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. The figure above shows the graph of a function \( g \). Which of the following statements is correct?

The rate of change of \( g \) is positive and decreasing.

The rate of change of \( g \) is positive and increasing.

The rate of change of \( g \) is negative and increasing.

The rate of change of \( g \) is negative and decreasing.
2. Selected values for the function \( f(x) \) are shown in the table above. Find the average rate of change for \( f(x) \) from \( x = 1 \) to \( x = 8 \).

\( x \) −3 1 4 8

\( f(x) \) 0 −2 7 3

\( \frac{5}{7} \)

\( -\frac{5}{7} \)

\( \frac{5}{8} \)

\( \frac{1}{7} \)
3. The graph of the exponential function \( g \) has the following end behaviors: \[ \lim_{x \to -\infty} g(x) = 0 \quad \text{and} \quad \lim_{x \to \infty} g(x) = -\infty \] Which of the following could be an equation for \( g \)?

\( g(x) = -3\left(\frac{1}{2}\right)^x \)

\( g(x) = -\frac{1}{2}(3)^x \)

\( g(x) = 3\left(\frac{1}{2}\right)^x \)

\( g(x) = \frac{1}{2}(3)^x \)
4. Zebra mussels are an invasive, fingernail-sized mollusk that are infesting some of the freshwater lakes in North America.Their population increases on average at a rate of 5% per day. Let 700 be the amount of zebra mussels in a certain lake at time \( d = 0 \) days. Which of the following functions \( g \) models the amount of zebra mussels after \( t \) weeks, where 700 is the amount of zebra mussels at time \( t = 0 \)?

\( g(t) = 700(1.05)^{t/7} \)

\( g(t) = 700(1.05)^{7t} \)

\( g(t) = 700\left(1.05^{1/7}\right)^{7t} \)

\( g(t) = 700\left(1.05^{7}\right)^{t/7} \)
5. Given the functions \( f(x) = -\sqrt{x} \) and \( g(x) = x - 2 \), which of the following graphs could represent \( y = g(f(x)) \)?

(A)

(B)

(C)

(D)
6. What is the value of the expression \[ \log_{3} 27 + \log_{4}\left(\frac{1}{2}\right)? \]

\( \frac{3}{2} \)

\( \frac{5}{2} \)

\( \frac{7}{2} \)

\( \frac{1}{2} \)
7. The graph of the piecewise-linear function \( f \) is shown in the figure. Let \( g \) be the inverse function of \( f \). What is the minimum value of \( g \)?

\( -4 \)

\( -3 \)

\( 2 \)

\( 3 \)
8. Given that \( f(x) = cx - 3 \) and \( g(x) = cx + 5 \) are both defined on the set of all real numbers and \( c \) is a constant, what is the value of \( c \) if \[ (f \circ g)(x) = (g \circ f)(x) \] for all values of \( x \)?

\( c = -1 \)

\( c = 0 \)

\( c = 1 \)

\( c = 2 \)
9. The rational function \( f \) is defined by \[ f(x) = \frac{3x^{4} - 5x^{2} + 7}{6x^{2} + x - 4}. \] Which of the following claim and explanation statements about the graph of \( f \) is true?

The graph of \( f \) has a horizontal asymptote because the degree of the numerator is equal to the degree of the denominator.

The graph of \( f \) has a horizontal asymptote because the leading coefficient of the numerator is greater than the leading coefficient of the denominator.

The graph of \( f \) does not have a horizontal asymptote because the degree of the numerator is greater than the degree of the denominator.

The graph of \( f \) does not have a horizontal asymptote because the leading coefficient of the numerator is less than the leading coefficient of the denominator.
10. The function \( g \) is defined by \( g(x) = \sin\left(x + \frac{\pi}{3}\right) \). The solutions to which of the following equations on the interval \( 0 \le x \le 2\pi \) are the solutions to \( g(x) = 1 \) on the interval \( 0 \le x \le 2\pi \)?

\( \sin x + \sqrt{3}\cos x = 2 \)

\( \sin x - \sqrt{3}\cos x = 2 \)

\( \sqrt{3}\sin x + \cos x = 2 \)

\( \sqrt{3}\sin x - \cos x = 2 \)

12. Let \( r \) be the rational function given by \[ r(x) = \frac{(x-3)(x+2)^2(x+5)(x-4)} {(x+2)(x+5)(x-4)^2}. \] Which of the following descriptions about the graph of \( r(x) \) is correct?

The graph of \( r(x) \) has one vertical asymptote and two holes.

The graph of \( r(x) \) has two vertical asymptotes and one hole.

The graph of \( r(x) \) has two vertical asymptotes and two holes.

The graph of \( r(x) \) has three vertical asymptotes and no holes.
13. Let \( f \) be a sinusoidal function. The graph of \( y = f(x) \) is given in the \( xy \)-plane. What is the amplitude of \( f \)?

\( 1 \)

\( 3 \)

\( 4 \)

\( 6 \)
14. A regression model was created for the data in the graph above (left). The residual plot for the model is given above (right). Which of the following statements about the regression model is best?

A quadratic regression model was used and the model is appropriate.

A quadratic regression model was used and the model is not appropriate.

An exponential regression model was used and the model is appropriate.

An exponential regression model was used and the model is not appropriate.
15. Let \( g(x) = 2\sin(x)\cos(x) \) and \( h(x) = -\sin(x) \). In the \( xy \)-plane, what are the \( x \)-coordinates of the points of intersection of the graphs of \( g \) and \( h \) for \( 0 \le x < 2\pi \)?

\( x = \frac{2\pi}{3} \) and \( x = \frac{4\pi}{3} \) only

\( x = \frac{\pi}{2},\; x = \frac{3\pi}{2},\; x = \frac{7\pi}{6},\; x = \frac{11\pi}{6} \)

\( x = 0,\; x = \pi,\; x = \frac{\pi}{3},\; x = \frac{5\pi}{3} \)

\( x = 0,\; x = \pi,\; x = \frac{2\pi}{3},\; x = \frac{4\pi}{3} \)
16. The table shows values for a function \( f \) at selected values of \( x \). Which of the following claim and explanation statements best fits these data?

\( x \) \( f(x) \)

1 \( -11 \)

2 \( -7 \)

3 \( -4 \)

4 \( -3 \)

The graph of \( f \) is concave up, because the rate of change over consecutive equal-length input-value intervals is positive.

The graph of \( f \) is concave up, because the rate of change over consecutive equal-length input-value intervals is increasing.

The graph of \( f \) is concave down, because the rate of change over consecutive equal-length input-value intervals is negative.

The graph of \( f \) is concave down, because the rate of change over consecutive equal-length input-value intervals is decreasing.
17. It is known for a function \( f \) that \[ \lim_{x \to \infty} f(x) = -\infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = \infty. \] Which of the following could represent the equation for \( f \)?

\( f(x) = 7x^6 - 5x^4 - 3x^2 + 11 \)

\( f(x) = -2x^5 + 8x^3 - 10x^2 - 20 \)

\( f(x) = -3x^4 + 13x^3 + 20x^2 - 4x + 1 \)

\( f(x) = 8x^2 - 14x + 9 \)
18. A robotic ant is designed to cross a table in a sinusoidal pattern, making a wave pattern as it travels from left to right. The table has a length of 1,800 mm and a width of 800 mm. Standing on one side of the table, the values \( x = 0 \) and \( x = 1{,}800 \) represent the left and right sides of the table, respectively. The values \( y = 0 \) and \( y = 800 \) represent the nearest and furthest sides of the table, respectively. The path of the robotic ant begins on the left side of the table, \( x = 0 \), and completes one period of a sinusoidal function by ending on the right side of the table, \( x = 1{,}800 \). During its path, the robotic ant reaches its maximum distance from the near side of the table of \( y = 750 \) before reaching its minimum distance of \( y = 150 \). If \( y = f(x) \) models the path of the robotic ant, which of the following could define \( f(x) \)?

\( 300 \sin\!\left(\frac{\pi}{900}x\right) + 450 \)

\( 300 \sin(3600\pi x) + 450 \)

\( 600 \sin\!\left(\frac{\pi}{900}x\right) + 450 \)

\( 600 \sin(3600\pi x) + 450 \)
19. The polynomial function \( g \) is given by \[ g(x) = (x - 6)(x^2 + 2x + 2). \] Which of the following describes the zeros of \( g \)?

\( g \) has exactly two distinct real zeros.

\( g \) has exactly three distinct real zeros.

\( g \) has exactly one distinct real zero and no non-real zeros.

\( g \) has exactly one distinct real zero and two non-real zeros.
20. A portion of the graph of the periodic function \( g \) is shown above, where the period of \( g \) is 11. What is the value of \( g(42) \)?

1

2

3

5
21. A polynomial function \( p \) is given by \[ p(x) = ax^n - 3x^2 + 4x - 1, \] where \( a \) and \( n \) are constants and \( n > 2 \). If the end behavior of \( p \) is given by the statements \[ \lim_{x \to -\infty} p(x) = \infty \quad \text{and} \quad \lim_{x \to \infty} p(x) = -\infty, \] which of the following could be the values of \( a \) and \( n \)?

\( a = -4, \; n = 5 \)

\( a = -4, \; n = 6 \)

\( a = 3, \; n = 7 \)

\( a = 3, \; n = 8 \)
22. An exponential function \( y = f(x) \) is shown. Which of the following equations could represent \( f(x) \)?

\( f(x) = 2(2.25)^{x-1} \)

\( f(x) = \frac{8}{9}(2.25)^x \)

\( f(x) = 2(1.5)^{x-1} \)

\( f(x) = 1(1.5)^x \)
23. The function \( f \) is given by \[ f(x) = x^3 - 3x^2 - 10x, \] and the function \( g \) is given by \[ g(x) = x^2 - 6x + 5. \] Let \( h \) be the function given by \[ h(x) = \frac{f(x)}{g(x)}. \] What are all intervals on which \( h(x) > 0 \)?

\( (-2,0) \cup (5,\infty) \)

\( (-2,0) \cup (1,\infty) \)

\( (-2,0) \cup (1,5) \cup (5,\infty) \)

\( (-\infty,-2) \cup (0,1) \cup (5,\infty) \)
24. The rational function \(r\) has a vertical asymptote at \(x=1\) and a hole at \(x=-1\), as shown above. Which of the following could be an expression for \(r(x)\)?

\( -\dfrac{(x-2)(x+1)}{(x-1)(x+1)} \)

\( -\dfrac{(x-2)(x-1)}{(x-1)(x+1)} \)

\( \dfrac{(x-2)(x+1)}{(x-1)(x+1)} \)

\( \dfrac{(x-2)(x-1)}{(x-1)(x+1)} \)
25. A polynomial function \(h\) is given by \(h(x)=-2x(x+3)(x-2)^2\). Which of the following could be the graph of \(h(x)\)?

(A)

(B)

(C)

(D)

27. The point \(A\) has polar coordinates \(\left(4,\dfrac{7\pi}{6}\right)\). Which of the following also gives the location of point \(A\) in polar coordinates?

\(\left(4,-\dfrac{11\pi}{6}\right)\)

\(\left(4,-\dfrac{5\pi}{6}\right)\)

\(\left(-4,-\dfrac{\pi}{6}\right)\)

\(\left(-4,-\dfrac{5\pi}{6}\right)\)
28. The figure shows the graph of the polar function \(r=f(\theta)\), for \(0<\theta<2\pi\), in the polar coordinate system. Which of the following could be an expression for \(f(\theta)\)?

\(6\cos(2\theta)\)

\(6\cos(4\theta)\)

\(6\sin(2\theta)\)

\(6\sin(4\theta)\)

\( x \)	−3	1	4	8
\( f(x) \)	0	−2	7	3

\( x \)	\( a \)	\( -4 \)	\( -1 \)	\( 1 \)	\( b \)	\( 12 \)	\( 17 \)
\( g(x) \)	\( -17 \)	\( -11 \)	\( a + b \)	\( c \)	\( 11 \)	\( 17 \)	\( 23 \)

\( x \)	\( f(x) \)
1	\( -11 \)
2	\( -7 \)
3	\( -4 \)
4	\( -3 \)

\(x\)	\(k(x)\)
\(45\)	\(1\)
\(15\)	\(2\)
\(5\)	\(3\)
\(\dfrac{5}{3}\)	\(4\)
\(\dfrac{5}{9}\)	\(5\)

Part-B

Show Calculator

1. A geometric sequence \( g_n \) has known terms \( g_4 = 9 \) and \( g_7 = 30.375 \). What is the value of \( g_{12} \)?

66.000

85.500

129.746

230.660
2. The table presents values for \(f\) at selected values of \(x\). A cubic regression \(y=ax^3+bx^2+cx+d\) is used to model these data. What is the value of \(f(4)\) predicted by the cubic function model?

\(x\) 1 3 4 7 8

\(f(x)\) 1.137 24.080 60.449 366.761 553.928

30.791

60.449

61.382

62.018
3. A complex number is represented by a point in the complex plane. The complex number has the rectangular coordinates \((-1,3)\). Which of the following is one way to express the complex number using its polar coordinates \((r,\theta)\)?

\( (\sqrt{10}\cos(-1.249)) + i(\sqrt{10}\sin(-1.249)) \)

\( (\sqrt{10}\cos(1.249)) + i(\sqrt{10}\sin(1.249)) \)

\( (\sqrt{10}\cos(1.893)) + i(\sqrt{10}\sin(1.893)) \)

\( (\sqrt{10}\cos(4.391)) + i(\sqrt{10}\sin(4.391)) \)
4. The table above gives values for the function \(k\) at selected values of \(x\). Which of the following graphs could represent these data in a semi‑log plot, where the vertical axis is logarithmically scaled?

\(x\) 1 2 3 4 5

\(k(x)\) 25 55 121 266 586

(A)

(B)

(C)

(D)
5. The function \(f\) is defined by \[ f(x)=a\cos(b(x+c))+d, \] for constants \(a,b,c,\) and \(d\). In the \(xy\)-plane, the points \((3,4)\) and \((5,8)\) represent consecutive minimum and maximum values, respectively, on the graph of \(f\). What are the values of \(b\) and \(d\)?

\(b=\dfrac{\pi}{2}\) and \(d=4\)

\(b=\dfrac{\pi}{2}\) and \(d=6\)

\(b=\pi\) and \(d=4\)

\(b=\pi\) and \(d=6\)
6. Let \(g(x)=3.5-4.7\sin(x)\) and \(h(x)=\csc(1.8x)\). In the \(xy\)-plane, what are the \(x\)-coordinates of the points of intersection of the graphs of \(g\) and \(h\) for \(0\le x\le \dfrac{\pi}{2}\)?

\(x=0.515\) only

\(x=0.616\) only

\(x=0.241\) and \(x=0.487\)

\(x=0.451\) and \(x=1.447\)
7. The height above ground of a rider on a Ferris wheel can be modeled by the function \(H\) given by \[ H(t)=42\sin\!\left(\frac{1}{20}(t+30\pi)\right)+44.5, \] where \(t\) is the time, in seconds, once the ride begins for \(0\le t\le120\). Which of the following best describes the behavior of \(H(t)\) when \(t=115\) seconds?

The height above ground of the rider is increasing at an increasing rate.

The height above ground of the rider is increasing at a decreasing rate.

The height above ground of the rider is decreasing at an increasing rate.

The height above ground of the rider is decreasing at a decreasing rate.
8. The graph of \(f\) is shown above. Which of the following intervals gives the domain of \(f^{-1}\), the inverse of \(f\)?

\([3,10]\)

\([2,8]\)

\(\left[\dfrac{1}{8},\dfrac{1}{2}\right]\)

\(\left[\dfrac{1}{10},\dfrac{1}{3}\right]\)
9. The function \(P\) is given by \[ P(t)=\ln(c^2 t), \] where \(c\) is a positive constant. If \(P(3)=6\), what is the value of \(P(10)\)?

6.105

7.204

7.764

11.596
10. The table presents values for a function \(h\) at selected values of \(x\). An exponential regression \(y=ab^{x}\) is used to model these data. The residual in the model when \(x=5\) is \(0.512\). What is \(h(5)\)?

\(x\) \(h(x)\)

0 10

1 7

3 3

4 2

6 0.4

0.378

0.890

1.402

1.488
11. Students taking a final exam in a large college math class turn in their exams as they complete them.The rate that exams are turned in is modeled by the function \(E\), where \[ E(t)=9.1\sin(0.18t)+12.63 \] for \(0\le t\le30\). \(E(t)\) is measured in exams per minute, and \(t\) is measured in minutes since the first exam was turned in. Based on the model, at what value of \(t\) does the rate of exams being turned in reach its maximum?

\(t=8.727\)

\(t=21.730\)

\(t=26.180\)

\(t=30\)
12. A toy airplane travels counterclockwise in a circular pathway of radius \(5\) feet at a constant rate. The figure above displays the path and direction of the toy airplane in the \(xy\)-plane. At time \(t=0\) seconds, the plane is at the point \((5,0)\). At time \(t=60\) seconds, the toy airplane has completed exactly \(10\) complete rotations around the circular pathway. A sinusoidal function is used to model the \(x\)-coordinate of the position of the plane as a function of time \(t\) in seconds. Which of the following functions is an appropriate model for this situation?

\(f(t)=5\cos\!\left(\dfrac{\pi}{3}t\right)\)

\(f(t)=5\cos(12\pi t)\)

\(f(t)=5\sin\!\left(\dfrac{\pi}{3}t\right)\)

\(f(t)=5\sin(12\pi t)\)

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)

\(x\)	\(h(x)\)
0	10
1	7
3	3
4	2
6	0.4