Practise Test

Part-A Instructions

Time — 1 hour and 20 minutes

28 Questions

NO CALCULATOR IS ALLOWED FOR THIS PART OF THE EXAM.

Directions:

Solve each problem using your own scratch paper for rough work.
After reviewing the answer choices, select the one best answer by clicking the corresponding option (A, B, C, or D).
Only your selected online response will be recorded and scored.
You may change your answer before submitting the test.
You have to answer atleast one question to be able to submit the test.
Manage your time wisely — do not spend too much time on any single problem.
Work efficiently, focus on accuracy, and attempt as many questions as possible within the time limit.
Once you submit you cannot return to this section

In this exam

Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.
Angle measures for trigonometric functions are assumed to be in radians.

Part-A

1. Consider the functions $g$ and $h$ given by $g(x)=4^x$ and $h(x)=16^{\,x+2}$. In the $xy$-plane, what is the $x$-coordinate of the point of intersection of the graphs of $g$ and $h$?

-4

-2

0

2
2. The function f is given by $ f(x) = \log_{2} x $. What input value in the domain of f yields an output value of 4?

32

16

2

1/2
3. The function k is given by $ k(\theta) = 2\sin \theta $. What are all values of $ \theta $, for $ 0 \le \theta < 2\pi $, where $ k(\theta) = -1 $?

$ \theta = \frac{\pi}{6} $ and $ \theta = \frac{5\pi}{6} $.

$ \theta = \frac{\pi}{3} $ and $ \theta = \frac{5\pi}{3} $.

$ \theta = \frac{2\pi}{3} $ and $ \theta = \frac{4\pi}{3} $.

$ \theta = \frac{7\pi}{6} $ and $ \theta = \frac{11\pi}{6} $.
4. Which of the following expressions is equivalent to $ \log_3(x^5) $ ?

$ \log_{3} 5 + \log_{3} x $

$ \log_3 5 \cdot \log_3 x $

$ 5\log_{3} x $

$ \dfrac{\log_{3} x}{\log_{3} 5} $
5. Which of the following statements is true about the exponential function $ h $ given by $ h(x) = -3 \cdot 4^x $?

$ h $ is always increasing, and the graph of $ h $ is always concave up.

$ h $ is always increasing, and the graph of $ h $ is always concave down.

$ h $ is always decreasing, and the graph of $ h $ is always concave up.

$ h $ is always decreasing, and the graph of $ h $ is always concave down.
6. In the xy-plane, the graph of a rational function f has a hole at x = 2. Input values of f sufficiently close to 2 correspond to output values arbitrarily close to 6. Which of the following could define f(x)?

$$ f(x) = \frac{6(x - 2)(x + 3)}{(x - 3)(x - 2)} $$

$$ f(x) = \frac{(x - 2)(x + 4)}{(x - 2)(x - 1)} $$

$$ f(x) = \frac{(x - 6)(x + 4)}{(x - 6)(x - 1)} $$

$$ f(x) = \frac{(x + 1)(x + 6)}{(x - 1)(x + 2)} $$
7. At time $ t = 0 $, water begins pouring into an empty container at a constant rate. The water pours into the container until it is full. The situation is modeled by the given graph, where time, in seconds, is the independent variable and the depth of water in the container, in centimeters, is the dependent variable. For which of the following containers would the graph be appropriate?

(A)

(B)

(c)

(D)
8. At time $ t = 0 $ years, the population of a certain city was $ 23{,}144 $. During each of the next 10 years, the population decreased by $ 4\% $ per year. Based on this information, which of the following models the population as a function of time $ t $, in years, for $ 0 \le t \le 10 $?

$ 23{,}144 - 0.04t $

$ 23{,}144 - 0.96t $

$ 23{,}144(0.96)^t $

$ 23{,}144(1.04)^t $
9. Where both expressions are defined, which of the following is equivalent to $ \dfrac{\sec^2 x - 1}{\sec^2 x} $?

$ \dfrac{\tan^2 x}{\sin^2 x} $

$ \dfrac{\tan^2 x}{\cos^2 x} $

$ \sin^2 x $

$ \cos^2 x $
10. The location of point $ X $ in polar coordinates $ (r, \theta) $ is$ \left(1, \frac{5\pi}{6}\right) $. Which of the following describes the location of point $ X $ in rectangular coordinates $ (x, y) $?

$ \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right) $

$ \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) $

$ \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) $

$ \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) $
11. The figure shows the graph of an exponential decay function $ f $. The coordinates of two of the points are labeled. If $ y = f(x) $, what is the y-coordinate of the point on the graph where $ x = 0 $?

40

30

20

15
12. The table gives values for a function g at selected values of x. Which of the following statements is true?

x g(x)

-1 -3

0 -2

1 1

2 6

3 13

g is best modeled by a linear function, because the average rate of change over any length input-value interval is constant.

g is best modeled by a quadratic function, because the average rates of change over consecutive equal-length input-value intervals are constant.

g is best modeled by a linear function, because the successive second differences of the output values over equal-interval input values are constant.

g is best modeled by a quadratic function, because the successive second differences of the output values over equal-interval input values are constant.
13. The function $ f $ is given by $ f(t) = e^{t} $, and the function $ g $ is given by $ g(t) = 7\ln t $. If the function $ h $ is given by $ h(t) = (f \circ g)(t) $, which of the following is an expression for $ h(t) $, for $ t > 0 $?

$ 7t $

$ t^{7} $

$ t e^{7} $

$ 7^{t} $
14. The binomial theorem can be used to expand the polynomial function $ p $, given by $ p(x) = (x - 3)^5 $. What is the coefficient of the $ x^3 $ term in the expanded polynomial?

$ (-3)^3 \cdot 10 $

$ (-3)^2 \cdot 10 $

$ (-3)^3 \cdot 5 $

$ (-3)^2 \cdot 5 $
15. The function g is given by $ g(x) = 2\cos(\pi x) + 1 $. Which of the following is the graph of g for $ 0 \le x \le 4 $?

(A)

(B)

(C)

(D)
16. The figure shows the graph of a function g in the $xy$-plane with four labeled points. It is known that a relative maximum of g occurs at point A, and the only point of inflection of the graph of g is point C.Of the following points, at which is the rate of change of g the least?

A

B

C

D

18. The amount of water used each day in an office building, measured in hundreds of gallons, is modeled by the function $ g $ defined by $ g(t) = 5\sin(0.8(t + 2)) + 25 $, for integer values of $ t $ with $ 0 \le t \le 365 $ days. Using actual data over time, it was determined that the model underestimates the amount of water used each day by 800 gallons. Based on this information, which of the following functions is a better model for the amount of water used each day, measured in hundreds of gallons?

$ f(t) = 5\sin(0.8(t + 10)) + 25 $

$ h(t) = 5\sin(0.8(t + 2)) + 33 $

$ k(t) = 5\sin(6.4(t + 2)) + 25 $

$ m(t) = 40\sin(0.8(t + 2)) + 25 $
19. In the xy-plane, two different angles $ \alpha $ and $ \beta $ are in standard position and share a terminal ray. Based on this information, which of the following gives possible values for $ \alpha $ and $ \beta $?

$ \alpha = -\frac{\pi}{4} $ and $ \beta = -\frac{7\pi}{4} $

$ \alpha = \frac{3\pi}{5} $ and $ \beta = -\frac{3\pi}{5} $

$ \alpha = \frac{2\pi}{3} $ and $ \beta = \frac{8\pi}{3} $

$ \alpha = \frac{5\pi}{6} $ and $ \beta = -\frac{\pi}{6} $
20. In the xy-plane, the function $ h $, given by $ h(x) = 3^{(x+2)} $, is a horizontal translation of the exponential function $ f $, given by $ f(x) = 3^x $. Which of the following is an equivalent form for $ h(x) $ that expresses $ h $ as a vertical dilation of $ f $?

$ h(x) = 3^{x/2} $

$ h(x) = 9 \cdot 3^x $

$ h(x) = 9 \cdot \left(\frac{1}{3}\right)^x $

$ h(x) = 9 + 3^x $
21. The table gives values for a polynomial function $ g $ at selected values of $ x $. If $ a < b $, then $ g(a) > g(b) $ for all $ a $ and $ b $ in the interval $ 3 < x < 7 $. Which of the following could be true about the graph of $ g $ on the interval $ 3 < x < 7 $?

$ x $ 3 4 5 6 7

$ g(x) $ $-11$ $-19$ $-29$ $-41$ $-55$

The graph of $ g $ is concave down because the function is decreasing, and the average rate of change over equal-length input-value intervals is increasing.

The graph of $ g $ is concave up because the function is decreasing, and the average rate of change over equal-length input-value intervals is increasing.

The graph of $ g $ is concave down because the function is decreasing, and the average rate of change over equal-length input-value intervals is decreasing.

The graph of $ g $ is concave up because the function is decreasing, and the average rate of change over equal-length input-value intervals is decreasing.
22. The function $ f $ is given by $ f(x) = 2\cos(\pi x) + 3 $. The graph of $ f $ is mapped to the graph of $ g $ in the same xy-plane by a horizontal translation of the graph of $ f $ by $ \frac{\pi}{2} $ units to the right. Which of the following is an expression for $ g(x) $?

$ 2\cos\!\left(\pi\left(x - \frac{1}{2}\right)\right) + 3 $

$ 2\cos\!\left(\pi\left(x - \frac{\pi}{2}\right)\right) + 3 $

$ 2\cos(\pi x) + 3 - \frac{\pi}{2} $

$ 2\cos(\pi x) + 3 + \frac{\pi}{2} $
23. The function $ C $ models temperature, in degrees Celsius, as a function of time $ t $, in hours, for $ t \ge 0 $. The function $ P $ models electricity usage, in kilowatts, as a function of temperature, in degrees Celsius. Let $ K $ be the composition function defined by $ K(t) = P(C(t)) $. Which of the following statements is true about function $ K $?

$ K $ models electricity usage as a function of time.

$ K $ models temperature as a function of electricity usage.

$ K $ models time as a function of electricity usage.

$ K $ models electricity usage as a function of temperature.
24. The figure shows a circle centered at the origin with an angle of measure $ \theta $ radians in standard position and point $ P $ on the circle. The terminal ray of the angle intersects the circle at point $ Q $. The length of arc $ \overline{PQ} $ is 6 units. Which of the following gives the distance of point $ Q $ from the y-axis?

$ \cos\!\left(\frac{6}{5}\right) $

$ \sin\!\left(\frac{6}{5}\right) $

$ 5\cos\!\left(\frac{6}{5}\right) $

$ 5\sin\!\left(\frac{6}{5}\right) $
25. In the xy-plane, the graph of which of the following functions has a vertical asymptote at $ x = \frac{3\pi}{4} $?

$ f(x) = \cot x $

$ f(x) = \cot\!\left(x - \frac{\pi}{2}\right) $

$ f(x) = \cot\!\left(x - \frac{\pi}{4}\right) $

$ f(x) = \cot\!\left(x + \frac{\pi}{4}\right) $
26. The function $r$ is given by \[ r(x)=\frac{x^{2}-x-2}{(x+1)^{2}(x-2)}. \] In the $xy$-plane, which of the following is true about holes in the graph of $r$?

There are holes at $x=-1$ and $x=2$ because the multiplicity of $-1$ in the denominator is greater than the multiplicity of $-1$ in the numerator, and because the multiplicity of $2$ in the

There are holes at $x=-1$ and $x=2$ because the multiplicity of $-1$ in the numerator is equal to the multiplicity of $-1$ in the denominator, and because the multiplicity of $2$ in the nume

There is a hole at $x=2$ only because the multiplicity of $-1$ in the denominator is greater than the multiplicity of $-1$ in the numerator, and because the multiplicity of $2$ in the numerato

There is a hole at $x=2$ only because the multiplicity of $-1$ in the numerator is equal to the multiplicity of $-1$ in the denominator, and because the multiplicity of $2$ in the numerator is
27. A regression model $ S $ is constructed for a data set. The residuals from the regression are plotted and labeled. Based on the vertical axis of the residual plot (not shown), point $ A $ is located at $ 1.3 $ and point $ B $ is located at $ -2.5 $. Which of the following statements is true about the model and the estimates produced by the model that correspond to $ A $ and $ B $?

The model produces an overestimate at $ A $ and an underestimate at $ B $. Based on the absolute values of the residuals, there is a greater error in the model with $ B $ than with $ A $.

The model produces an underestimate at $ A $ and an overestimate at $ B $. Based on the absolute values of the residuals, there is a greater error in the model with $ B $ than with $ A $.

The model produces an overestimate at $ A $ and an underestimate at $ B $. Based on the absolute values of the residuals, there is a greater error in the model with $ A $ than with $ B $.

The model produces an underestimate at $ A $ and an overestimate at $ B $. Based on the absolute values of the residuals, there is a greater error in the model with $ A $ than with $ B $.
28. A portion of the graph of the polar function $ r = f(\theta) $, where $ f(\theta) = 2 - 4\cos\theta $, is shown in the polar coordinate system for $ a \le \theta \le b $. If $ 0 \le a < b < 2\pi $, which of the following are the values for $ a $ and $ b $?

$ a = 0 $ and $ b = \frac{\pi}{3} $

$ a = 0 $ and $ b = \frac{\pi}{6} $

$ a = \pi $ and $ b = \frac{4\pi}{3} $

$ a = \pi $ and $ b = \frac{7\pi}{6} $

x	g(x)
-1	-3
0	-2
1	1
2	6
3	13

Point	A	B	C	D
\(r\)	2	2	-2	-2
\(\theta\)	\(-\frac{\pi}{4}\)	\(\frac{3\pi}{4}\)	\(\frac{5\pi}{4}\)	\(\frac{7\pi}{4}\)

\( x \)	3	4	5	6	7
\( g(x) \)	\(-11\)	\(-19\)	\(-29\)	\(-41\)	\(-55\)

Part-B

Show Calculator

1. The table gives values for a polynomial function f at selected values of x. What is the average rate of change of f over the closed interval [1,4]?

x 0 1 2 3 4

f(x) 18 16 6 0 10

-2

-1/2

2

13

3. The figure shows the graph of function $g$ for $0 \le x \le 13$. The endpoints of the interval are labeled with points $A$ and $E$. Two other extrema for $g$ are labeled with points $B$ and $D$. Point $C$ is the only point of inflection of the graph of $g$ for $0 \le x \le 13$. Let $t_A$, $t_B$, $t_C$, $t_D$, and $t_E$ represent the $x$-coordinates at those points. Of the following, on which intervals is the rate of change of $g$ decreasing?

$[t_A, t_B]$ only

$[t_D, t_E]$ only

$[t_A, t_B]$ and $[t_D, t_E]$

$[t_C, t_D]$ and $[t_D, t_E]$
4. The figure shows a circle centered at the origin with an angle of measure radians in standard position. The terminal ray of the angle intersects the circle at point . If , what is the slope of the line that passes through the origin and ?

$ \frac{4}{3} $

$ \frac{4}{5} $

$ \frac{3}{4} $

$ \frac{3}{5} $
5. A set of data is represented using a semi-log plot (not shown), in which the vertical axis is logarithmically scaled. The points on the semi-log plot appear to follow a decreasing linear pattern. Which of the following function types best models the set of data?

Linear

Exponential growth

Exponential decay

Logarithmic
6. The function C models the cost, in dollars, for producing x items and is given by $$C(x) = \frac{1000 + bx}{x},$$ where b is a constant. It is known that the cost is 115 to produce 10 items and 65 to produce 20 items. What is the average rate of change of C as x changes from x = 30 to x = 40?

−$0.83 per item

−$5 per item

−$8.33 per item

−$50 per item
7. For time $t$ hours, $0 \le t \le 2$, the number of people inside a large shopping center is changing at a rate modeled by the function $P$ given by $P(t) = t^{3} - 4t^{2} + 3t + 1$, where $P(t)$ is measured in hundreds of people per hour. Which of the following gives time $t$ and reasoning for when the number of people inside the shopping center is at its maximum?

$t = 0.451$, because the rate of change in the number of people inside the shopping center changes from increasing to decreasing.

$t = 0.451$, because the rate of change in the number of people inside the shopping center changes from positive to negative.

$t = 1.445$, because the rate of change in the number of people inside the shopping center changes from increasing to decreasing.

$t = 1.445$, because the rate of change in the number of people inside the shopping center changes from positive to negative.

9. The table gives the weight $y$, in pounds, of an animal for selected ages$x$, in years. A logarithmic regression is used to model these data. What is the weight of the animal, to the nearest pound, predicted by the logarithmic function model at age $4.5$ years?

$x$ (years) 1.5 2 2.5 3 4 6

$y$ (pounds) 2420 3150 3615 4105 4835 6355

5073

5203

5333

5345
10. The polar function $r=f(\theta)$, where $f(\theta)=1+2\cos\theta$, is graphed in the polar coordinate system for $0 \le \theta \le 2\pi$. On which of the following intervals of $\theta$ is the distance between the point with polar coordinates $(f(\theta),\theta)$ and the origin decreasing?

(0, 2.094) only

(2.094, 4.189)

(0, 2.094) and (3.142, 4.189)

(2.094, 3.142) and (4.189, 6.283)
11. On a given day, the number of people, in thousands, that have entered a museum is modeled by the function $h$ given by $h(t)=4.217\tan^{-1}(0.7t-0.026)$, where $t$ is measured in hours and $0 \le t \le 8$. Based on the model, at what time $t$ did person number $4000$ enter the museum?

t = 1.121

t = 2.029

t = 5.165

t = 6.623
12. For $0 \le t \le 16$, the rate at which customers arrive at a restaurant on a given day is modeled by the function $R$, where $R(t)$ is measured in customers per hour and $t$ is measured in hours since the restaurant opened. The function $R$ is increasing for $0 < t < 4$ and $8 < t < 12$, and $R$ is decreasing for $4 < t < 8$ and $12 < t < 16$. The function $N$ models the total number of customers who have arrived at the restaurant since it opened, up to time $t$. Which of the following could be the graph of $y = N(t)$ for $0 \le t \le 16$?

(A)

(B)

(C)

(D)

Year (time t)	Length of High-Speed Railways (thousands of kilometers)
2010 (t = 0)	1
2013 (t = 3)	7
2016 (t = 6)	16
2019 (t = 9)	25
2022 (t = 12)	38

\(x\)	-5.1	-5.01	-5.001	-5	-4.999	-4.99	-4.9
\(f(x)\)	-10	-100	-1000	undefined	1000	100	10

Practise Test

Part-A Instructions

Time — 1 hour and 20 minutes

Part-B Instructions

Time — 40 minutes

Part-A

Part-B

Section II : Free Response Questions (FRQs)

x	0	1	2	3	4
f(x)	18	16	6	0	10

\(x\) (years)	1.5	2	2.5	3	4	6
\(y\) (pounds)	2420	3150	3615	4105	4835	6355