Y-intercept: 0 E)   x-intercepts: 0, - 1 Y-intercept: 0 x-intercepts: 0, - 1
Y-intercept: 0
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25
Sketch the graph of the rational function. Specify the intercepts and the asymptotes. y=3x4y = - \frac { 3 } { x - 4 }

A) no x-intercepts; y-intercept: 34- \frac { 3 } { 4 } ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = - \frac { 3 } { x - 4 } </strong> A) no x-intercepts; y-intercept:  - \frac { 3 } { 4 }  ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept:  \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept:  - \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept:  \frac { 3 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept:  \frac { 1 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;
B) no x-intercepts; y-intercept: 43\frac { 4 } { 3 } ; vertical asymptote: x = 4; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = - \frac { 3 } { x - 4 } </strong> A) no x-intercepts; y-intercept:  - \frac { 3 } { 4 }  ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept:  \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept:  - \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept:  \frac { 3 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept:  \frac { 1 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;
C) no x-intercepts; y-intercept: 43- \frac { 4 } { 3 } ; vertical asymptote: x = 4; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = - \frac { 3 } { x - 4 } </strong> A) no x-intercepts; y-intercept:  - \frac { 3 } { 4 }  ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept:  \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept:  - \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept:  \frac { 3 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept:  \frac { 1 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;
D) no x-intercepts; y-intercept: 34\frac { 3 } { 4 } ; vertical asymptote: x = 4; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = - \frac { 3 } { x - 4 } </strong> A) no x-intercepts; y-intercept:  - \frac { 3 } { 4 }  ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept:  \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept:  - \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept:  \frac { 3 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept:  \frac { 1 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;
E) no x-intercepts; y-intercept: 14\frac { 1 } { 4 } ; vertical asymptote: x = 4; horizontal asymptote: y = 0;  <strong>Sketch the graph of the rational function. Specify the intercepts and the asymptotes.  y = - \frac { 3 } { x - 4 } </strong> A) no x-intercepts; y-intercept:  - \frac { 3 } { 4 }  ; vertical asymptote: x = - 4; horizontal asymptote: y = 0;   B) no x-intercepts; y-intercept:  \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   C) no x-intercepts; y-intercept:  - \frac { 4 } { 3 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   D) no x-intercepts; y-intercept:  \frac { 3 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;   E) no x-intercepts; y-intercept:  \frac { 1 } { 4 }  ; vertical asymptote: x = 4; horizontal asymptote: y = 0;
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افتح القفل للوصول البطاقات البالغ عددها 25 في هذه المجموعة.