Question 1

A small theater has a seating capacity of 2000.When the ticket price is $20,attendance is 1500.For each $1 decrease in price,attendance increases by 50.The revenue R of the theater as a function of ticket price x is as follows $R ( x ) = - 50 x ^ { 2 } + 2500 x , 10 \leq x \leq 20$ What ticket price will yield a maximum revenue?

Accepted Answer

To find the ticket price that yields maximum revenue, we can use the vertex formula for a parabola, $x = -\frac{b}{2a}$, where $a = -50$ and $b = 2500$. Plugging these values in gives $x = -\frac{2500}{2(-50)} = 25$. Therefore, a ticket price of $25 will yield maximum revenue.

Question 2

Determine the x-intercept(s)of the quadratic function $f ( x ) = x ^ { 2 } + 8 x + 12$ .

Accepted Answer

To find the x-intercepts, set $f(x) = 0$: $x^2 + 8x + 12 = 0$. Factoring gives $(x + 2)(x + 6) = 0$, leading to $x = -2$ and $x = -6$, so the x-intercepts are $(-2, 0)$ and $(-6, 0)$.

Question 3

Compare the graph of $n ( x ) = 7 ( x - 3 ) ^ { 2 } + 6$ with $n ( x ) = x ^ { 2 }$ .

Accepted Answer

The form $n(x) = 7(x - 3)^2 + 6$ indicates a transformation of the base graph $n(x) = x^2$. The term $(x - 3)$ indicates a horizontal shift to the right by 3 units (since it's in the form $x - h$, where $h$ is positive), the $+6$ indicates a vertical shift upward by 6 units, and the coefficient $7$ indicates a vertical stretch by a factor of 7, not a shrink.

Question 4

The total revenue R earned (in thousands of dollars)from manufacturing handheld video games is given by? $R ( p ) = - 25 p ^ { 2 } + 1200 p$ ? where p is the price per unit (in dollars).
?
Find the revenue when the price per unit is $20.
?

Accepted Answer

To find the revenue when the price per unit is $20, substitute $p = 20$ into the revenue function $R(p) = -25p^2 + 1200p$. This gives $R(20) = -25(20)^2 + 1200(20) = -25(400) + 24000 = -10000 + 24000 = 14000$. Since the revenue is given in thousands of dollars, $14,000 translates to $14,000,000.

Question 5

From the graph of the quadratic function $f ( x ) = ( x + 6 ) ^ { 2 } - 1$ ,determine the equation of the axis of symmetry.

Accepted Answer

The axis of symmetry for a quadratic function in the form $f(x) = (x + h)^2 + k$ is given by $x = -h$. For $f(x) = (x + 6)^2 - 1$, $h = -6$, so the axis of symmetry is $x = -6$.

Question 6

The profit P (in hundreds of dollars)that a company makes depends on the amount x (in hundreds of dollars)the company spends on advertising according to the model? $P = 230 + 30 x - 0.5 x ^ { 2 }$ ? What expenditure for advertising will yield a maximum profit?
?

Accepted Answer

The answer of The profit P (in hundreds of dollars)that...

Question 7

The total revenue R earned per day (in dollars)from a pet-sitting service is given by? $R ( p ) = - 12 p ^ { 2 } + 150 p$ ? where p is the price charged per pet (in dollars).
?
Find the revenue when the price per pet is $3.
?

Accepted Answer

To find the revenue when the price per pet is $3, substitute $p = 3$ into the revenue function: $R(3) = -12(3)^2 + 150(3) = -12(9) + 450 = -108 + 450 = 342$.

Question 8

Find the values of b such that the function $f ( x ) = x ^ { 2 } + b x - 25$ has the given minimum value -74. ?

Accepted Answer

The answer of Find the values of b such that...

Question 9

Determine the x-intercept(s)of the quadratic function $$f ( x ) = x ^ { 2 } + 2 x + 2$$ .

Accepted Answer

To find the x-intercepts of the quadratic function $f(x) = x^2 + 2x + 2$, we set $f(x) = 0$ and solve for $x$. This gives us the equation $x^2 + 2x + 2 = 0$. Using the discriminant $b^2 - 4ac$ from the quadratic formula, where $a = 1$, $b = 2$, and $c = 2$, we find that the discriminant is $2^2 - 4(1)(2) = 4 - 8 = -4$. Since the discriminant is negative, this means the quadratic equation has no real solutions and therefore the function has no x-intercepts.

Question 10

Determine the vertex of the graph of the quadratic function $f ( x ) = x ^ { 2 } + 5 x + \frac { 29 } { 4 }$ .

Accepted Answer

The vertex of a quadratic function in the form $f(x) = ax^2 + bx + c$ can be found using the formula $(-b/2a, f(-b/2a))$. For the given function $f(x) = x^2 + 5x + \frac{29}{4}$, $a = 1$ and $b = 5$, so the x-coordinate of the vertex is $-5/(2*1) = -5/2$. Plugging this back into the function gives the y-coordinate as $f(-5/2) = (-5/2)^2 + 5(-5/2) + \frac{29}{4} = \frac{25}{4} - \frac{25}{2} + \frac{29}{4} = 1$. Therefore, the vertex is $\left(-\frac{5}{2}, 1\right)$.

Question 11

Find the values of b such that the function $f ( x ) = - x ^ { 2 } + b x - 75$ has the given maximum value 94. ?

Accepted Answer

The answer of Find the values of b such that...

Question 12

A small theater has a seating capacity of 2000.When the ticket price is $20,attendance is 1500.For each $1 decrease in price,attendance increases by 125.Write the revenue R of the theater as a function of ticket price x. ?

Accepted Answer

The answer of A small theater has a seating capacity...

Question 13

From the graph of the quadratic function $f ( x ) = - x ^ { 2 } + 12 x + 4$ ,determine the equation of the axis of symmetry.

Accepted Answer

The axis of symmetry for a quadratic function in the form $f(x) = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. For the given function $f(x) = -x^2 + 12x + 4$, $a = -1$ and $b = 12$, so the axis of symmetry is $x = -\frac{12}{2(-1)} = 6$.

Question 14

Find the values of b such that the function $f ( x ) = x ^ { 2 } + b x + 59$ has the given minimum value 10. ?

Accepted Answer

The answer of Find the values of b such that...

Question 15

Compare the graph of $n = \left[ - \frac { 1 } { 9 } ( x + 6 ) \right] ^ { 2 } - 1$ with $n ( x ) = x ^ { 2 }$ .

Accepted Answer

The correct transformation of the graph $n = \left[ - \frac { 1 } { 9 } ( x + 6 ) \right] ^ { 2 } - 1$ compared to $n(x) = x^2$ involves a shift to the right by 6 units (due to $x + 6$), a downward shift by 1 unit (due to $-1$), and a vertical shrink by a factor of $\frac{1}{81}$ (since squaring the $-\frac{1}{9}$ results in $\frac{1}{81}$, affecting the vertical stretch/shrink of the graph).

Question 16

The total revenue R earned per day (in dollars)from a pet-sitting service is given by? $R ( p ) = - 10 p ^ { 2 } + 120 p$ ? where p is the price charged per pet (in dollars).
?
Find the price that will yield a maximum revenue.
?

Accepted Answer

The answer of The total revenue R earned per day...

Question 17

Write the quadratic function $f ( x ) = x ^ { 2 } + 2 x + 5$ in standard form.

Accepted Answer

The standard form of a quadratic function is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. Completing the square for $f(x) = x^2 + 2x + 5$, we add and subtract $(\frac{2}{2})^2 = 1$, getting $f(x) = (x^2 + 2x + 1) + 4 = (x + 1)^2 + 4$, which matches choice D.

Question 18

Find the standard form of the quadratic function shown below:?  ?

Accepted Answer

The answer of Find the standard form of the quadratic...

Question 19

Find the values of b such that the function $f ( x ) = - x ^ { 2 } + b x - 16$ has the given maximum value 65. ?

Accepted Answer

The maximum value of a quadratic function $f(x) = ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$. For the given function $f(x) = -x^2 + bx - 16$, $a = -1$, $b = b$, and $c = -16$. The maximum value is given as 65. To find the value of $b$, we use the vertex form of a quadratic function, which gives the maximum value when $a < 0$. The vertex form is $f(x) = a(x - h)^2 + k$, where $k$ is the maximum value, and $h$ is the x-coordinate of the vertex. Since $k = 65$, we can set up the equation for the maximum value:$$65 = -\left(-\frac{b}{2(-1)}\right)^2 + b\left(-\frac{b}{2(-1)}\right) - 16$$Simplifying, we find:$$65 = -\frac{b^2}{4} + \frac{b^2}{2} - 16$$$$65 = \frac{b^2}{4} - 16$$$$65 + 16 = \frac{b^2}{4}$$$$81 = \frac{b^2}{4}$$$$324 = b^2$$$$b = \pm18$$Therefore, the correct choice is $b = \pm18$, which corresponds to choice A.

Question 20

The total revenue R earned (in thousands of dollars)from manufacturing handheld video games is given by? $R ( p ) = - 25 p ^ { 2 } + 1900 p$ ? where p is the price per unit (in dollars).
?
Find the unit price that will yield a maximum revenue.
?

Accepted Answer

The answer of The total revenue R earned (in thousands...

Quiz 11: Quadratic Functions and Models

Determine the x-intercept(s) of the quadratic function $f ( x ) = x ^ { 2 } + 8 x + 12$ .

Compare the graph of $n ( x ) = 7 ( x - 3 ) ^ { 2 } + 6$ with $n ( x ) = x ^ { 2 }$ .

The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given by? $R ( p ) = - 25 p ^ { 2 } + 1200 p$ ? where p is the price per unit (in dollars) . ? Find the revenue when the price per unit is $20. ?

From the graph of the quadratic function $f ( x ) = ( x + 6 ) ^ { 2 } - 1$ ,determine the equation of the axis of symmetry.

The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) the company spends on advertising according to the model? $P = 230 + 30 x - 0.5 x ^ { 2 }$ ? What expenditure for advertising will yield a maximum profit? ?

The total revenue R earned per day (in dollars) from a pet-sitting service is given by? $R ( p ) = - 12 p ^ { 2 } + 150 p$ ? where p is the price charged per pet (in dollars) . ? Find the revenue when the price per pet is $3. ?

Find the values of b such that the function $f ( x ) = x ^ { 2 } + b x - 25$ has the given minimum value -74. ?

Determine the x-intercept(s) of the quadratic function $f ( x ) = x ^ { 2 } + 2 x + 2$ .

Determine the vertex of the graph of the quadratic function $f ( x ) = x ^ { 2 } + 5 x + \frac { 29 } { 4 }$ .

Find the values of b such that the function $f ( x ) = - x ^ { 2 } + b x - 75$ has the given maximum value 94. ?

A small theater has a seating capacity of 2000.When the ticket price is $20,attendance is 1500.For each $1 decrease in price,attendance increases by 125.Write the revenue R of the theater as a function of ticket price x. ?

From the graph of the quadratic function $f ( x ) = - x ^ { 2 } + 12 x + 4$ ,determine the equation of the axis of symmetry.

Find the values of b such that the function $f ( x ) = x ^ { 2 } + b x + 59$ has the given minimum value 10. ?

Compare the graph of $n = \left[ - \frac { 1 } { 9 } ( x + 6 ) \right] ^ { 2 } - 1$ with $n ( x ) = x ^ { 2 }$ .

The total revenue R earned per day (in dollars) from a pet-sitting service is given by? $R ( p ) = - 10 p ^ { 2 } + 120 p$ ? where p is the price charged per pet (in dollars) . ? Find the price that will yield a maximum revenue. ?

Write the quadratic function $f ( x ) = x ^ { 2 } + 2 x + 5$ in standard form.

Find the values of b such that the function $f ( x ) = - x ^ { 2 } + b x - 16$ has the given maximum value 65. ?

The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given by? $R ( p ) = - 25 p ^ { 2 } + 1900 p$ ? where p is the price per unit (in dollars) . ? Find the unit price that will yield a maximum revenue. ?

Quiz 11: Quadratic Functions and Models

Determine the x-intercept(s) of the quadratic function f(x) =x2+8x+12f ( x ) = x ^ { 2 } + 8 x + 12f(x) =x2+8x+12 .

Compare the graph of n(x) =7(x−3) 2+6n ( x ) = 7 ( x - 3 ) ^ { 2 } + 6n(x) =7(x−3) 2+6 with n(x) =x2n ( x ) = x ^ { 2 }n(x) =x2 .

The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given by? R(p) =−25p2+1200pR ( p ) = - 25 p ^ { 2 } + 1200 pR(p) =−25p2+1200p ? where p is the price per unit (in dollars) . ? Find the revenue when the price per unit is $20. ?

From the graph of the quadratic function f(x) =(x+6) 2−1f ( x ) = ( x + 6 ) ^ { 2 } - 1f(x) =(x+6) 2−1 ,determine the equation of the axis of symmetry.

The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) the company spends on advertising according to the model? P=230+30x−0.5x2P = 230 + 30 x - 0.5 x ^ { 2 }P=230+30x−0.5x2 ? What expenditure for advertising will yield a maximum profit? ?

The total revenue R earned per day (in dollars) from a pet-sitting service is given by? R(p) =−12p2+150pR ( p ) = - 12 p ^ { 2 } + 150 pR(p) =−12p2+150p ? where p is the price charged per pet (in dollars) . ? Find the revenue when the price per pet is $3. ?

Find the values of b such that the function f(x) =x2+bx−25f ( x ) = x ^ { 2 } + b x - 25f(x) =x2+bx−25 has the given minimum value -74. ?

Determine the x-intercept(s) of the quadratic function f(x) =x2+2x+2f ( x ) = x ^ { 2 } + 2 x + 2f(x) =x2+2x+2 .

Determine the vertex of the graph of the quadratic function f(x) =x2+5x+294f ( x ) = x ^ { 2 } + 5 x + \frac { 29 } { 4 }f(x) =x2+5x+429​ .

Find the values of b such that the function f(x) =−x2+bx−75f ( x ) = - x ^ { 2 } + b x - 75f(x) =−x2+bx−75 has the given maximum value 94. ?

A small theater has a seating capacity of 2000.When the ticket price is $20,attendance is 1500.For each $1 decrease in price,attendance increases by 125.Write the revenue R of the theater as a function of ticket price x. ?

From the graph of the quadratic function f(x) =−x2+12x+4f ( x ) = - x ^ { 2 } + 12 x + 4f(x) =−x2+12x+4 ,determine the equation of the axis of symmetry.

Find the values of b such that the function f(x) =x2+bx+59f ( x ) = x ^ { 2 } + b x + 59f(x) =x2+bx+59 has the given minimum value 10. ?

Compare the graph of n=[−19(x+6) ]2−1n = \left[ - \frac { 1 } { 9 } ( x + 6 ) \right] ^ { 2 } - 1n=[−91​(x+6) ]2−1 with n(x) =x2n ( x ) = x ^ { 2 }n(x) =x2 .

The total revenue R earned per day (in dollars) from a pet-sitting service is given by? R(p) =−10p2+120pR ( p ) = - 10 p ^ { 2 } + 120 pR(p) =−10p2+120p ? where p is the price charged per pet (in dollars) . ? Find the price that will yield a maximum revenue. ?

Write the quadratic function f(x) =x2+2x+5f ( x ) = x ^ { 2 } + 2 x + 5f(x) =x2+2x+5 in standard form.

Find the standard form of the quadratic function shown below:? ?

Find the values of b such that the function f(x) =−x2+bx−16f ( x ) = - x ^ { 2 } + b x - 16f(x) =−x2+bx−16 has the given maximum value 65. ?

The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given by? R(p) =−25p2+1900pR ( p ) = - 25 p ^ { 2 } + 1900 pR(p) =−25p2+1900p ? where p is the price per unit (in dollars) . ? Find the unit price that will yield a maximum revenue. ?

Determine the x-intercept(s) of the quadratic function $f ( x ) = x ^ { 2 } + 8 x + 12$ .

Compare the graph of $n ( x ) = 7 ( x - 3 ) ^ { 2 } + 6$ with $n ( x ) = x ^ { 2 }$ .

The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given by? $R ( p ) = - 25 p ^ { 2 } + 1200 p$ ? where p is the price per unit (in dollars) . ? Find the revenue when the price per unit is $20. ?

From the graph of the quadratic function $f ( x ) = ( x + 6 ) ^ { 2 } - 1$ ,determine the equation of the axis of symmetry.

The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) the company spends on advertising according to the model? $P = 230 + 30 x - 0.5 x ^ { 2 }$ ? What expenditure for advertising will yield a maximum profit? ?

The total revenue R earned per day (in dollars) from a pet-sitting service is given by? $R ( p ) = - 12 p ^ { 2 } + 150 p$ ? where p is the price charged per pet (in dollars) . ? Find the revenue when the price per pet is $3. ?

Find the values of b such that the function $f ( x ) = x ^ { 2 } + b x - 25$ has the given minimum value -74. ?

Determine the x-intercept(s) of the quadratic function $f ( x ) = x ^ { 2 } + 2 x + 2$ .

Determine the vertex of the graph of the quadratic function $f ( x ) = x ^ { 2 } + 5 x + \frac { 29 } { 4 }$ .

Find the values of b such that the function $f ( x ) = - x ^ { 2 } + b x - 75$ has the given maximum value 94. ?

From the graph of the quadratic function $f ( x ) = - x ^ { 2 } + 12 x + 4$ ,determine the equation of the axis of symmetry.

Find the values of b such that the function $f ( x ) = x ^ { 2 } + b x + 59$ has the given minimum value 10. ?

Compare the graph of $n = \left[ - \frac { 1 } { 9 } ( x + 6 ) \right] ^ { 2 } - 1$ with $n ( x ) = x ^ { 2 }$ .

The total revenue R earned per day (in dollars) from a pet-sitting service is given by? $R ( p ) = - 10 p ^ { 2 } + 120 p$ ? where p is the price charged per pet (in dollars) . ? Find the price that will yield a maximum revenue. ?

Write the quadratic function $f ( x ) = x ^ { 2 } + 2 x + 5$ in standard form.

Find the values of b such that the function $f ( x ) = - x ^ { 2 } + b x - 16$ has the given maximum value 65. ?

The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given by? $R ( p ) = - 25 p ^ { 2 } + 1900 p$ ? where p is the price per unit (in dollars) . ? Find the unit price that will yield a maximum revenue. ?