Practise Test

1. A polynomial function \(p\) is given by \(p(x) = -x(x-4)(x+2)\). What are all intervals on which \(p(x)\ge 0\)?

2. The function \(f\) is given by \(f(x)=9\cdot 25^{x}\). Which of the following is an equivalent form for \(f(x)\) ?

3. The figure shows the graph of a function \(f\). The zero and extrema for \(f\) are labeled, and the point of inflection of the graph of \(f\) is labeled. Let \(A, B, C, D,\) and \(E\) represent the \(x\)-coordinates at those points. Of the following, on which interval is \(f\) increasing and the graph of \(f\) concave down?

4. The function \(f\) is given by \(f(x)=5x^{6}-2x^{3}-3\). Which of the following describes the end behavior of \(f\)?

5. Let \(x\) and \(y\) be positive constants. Which of the following is equivalent to \(2\ln x - 3\ln y\)?

6. The polynomial function \(p\) is given by \(p(x) = (x+3)(x^{2}-2x-15)\). Which of the following describes the zeros of \(p\)?

7. In the \(xy\)-plane, the graph of a rational function \(f\) has a vertical asymptote at \(x=-5\). Which of the following could be an expression for \(f(x)\)?

8. Let \( f \) be a sinusoidal function. The graph of \( y = f(x) \) is given in the \( xy \)-plane. What is the period of \( f \)?

9. The graph of \(y=f(x)\), consisting of four line segments and a semicircle, is shown for \(-3 \le x \le 3\). Which of the following is the transformed graph for \(y=f(x+1)-2\)?

10.

The exponential function \(f\) is defined by \(f(x)=ab^{x}\), where \(a\) and \(b\) are positive constants. The table gives values of \(f(x)\) at selected values of \(x\). Which of the following statements is true?

11. The function \(f\) is given by \(f(x)=x^{2}+1\), and the function \(g\) is given by \(g(x)=\dfrac{x-3}{x}\). Which of the following is an expression for \(f(g(x))\)?

12.

The table shows values for a function \(g\) at selected values of \(x\). Which of the following claim and explanation statements best fits these data?

13. The function \(g\) is given by \(g(x)=7\sin(2x)\). Which of the following is an equivalent form for \(g(x)\)?

14. The function \(g\) has the property that for each time the input values double, the output values increase by 1. Which of the following could be the graph of \(y=g(x)\) in the \(xy\)-plane?

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

16. The figure shows a circle centered at the origin with an angle of measure \(\theta\) radians in standard position. The terminal ray of the angle intersects the circle at point \(P\), and point \(R\) also lies on the circle. The coordinates of \(P\) are \((x,y)\), and the coordinates of \(R\) are \((x,-y)\). Which of the following is true about the sine of \(\theta\)?

17. The function \(f\) is given by \(f(x)=3x^{2}+2x+1\). The graph of which of the following functions is the image of the graph of \(f\) after a vertical dilation of the graph of \(f\) by a factor of \(2\)?

18.

The table gives values for the functions \(f\) and \(g\) at selected values of \(x\). Functions \(f\) and \(g\) are defined for all real numbers. Let \(h\) be the function defined by \(h(x)=f(g(x))\). What is the value of \(h(0)\)?

19. The functions \(f\) and \(g\) are defined for all real numbers such that \(g(x)=f(2(x-4))\). Which of the following sequences of transformations maps the graph of \(f\) to the graph of \(g\) in the same \(xy\)-plane?

20. The figure shows the graph of a trigonometric function \(f\). Which of the following could be an expression for \(f(x)\)?

21. Let \(f\) be a rational function that is graphed in the \(xy\)-plane. Consider \(x=1\) and \(x=7\). The polynomial in the numerator of \(f\) has a zero at \(x=1\) and does not have a zero at \(x=7\). The polynomial in the denominator of \(f\) has zeros at both \(x=1\) and \(x=7\). The multiplicities of the zeros at \(x=1\) in the numerator and in the denominator are equal. Which of the following statements is true?

22.

The increasing function \(P\) gives the number of followers, in thousands, for a new musical group on a social media site. The table gives values of \(P(t)\) for selected values of \(t\), in months, since the musical group created their account on this social media site. If a model is constructed to represent these data, which of the following best applies to this situation?

23. The polynomial function \(k\) is given by \(k(x)=ax^{4}-bx^{3}+15\), where \(a\) and \(b\) are nonzero real constants. Each of the zeros of \(k\) has multiplicity \(1\). In the \(xy\)-plane, an \(x\)-intercept of the graph of \(k\) is \((17.997,0)\). A zero of \(k\) is \(-0.478-0.801i\). Which of the following statements must be true?

24. Consider the functions \(f\) and \(g\) given by \(f(x)=\log_{10}(x-1)+\log_{10}(x+3)\) and \(g(x)=\log_{10}(x+9)\). In the \(xy\)-plane, what are all \(x\)-coordinates of the points of intersection of the graphs of \(f\) and \(g\)?

25. What are all values of \(\theta\), for \(0 \le \theta < 2\pi\), where \(2\sin^{2}\theta = -\sin\theta\)?

26. The figure shows the graph of the polar function \(r=f(\theta)\), where \(f(\theta)=4\cos(2\theta)\), in the polar coordinate system for \(0 \le \theta \le 2\pi\). There are five points labeled \(A, B, C, D,\) and \(E\). If \(f\) is restricted to \(0 \le \theta \le \frac{\pi}{2}\), the portion of the given graph that remains consists of two pieces. One of those pieces is the portion of the graph in Quadrant I from \(C\) to \(E\). Which of the following describes the other remaining piece?

27. A physical therapy center has a bicycle that patients use for exercise. The height, in inches (in.), of the bicycle pedal above level ground periodically increases and decreases when used. The figure gives the position of the pedal \(P\) at a height of \(12\) inches above the ground at time \(t=0\) seconds. The pedal’s \(8\)-inch arm defines the circular motion of the pedal. If a patient pedals \(1\) revolution per second, which of the following could be an expression for \(h(t)\), the height, in inches, of the bicycle pedal above level ground at time \(t\) seconds?

28. Consider the graph of the polar function \(r=f(\theta)\), where \(f(\theta)=1+2\sin\theta\), in the polar coordinate system for \(0 \le \theta \le 2\pi\). Which of the following statements is true about the distance between the point with polar coordinates \((f(\theta),\theta)\) and the origin?

1. The figure shows a swimming pool filled with water. A pump is used to remove water from the pool until the pool is empty. When the pump is running, the rate at which the volume of water in the pool decreases is constant. During the first two hours, the pump works slower than usual due to a broken piece. Then the pump stops working. The broken piece is replaced, and the pump works at its usual rate until the pool is completely emptied of water. The entire process of emptying the pool takes six hours. Which of the following graphs could depict this situation, where time, in hours, is the independent variable, and the volume of water in the pool, in gallons, is the dependent variable?

2. In a certain simulation, the population of a bacteria colony can be modeled using a geometric sequence, where the first day of the simulation is day \(1\). The population on day \(4\) was \(4{,}000\) bacteria, and the population on day \(8\) was \(49{,}000\) bacteria. What was the population of the colony on day \(6\) based on the simulation?

3. The rate of people entering a subway car on a particular day is modeled by the function \(R\), where \[ R(t) = 0.03t^{3} - 0.846t^{2} + 6.587t + 1.428 \] for \(0 \le t \le 20\). \(R(t)\) is measured in people per hour, and \(t\) is measured in hours since the subway began service for the day. Based on the model, at what value of \(t\) does the rate of people entering the subway car change from increasing to decreasing?

4.

The table gives values for a function \(m\) at selected values of \(t\). Which of the following graphs could represent these data in a semi-log plot, where the vertical axis is logarithmically scaled?

5. At a coastal city, the height of the tide, in feet (ft), is modeled by the function \(h\), defined by \[ h(t) = 6.3\cos\!\left(\frac{\pi t}{6}\right) + 7.5 \] for \(0 \le t \le 12\) hours Based on the model, which of the following is true?

6. The function \(f\) is defined by \(f(x)=a\sin(b(x+c))+d\), for constants \(a, b, c,\) and \(d\). In the \(xy\)-plane, the points \((2,2)\) and \((4,4)\) represent a minimum value and a maximum value, respectively, on the graph of \(f\). What are the values of \(a\) and \(d\)?

7. The function \(S\) is given by \[ S(t)=\frac{500{,}000}{1+0.4e^{kt}}, \] where \(k\) is a constant. If \(S(4)=300{,}000\), what is the value of \(S(12)\)?

8. Two function models \(k\) and \(m\) are constructed to represent the sales of a product at a group of grocery stores. Both \(k(t)\) and \(m(t)\) represent the sales of the product, in thousands of units, after weeks for \(t \ge 2\). If \(k(t)=14-2.885\ln t\) and \(m(t)= -t+14\), what is the first time \(t\) that sales predicted by the logarithmic model will be \(0.1\) thousand units more than sales predicted by the linear model?

9. The function \(f\) is given by \(f(x)=2\sin(4x)+\cos(2x)\). Using the period of \(f\), which of the following is the number of complete cycles of the graph of \(f\) in the \(xy\)-plane on the interval \(0 \le x \le 1000\)?

10. The graph of the piecewise-linear function \(f\) is shown in the figure. Let \(g\) be the inverse function of \(f\). What is the maximum value of \(g\)?

11. The function \(f\) is given by \[ f(x)=\sin(2.25x+0.2). \] The function \(g\) is given by \(g(x)=f(x)+0.5\). What are the zeros of \(g\) on the interval \(0 \le x \le \pi\)?

12. A large wheel of radius \(2\) feet is rotated at a constant rate. The figure provides a representation of the wheel in the \(xy\)-plane with the direction of rotation indicated. At time \(t=0\) minutes, the wheel begins to rotate. Point \(P\) on the wheel is at the “Start” position in the figure. At time \(t=20\) minutes, \(120\) rotations of the wheel have been completed, and \(P\) is in the same position as it was at time \(t=0\). A sinusoidal function is used to model the \(y\)-coordinate of the position of \(P\) as a function of time \(t\) in minutes. Which of the following functions is an appropriate model for this situation?