Question 1

Suppose the amount of carbon dioxide (in pounds per 15,000 miles) released by a typical sport utility vehicle (SUV) depends on its fuel efficiency according to the formula  
 $10 x ^ { 2 } - 580 x + 30,803$  
Where x is a fuel efficiency of an SUV in miles per gallon. According to the model, what is the fuel efficiency of an SUV that has the least carbon dioxide pollution 
 
A)14 miles per gallon
B) 19 miles per gallon
C) 18 miles per gallon
D) 32 miles per gallon
E) 29 miles per gallon

Accepted Answer

The answer of Suppose the amount of carbon dioxide (in...

Question 2

The following chart shows the value of trade between two countries for the period 1994 - 2004 ( $t = 0$ represents 1994).     
Which of the following models best approximates the data given  (Try to answer this without actually computing values.)
 
A) $f ( t ) = 2 t ^ { 2 } - 4 t - 45$
B) $f ( t ) = - 2 t ^ { 2 } - 4 t - 35$
C) $f ( t ) = - 2 t ^ { 2 } + 4 t - 45$
D) $f ( t ) = - 2 t ^ { 2 } - 4 t + 35$
E) $f ( t ) = 2 t ^ { 2 } - 4 t + 35$

Accepted Answer

The answer of The following chart shows the value of...

Question 3

Find the vertex of the graph of the quadratic function.   $- x ^ { 2 } + 12 x - 36$  
A) $( 6,0 )$
B) $( - 6,6 )$
C) $( - 6,0 )$
D) $( 0,6 )$
E) $( - 6 , - 6 )$

Accepted Answer

The vertex of a quadratic function in the form $f(x) = ax^2 + bx + c$ can be found using the formula $x = -\frac{b}{2a}$. For the given function $-x^2 + 12x - 36$, $a = -1$ and $b = 12$, so the x-coordinate of the vertex is $x = -\frac{12}{2(-1)} = 6$. To find the y-coordinate, substitute $x = 6$ into the original equation: $-6^2 + 12(6) - 36 = -36 + 72 - 36 = 0$. Therefore, the vertex is $(6, 0)$.

Question 4

Model the data using an exponential function $$f ( x ) = A b ^ { x }$$ . $$\begin{array} { | l | l | l | l | } 
\hline \boldsymbol { x } & 0 & 1 & 2 \
\hline f ( x ) & 300 & 75 & 18.75 \
\hline
\end{array}$$

Accepted Answer

The answer of Model the data using an exponential function...

Question 5

For the following demand equation, find the largest possible revenue.  $q = - 4 p + 6,400$  
A)3,200
B) 5,120,000
C) 7,680,000
D) 800
E) 2,560,000

Accepted Answer

Revenue is maximized when price is set at the midpoint of the demand curve. Solving for price when $q = 0$ and $q = 6,400$, the midpoint is $p = 800$. Substituting $p = 800$ into the revenue equation $R = p \times q$, we get $R = 800 \times 3,200 = 2,560,000$.

Question 6

The given table corresponds to the function $$f ( x ) = 0.042 \left( 4.2 - \frac { 5 } { 1.7 } \right) ^ { - x }$$ . $$\begin{array} { | l | l | l | l | l | l | l | l | } 
\hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \
\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.084 & 0.067 & 0.067 & 1 & 0.033 & 0.027 & 0.021 \
\hline
\end{array}$$

Accepted Answer

The values of $f(x)$ in the table do not match the output of the function $f(x) = 0.042(4.2 - \frac{5}{1.7})^{-x}$ when calculated for each given $x$ value.

Question 7

The U.S. population was 170 million in 1970 and 250 million in 1995. Assuming exponential population growth, what will the population be in the year 2025 
 
Round your answer to the nearest million.
 
__________ million

Accepted Answer

The answer of The U.S. population was 170 million in...

Question 8

The fuel efficiency (in miles per gallon) of a sport utility vehicle (SUV) depends on its weight according to the formula  
 $E = 0.00005 x ^ { 2 } - 0.3 x + 41$  
Where x is the weight of an SUV in pounds. According to the model, what is the weight of the least fuel-efficient SUV 
 
A)3,000 pounds
B) 3,100 pounds
C) 2,000 pounds
D) 6,000 pounds
E) 491 pounds

Accepted Answer

The answer of The fuel efficiency (in miles per gallon)...

Question 9

Find the y-intercept(s) of the graph of the quadratic function.  $3 x ^ { 2 } - 30 x + 75$  
A) $( 0 , - 75 )$
B)   $( 5,0 ) , ( 5,0 )$
C) $( 0,5 ) , ( 0,5 )$
D) $( 75,0 )$
E) $( 0,75 )$

Accepted Answer

The y-intercept of a function is found by setting $x = 0$. Substituting $x = 0$ into the given function $3x^2 - 30x + 75$ gives $3(0)^2 - 30(0) + 75 = 75$, so the y-intercept is $(0, 75)$.

Question 10

Which of the following five functions will be smallest for large values of x   
Select the correct answer.
 
A) $f ( x ) = x ^ { - 36 }$
B) $f ( x ) = x ^ { - 216 }$
C) $g ( x ) = 6 ^ { - x }$
D) $f ( x ) = x ^ { - 6 }$
E) $h ( x ) = x ^ { - 100 }$

Accepted Answer

For large values of x, exponential functions decrease faster than polynomial functions with negative exponents. $g(x) = 6^{-x}$ will be the smallest.

Question 11

The given table corresponds to the function $$f ( x ) = e ^ { \frac { x } { 5 } }$$ . $$\begin{array} { | l | l | l | l | l | l | l | l | } 
\hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \
\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.549 & 0.67 & 0.819 & 1 & 1.221 & 1.492 & 1.822 \
\hline
\end{array}$$

Accepted Answer

The values in the table match the outputs of the function $f(x) = e^{\frac{x}{5}}$ when the corresponding $x$ values are substituted into the function.

Question 12

The given table corresponds to the function $f ( x ) = 10 ^ { x }$ . $$\begin{array} { | l | l | l | l | l | l | l | l | } 
\hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \
\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.001 & 0.01 & 0.1 & 1 & 10 & 100 & 1,000 \
\hline
\end{array}$$

Accepted Answer

The values of $f(x) = 10^x$ correctly correspond to the powers of 10 for each given value of $x$ in the table.

Question 13

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $$\begin{array} { | l | l | l | } 
\hline \boldsymbol { x } & 1 & 2 \
\hline f ( x ) & 170 & 289 \
\hline
\end{array}$$

Accepted Answer

The answer of Model the data using an exponential function...

Question 14

Find the equation for the exponential function that passes through the pair of points $( 2,4 )$ and $( 7,1 )$ .
​ $y = A b ^ { x } ( b > 0 )$ ​
A = __________
​
b = __________
​
Round your answer to four decimal places.

Accepted Answer

The answer of Find the equation for the exponential function...

Question 15

The given table corresponds to the function $$f ( x ) = 10 ^ { - x }$$ . $$\begin{array} { | l | l | l | l | l | l | l | l | } 
\hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \
\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.001 & 0.01 & 0.1 & 1 & 10 & 100 & 1,000 \
\hline
\end{array}$$

Accepted Answer

The answer of The given table corresponds to the function...

Question 16

The given table corresponds to the function $$f ( x ) = 5 ^ { x } - 1$$ . $$\begin{array} { | l | l | l | l | l | l | l | l | } 
\hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \
\hline \boldsymbol { f } ( \boldsymbol { x } ) & - 0.992 & - 0.96 & - 0.8 & 0 & 4 & 24 & 124 \
\hline
\end{array}$$

Accepted Answer

The values in the table match the function $f(x) = 5^x - 1$ for each given $x$.

Question 17

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $$\begin{array} { | l | l | l | l | } 
\hline \boldsymbol { x } & 0 & 1 & 2 \
\hline \boldsymbol { f } ( \boldsymbol { x } ) & 10 & 20 & 40 \
\hline
\end{array}$$

Accepted Answer

The answer of Model the data using an exponential function...

Question 18

The given table corresponds to the function $f ( x ) = 10 ^ { x - 1 }$ . $\begin{array} { | l | l | l | l | l | l | l | l | } 
\hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \
\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.0001 & 0.001 & 0.01 & 0.1 & 1 & 10 & 100 \
\hline
\end{array}$

Accepted Answer

The values of $f(x) = 10^{x-1}$ correctly correspond to the given $x$ values in the table, as each $f(x)$ value is $10$ raised to the power of $(x-1)$, resulting in the sequence 0.0001, 0.001, 0.01, 0.1, 1, 10, 100 for $x$ values -3, -2, -1, 0, 1, 2, 3 respectively.

Question 19

For the following demand equation, find the largest possible revenue.  
 $q = - 4 p + 3,600$  
A)1,620,000
B)  450
C)  900
D)  810,000
E)  648,000

Accepted Answer

The answer of For the following demand equation, find the...

Question 20

For the following demand equation, express the total revenue R as a function of the price p per item.  $q = - 2 p + 1600$  
A) $R = - 2 p ^ { 2 } + 1600 p$
B) $R = - 2 p ^ { 2 } + 1600$
C) $R = 1598 p$
D) $R = - 2 + \frac { 1600 } { p }$
E) $R = - 2 p ^ { 2 }$

Accepted Answer

Total revenue $R$ is calculated as the product of price per item $p$ and quantity $q$, so $R = p \times q$. Substituting $q = -2p + 1600$ into this formula gives $R = p(-2p + 1600) = -2p^2 + 1600p$.

Quiz 2: Nonlinear Functions and Models

Find the vertex of the graph of the quadratic function. $- x ^ { 2 } + 12 x - 36$

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $\begin{array} { | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 \\\hline f ( x ) & 300 & 75 & 18.75 \\\hline\end{array}$

For the following demand equation, find the largest possible revenue. $q = - 4 p + 6,400$

The U.S. population was 170 million in 1970 and 250 million in 1995. Assuming exponential population growth, what will the population be in the year 2025 Round your answer to the nearest million. __________ million

The fuel efficiency (in miles per gallon) of a sport utility vehicle (SUV) depends on its weight according to the formula $E = 0.00005 x ^ { 2 } - 0.3 x + 41$ Where x is the weight of an SUV in pounds. According to the model, what is the weight of the least fuel-efficient SUV

Find the y-intercept(s) of the graph of the quadratic function. $3 x ^ { 2 } - 30 x + 75$

Which of the following five functions will be smallest for large values of x Select the correct answer.

The given table corresponds to the function $f ( x ) = 10 ^ { x }$ . $\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.001 & 0.01 & 0.1 & 1 & 10 & 100 & 1,000 \\\hline\end{array}$

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $\begin{array} { | l | l | l | } \hline \boldsymbol { x } & 1 & 2 \\\hline f ( x ) & 170 & 289 \\\hline\end{array}$

Find the equation for the exponential function that passes through the pair of points $( 2,4 )$ and $( 7,1 )$ . $y = A b ^ { x } ( b > 0 )$ A = b = Round your answer to four decimal places.

The given table corresponds to the function $f ( x ) = 10 ^ { - x }$ . $\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.001 & 0.01 & 0.1 & 1 & 10 & 100 & 1,000 \\\hline\end{array}$

The given table corresponds to the function $f ( x ) = 5 ^ { x } - 1$ . $\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & - 0.992 & - 0.96 & - 0.8 & 0 & 4 & 24 & 124 \\\hline\end{array}$

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $\begin{array} { | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 10 & 20 & 40 \\\hline\end{array}$

The given table corresponds to the function $f ( x ) = 10 ^ { x - 1 }$ . $\begin{array} { | l | l | l | l | l | l | l | l | } \hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 0.0001 & 0.001 & 0.01 & 0.1 & 1 & 10 & 100 \\\hline\end{array}$

For the following demand equation, find the largest possible revenue. $q = - 4 p + 3,600$

For the following demand equation, express the total revenue R as a function of the price p per item. $q = - 2 p + 1600$

Quiz 2: Nonlinear Functions and Models

The following chart shows the value of trade between two countries for the period 1994 - 2004 ( t=0t = 0t=0 represents 1994) . Which of the following models best approximates the data given (Try to answer this without actually computing values.)

Find the vertex of the graph of the quadratic function. −x2+12x−36- x ^ { 2 } + 12 x - 36−x2+12x−36

Model the data using an exponential function f(x)=Abxf ( x ) = A b ^ { x }f(x)=Abx . x012f(x)3007518.75\begin{array} { | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 \\\hline f ( x ) & 300 & 75 & 18.75 \\\hline\end{array}xf(x)​0300​175​218.75​​

For the following demand equation, find the largest possible revenue. q=−4p+6,400q = - 4 p + 6,400q=−4p+6,400

The U.S. population was 170 million in 1970 and 250 million in 1995. Assuming exponential population growth, what will the population be in the year 2025 Round your answer to the nearest million. __________ million

Find the y-intercept(s) of the graph of the quadratic function. 3x2−30x+753 x ^ { 2 } - 30 x + 753x2−30x+75

Which of the following five functions will be smallest for large values of x Select the correct answer.

Model the data using an exponential function f(x)=Abxf ( x ) = A b ^ { x }f(x)=Abx . x12f(x)170289\begin{array} { | l | l | l | } \hline \boldsymbol { x } & 1 & 2 \\\hline f ( x ) & 170 & 289 \\\hline\end{array}xf(x)​1170​2289​​

Find the equation for the exponential function that passes through the pair of points (2,4)( 2,4 )(2,4) and (7,1)( 7,1 )(7,1) . ​ y=Abx(b>0)y = A b ^ { x } ( b > 0 )y=Abx(b>0) ​ A = __________ ​ b = __________ ​ Round your answer to four decimal places.

Model the data using an exponential function f(x)=Abxf ( x ) = A b ^ { x }f(x)=Abx . x012f(x)102040\begin{array} { | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 10 & 20 & 40 \\\hline\end{array}xf(x)​010​120​240​​

For the following demand equation, find the largest possible revenue. q=−4p+3,600q = - 4 p + 3,600q=−4p+3,600

For the following demand equation, express the total revenue R as a function of the price p per item. q=−2p+1600q = - 2 p + 1600q=−2p+1600

Find the vertex of the graph of the quadratic function. $- x ^ { 2 } + 12 x - 36$

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $\begin{array} { | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 \\\hline f ( x ) & 300 & 75 & 18.75 \\\hline\end{array}$

For the following demand equation, find the largest possible revenue. $q = - 4 p + 6,400$

Find the y-intercept(s) of the graph of the quadratic function. $3 x ^ { 2 } - 30 x + 75$

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $\begin{array} { | l | l | l | } \hline \boldsymbol { x } & 1 & 2 \\\hline f ( x ) & 170 & 289 \\\hline\end{array}$

Find the equation for the exponential function that passes through the pair of points $( 2,4 )$ and $( 7,1 )$ . $y = A b ^ { x } ( b > 0 )$ A = b = Round your answer to four decimal places.

Model the data using an exponential function $f ( x ) = A b ^ { x }$ . $\begin{array} { | l | l | l | l | } \hline \boldsymbol { x } & 0 & 1 & 2 \\\hline \boldsymbol { f } ( \boldsymbol { x } ) & 10 & 20 & 40 \\\hline\end{array}$

For the following demand equation, find the largest possible revenue. $q = - 4 p + 3,600$

For the following demand equation, express the total revenue R as a function of the price p per item. $q = - 2 p + 1600$