Practise Test

1. The function \( h \) is given by \( h(x) = 8 \cdot 2^x \). For which of the following values of \( x \) is \( h(x) = 256 \)?

2. The figure shows the graph of the quartic polynomial function p. How many points of inflection does the graph of p have?

3. The function \(f\) is given by \(f(x) = e^{2x}\), and the function \(g\) is given by \(g(x) = \ln(3x)\). For \(x > 0\), which of the following is an expression for \(f(g(x))\)?

4. Consider the constant function \(f\) given by \(f(x) = -1\) and the function \(g\) given by \(g(x) = \log_3 x\). Let \(h\) be the function given by \(h(x) = g(x) - f(x)\). In the xy-plane, what is the x-intercept of the graph of \(h\)?

5. The figure shows the graph of an exponential function \(k\), where \(k(x) = b^x\) for \(b > 0\) and \(b \ne 1\). Consider the logarithmic function \(h\) (not shown) given by \(h(x) = \log_b x\). Of the following pairs of points, which are on the graph of \(h\)?

6. The table gives values for the invertible functions \( h \) and \( k \) at selected values of \( x \). What is the value of \( h^{-1}(k(3)) \)?

7. The function f is given by \( f(x) = 2^{(3x)} \). Which of the following statements describes characteristics of the graph of f in the \( xy \)-plane?

8. The function H models a periodic phenomenon. The maximum value of H is 8, which occurs at time \( t = 0 \) hours. The period of H is 12 hours. For \( 0 \le t \le 24 \) hours, which of the following could be an expression for \( H(t) \)?

9. A data set that appears exponential is modeled by the function \( k(t) = 7 \cdot 5^t \). The data are represented using a semi-log plot, where the vertical axis is logarithmically scaled with the natural logarithm. Which of the following could describe the appearance of the data in the semi-log plot?

10. The figure shows a sphere that can expand. As it expands, the volume inside the sphere and the radius of the sphere both increase. In particular, the radius increases at a decreasing rate with respect to the volume. Which of the following graphs could depict this situation, where volume, in cubic centimeters, is the independent variable and radius, in centimeters, is the dependent variable?

11. The location of a point in the plane is given by polar coordinates \( (-3, \frac{2\pi}{3}) \). Which of the following gives another representation for this point in polar coordinates?

12. The height of a point on a dolphin swimming in the ocean can be modeled by a sinusoidal function \( h \). For time \( t \), in seconds, if \( h(t) \) is positive, the point on the dolphin is above the surface the water, and if \( h(t) \) is negative, the point on the dolphin is below the surface of the water. As the dolphin swims, it jumps to a maximum height of 3 feet above the surface, then hits the surface, and dives to a minimum height of 3 feet below the surface. The dolphin then returns to the surface, and jumps to a maximum height of 3 feet above the surface. This cycle from maximum height to the next maximum height repeats every 4 seconds. Which of the following could be an expression for \( h(t) \)?

13. The table gives values for the function \( f \) at selected values of \( x \). Which of the following function types best models these data?

14. Which of the following is equivalent to the expression \[ 4\left(\cos^2\left(\frac{2\pi}{5}\right) - \sin^2\left(\frac{2\pi}{5}\right)\right)? \]

15. The population of people in a community is decreasing over time. The population can be modeled by the exponential decay function \( P \) given by \[ P(t) = 20{,}000(0.92)^t, \] where \( t \) is measured in years since 2020. Based on updated data, a new exponential decay function model for the population was constructed. The new model \( Q \) is given by \[ Q(t) = 20{,}000\big((0.9)(0.92)\big)^t. \] Which of the following describes the relationship between model \( Q \) and model \( P \)?

16. Consider a circle centered at the origin in the xy-plane. An angle of measure \( \frac{\pi}{4} \) radians in standard position has a terminal ray that intersects the circle at point \( P \). The angle is subtended by an arc of the circle in Quadrant I with length 40 units. What is the radius of the circle?

17. The function \( g \) is given by \( g(x) = \frac{1 + \sin x}{\cos x} \). On the interval \( [0, 2\pi] \), what are all zeros of \( g \)?

18. The number of people waiting in a line to enter a stadium can be modeled by the function \( b \) defined by \[ b(t) = 42\cos\!\left(\frac{\pi}{30}t\right) + 45 \] for \( 0 \le t \le 60 \), where \( t \) is the time, in minutes, since the stadium opened. Based on the graph of the model, which of the following is true?

19. The rational function \( f \) is given by \[ f(x) = \frac{x^{3} - x^{2} - 4x}{x^{2} - 4x} \] In the \( xy \)-plane, which of the following is the slope of a slant asymptote of the graph of \( f \)?

20. Let \( f \) be the function given by \( f(x) = \sin x \). The function \( g \) has the same amplitude as \( f \), and the period of \( g \) is \( \frac{1}{2} \) of the period of \( f \). Which of the following defines \( g \) in terms of \( f \)?

21. The binomial theorem can be used to expand an expression of the form \( (a + b)^n \). Which of the following is equivalent to \( (x + 3y)^5 \)?

22. The polar function\( r = f(\theta) \), where \( f(\theta) = 1 - 2\cos(-\theta) \), is graphed in the polar coordinate system. As \( \theta \) varies from \( \theta = \frac{\pi}{2} \) to \( \theta = \pi \), how is the distance between the origin and the point with polar coordinates \( (f(\theta), \theta) \) changing?

23. The function \( f \) is given by \( f(x) = x + 1 \), and the function \( g \) is given by \( g(x) = (x + 1)(3x - 4) \). Consider the rational function \( m \) defined by \( m(x) = \frac{f(x)}{g(x)} \). Which of the following is true about holes in the graph of \( m \) in the \( xy \)-plane?

24. Consider the functions \( g \) and \( h \) given by \( g(\theta) = \sec \theta \) and \( h(\theta) = \csc \theta \). Which of the following statements is true for both \( g \) and \( h \)?

25. Both nonzero functions \(f\) and \(g\) are invertible. The input values of \(f\) are times, in hours, and the output values of \(f\) are rates, in miles per hour. The input values of \(g\) are rates, in miles per hour, and the output values of \(g\) are costs, in dollars. For which of the following are the input values costs, in dollars, and the output values times, in hours?

26. The table gives values for a function \(f\) at selected values of \(x\). Which of the following is a table of values for the function \(h\) given by \(h(x) = f(x - 1) + 3\)?

27. A table of values is given for a data set that can be modeled with \( y \) as a function of \( x \). Two function models, linear and exponential, are constructed using regressions. Residual plots for the two regressions are shown. Which of the following statements is true?>

28. The figure shows the graph of the polar function \( r = f(\theta) \), for \( 0 \le \theta \le 2\pi \), in the polar coordinate system. Which of the following could be an expression for \( f(\theta) \)?

1. An object travels along a horizontal line with velocity given by the function \( v \). The graph of \( y = v(t) \), which consists of four line segments, is shown for \( 0 \le t \le 10 \). On which of the following intervals is the object's velocity decreasing over the entire interval?

2. Consider the sinusoidal functions \( f \) and \( g \), defined by \( f(x) = 3\cos(x + 2.78) \) and \( g(x) = \sin(x - 0.25) \). Which of the following describes a zero of the function \( h \) defined by \( h(x) = f(x) - g(x) \) on the interval \( [0, \pi] \)?

3. The figure shows the graph of a quartic polynomial function with zeros at \( A \), \( B \), and \( C \). Which of the following statements gives the multiplicity of the zero at \( C \) with correct reasoning?

4. In the \( xy \)-plane, the function \( h \) is graphed on the interval \( 0 \le x \le 10 \). The point \( (3, -2) \) is on the graph of \( y = h(x) \). Which of the following points is on the graph of \( y = 2h(x) - 1 \)?

5. Consider the piecewise function \( W \) defined by \[ W(t) = \begin{cases} 125 - 0.2t^2 & \text{for } 0 \le t < 20, \\ 445 - 20t & \text{for } 20 \le t \le 22. \end{cases} \] Which of the following is a correct statement about the average rates of change of \( W \)?

6. The figure shows a portion of the graph of a function \( f \). Which of the following conclusions is possible for \( f \)?

7. The polynomial function \( f \) is an odd function with domain \( -4 \le x \le 4 \). The table gives information about values of \( f(x) \) and the behavior of the function. What is the absolute maximum value of \( f \) on its domain?

8. The average monthly high temperature, in degrees Fahrenheit (°F), varies periodically for a certain city. The table gives the average monthly high temperature for selected months of the year. Based on a sinusoidal regression of these data, a model is constructed to predict average monthly high temperature as a function of time \( t \), the number of the month of the year. What is the average monthly high temperature predicted by the model for June \( (t = 6) \)?

9. The figure shows a square sheet of cardboard on the left and an open box on the right. The square sheet has side length 12 inches (in) and is used to make the open box by removing from each of the four corners a square of side length \( x \) inches, and folding up the sides. For what value of \( x \) does the box have the maximum possible volume? Note: The volume of a rectangular solid with length \( l \), width \( w \), and height \( h \) is \[ V = lwh. \]

10. In the polar coordinate system, the graph of a polar function \( r = f(\theta) \) is shown with a domain of all real values of \( \theta \) for \( 0 \le \theta < 2\pi \). On this interval of \( \theta \), the graph has no holes, passes through each point exactly one time, and as \( \theta \) increases, the graph passes through the labeled points \( A \), \( B \), \( C \), and \( D \), in that order. On which of the following intervals is the average rate of change of \( r \) with respect to \( \theta \) greatest?

11. A builder is trying to find the maximum area that can be bounded by a rectangular region on a certain plot of land. The builder determined four different areas based on the length \( x \) of one side of the rectangle, for four different rectangles. The table gives values of one side of the rectangle, in meters (m), and the corresponding area, in square meters (m\(^2\)). After noticing that as \( x \) increases, the area values increase and then decrease, the builder used a quadratic regression to produce a function \( A \) that models the area based on the constraints. Of the following, what is the maximum value of \( A(x) \), to the nearest integer, based on the model?

12. The function \( L \) is defined by \( L(t) = ae^{-0.0149t} + b \), where \( a \) and \( b \) are positive constants. If \( L(0) = 15 \) and \( \lim_{t \to \infty} L(t) = 1 \), for what value of \( t \) is \( L(t) = 6 \)?