AP PreCalculus Unit 1: Polynomial and Rational Functions Practice Exam

1. The table below shows values of a function $f$ for selected values of $x$. | $x$ | 2 | 4 | 6 | 8 | | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | 5 | 12 | 17 | 20 | Based on the data in the table, which of the following best describes the graph of $f$ on the interval $2 \le x \le 8$?

2. Water is being poured into a vase at a constant rate of volume per second. The vase consists of a wide cylindrical base that abruptly narrows into a thin cylindrical neck. Which of the following describes the behavior of the height of the water, $h(t)$, as a function of time $t$?

3. The graph of a function $g$ shows the relationship between two quantities, $x$ and $y$. On the interval $0 < x < 10$, the value of $y$ is increasing, but the graph is concave down. Which of the following is true about the average rate of change of $y$ with respect to $x$ on this interval?

4. [Skill: 2.A | Topic: 1.10] Consider the rational function $f$ defined by $f(x) = \frac{x(x-1)^2(x+4)^3}{x^2(x-1)(x+4)^3}$. Based on the multiplicities of the factors in the numerator and denominator, which of the following describes the behavior of the graph of $f$ at $x=0$, $x=1$, and $x=-4$?

5. [Skill: 1.C | Topic: 1.10] The function $r$ is given by $r(x) = \frac{2x^2 + 5x - 3}{x^2 + 2x - 3}$. The graph of $r$ has a hole at $x = -3$. What are the coordinates of the hole?

6. [Skill: 2.B | Topic: 1.10] The table below shows values of a rational function $g(x)$ for selected values of $x$ near $x=5$. | x | 4.9 | 4.99 | 4.999 | 5 | 5.001 | 5.01 | 5.1 | |---|---|---|---|---|---|---|---| | g(x) | 3.8 | 3.98 | 3.998 | Undefined | 4.002 | 4.02 | 4.2 | Based on the data in the table, which of the following best describes the graph of $g$ at $x=5$?

7. The function $f$ is defined by $f(x) = \frac{3x^2 - 8x + 10}{x - 2}$. Which of the following is an equivalent form of $f(x)$ generated by polynomial long division?

8. What is the coefficient of the $x^2$ term in the binomial expansion of $(3x - 2)^4$?

9. The function $r$ is given by $r(x) = \frac{x^3 + 4x^2 - 5x}{x^2 - 1}$. Which of the following statements about the graph of $r$ is best supported by rewriting $r(x)$ in factored form?

10. The function $g$ is defined by a transformation of the function $f$. The graph of $g$ is obtained by vertically compressing the graph of $f$ by a factor of $\frac{1}{3}$ and shifting it horizontally to the right by 5 units. Which of the following equations represents $g(x)$?

11. The domain of the function $f$ is the interval $[-6, 12]$ and the range of $f$ is the interval $[2, 8]$. Let $g$ be the function defined by $g(x) = -2f(3x) + 4$. Which of the following intervals is the range of $g$?

12. The graph of the function $f$ contains the point $(-4, 6)$. Let $g$ be the function defined by $g(x) = f(2(x-1)) - 3$. Which of the following points must be on the graph of $g$?

13. A landscaping company is designing rectangular flower beds for a city park. The company creates a table of values to estimate the cost of mulch required based on the width of the flower bed. The length of the flower bed is always 4 feet longer than the width. The table below shows the cost, $C(w)$, in dollars, for various widths, $w$, in feet. | Width $w$ (ft) | 5 | 10 | 15 | 20 | | :--- | :--- | :--- | :--- | :--- | | Cost $C(w)$ ($) | 45 | 140 | 285 | 480 | Based on the data, which type of function is most appropriate to model the cost $C(w)$, and why?

14. A scientist is conducting an experiment to measure the temperature of a chemical solution over time. The solution is initially heated at a steady rate for 10 minutes. At $t=10$ minutes, the heat source is turned off, and the solution is allowed to cool down at a constant rate for the next 20 minutes. Which of the following function types would be most appropriate to construct a model for the temperature $T(t)$ over the interval $0 \le t \le 30$?

15. A rectangular sheet of aluminum measures 24 inches by 40 inches. Squares with side length $x$ inches are cut from each of the four corners, and the resulting flaps are folded up to create an open-top box. Which of the following functions $V$ models the volume of the box, in cubic inches, and what is the valid domain for $x$ in this context?

16. A manufacturing company incurs a fixed startup cost of $5,000 for a new product line and a variable cost of $12 per unit produced. Let $n$ represent the number of units produced. Which of the following functions $A$ models the *average cost per unit*, in dollars, and what is the horizontal asymptote of the graph of $A$?

17. The concentration of a drug in a patient's bloodstream $t$ hours after injection is modeled by the function $C(t) = \frac{15t}{2t^2 + 18}$, where $C(t)$ is measured in milligrams per liter (mg/L). Based on the model, which of the following statements correctly describes the concentration of the drug?

18. The table below gives values for a polynomial function $f$ at selected values of $x$. | $x$ | 0 | 2 | 4 | 6 | | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | 5 | 13 | 29 | 53 | Let $R_A$ be the average rate of change of $f$ over the closed interval $[0, 2]$ and let $R_B$ be the average rate of change of $f$ over the closed interval $[4, 6]$. Which of the following statements correctly compares $R_A$ and $R_B$?

19. The volume of water in a tank, in gallons, is modeled by the function $V(t)$, where $t$ is measured in minutes. Which of the following is the best interpretation of the statement $\frac{V(10)-V(5)}{10-5} = -8$?

20. Let $g$ be the function defined by $g(x) = x^2 - 2x$. For what value of $k$, where $k > 3$, is the average rate of change of $g$ over the interval $[3, k]$ equal to 7?

21. [Skill: 2.B | Topic: 1.3] The table below shows values of a function $f$ for selected values of $x$. Based on the data in the table, which of the following best describes the function $f$? | $x$ | $f(x)$ | | :---: | :---: | | 0 | 4 | | 2 | 10 | | 4 | 24 | | 6 | 46 |

22. [Skill: 3.A | Topic: 1.3] Let $g$ be a function defined for all real numbers. The average rate of change of $g$ over the interval $[x, x+3]$ is given by the function $r(x) = -2x + 5$. Which of the following statements is true regarding the graph of $g$?

23. [Skill: 1.B | Topic: 1.3] The function $p$ is given by $p(x) = x^2 + 4x$. Which of the following linear functions models $A(x)$, the average rate of change of $p$ over the interval $[x, x+2]$?

24. Consider the polynomial function $p(x) = -4x^6 + 7x^3 - 2x + 11$. Which of the following statements is true regarding the global behavior of the function?

25. Let $f(x) = x^3 - 2x^2 + 5$. What is the average rate of change of $f$ over the interval $[-1, 2]$?

26. The function $f$ is a polynomial of degree 3 with real coefficients. The function has zeros at $x = -2$ and $x = 3 + i$. Which of the following expressions defines $f(x)$ ?

27. Let $g$ be the polynomial function defined by $g(x) = 3x^6 - 5x^4 + 2x^2 - 9$. Which of the following statements correctly identifies whether $g$ is an even or odd function and provides the correct justification?

28. A polynomial function $p$ with real coefficients has degree 5. The graph of $y=p(x)$ intersects the x-axis at exactly two distinct points, $x=1$ and $x=4$. It is known that $x=1$ is a zero of multiplicity 2 and $x=4$ is a zero of multiplicity 1. No other real zeros exist. What is the total number of non-real complex zeros of $p$?

29. The graph of a polynomial function $p(x)$ is shown in the figure. The graph passes through the origin, extends down into the third quadrant as $x$ moves left, and extends up into the first quadrant as $x$ moves right. Which of the following is true about the end behavior of $p(x)$?

30. Let $g(x) = a x^n + 4x^2 - 1$, where $a$ is a nonzero constant and $n$ is an integer greater than 2. If $\lim_{x \to -\infty} g(x) = \infty$ and $\lim_{x \to \infty} g(x) = \infty$, which of the following must be true about $a$ and $n$?

31. [Skill: 1.A | Topic: 1.7] The rational function $r$ is defined by $r(x) = \frac{8 - 2x^3}{4x^3 + 5x}$. Which of the following equations describes the horizontal asymptote of the graph of $r$?

32. [Skill: 2.A | Topic: 1.7] The function $f$ is given by $f(x) = \frac{3x(x - 2)}{x^3 + 1}$. Which of the following statements correctly describes the limit of $f(x)$ as $x$ approaches infinity?

33. [Skill: 3.B | Topic: 1.7] Let $g$ be a rational function defined by $g(x) = \frac{2x^4 + 5x}{x^3 - 2}$. Which of the following describes the end behavior of $g$ as $x \to -\infty$ using the quotient of the leading terms?

34. The rational function $f$ is given by $f(x) = \frac{(x-2)(x+4)(x-5)}{(x-2)(x+1)}$. Which of the following describes the zeros of $f$?

35. The table below gives values for the polynomial functions $g$ and $h$ at selected values of $x$. Let $r$ be the rational function defined by $r(x) = \frac{g(x)}{h(x)}$. | $x$ | $g(x)$ | $h(x)$ | |:---:|:---:|:---:| | 1 | 0 | 5 | | 2 | 0 | 0 | | 3 | 8 | 0 | | 4 | 0 | -3 | Based on the table, for which value(s) of $x$ does $r(x) = 0$?

36. Consider the rational function $k(x) = \frac{2x^3 + 6x^2}{x^2 + 3x}$. Which of the following statements correctly identifies the number of real zeros of $k$?

37. Let $f$ be the rational function defined by $f(x) = \frac{x^2 + 3x - 10}{x^2 - 4}$. Which of the following equations represents a vertical asymptote of the graph of $y = f(x)$?

38. The function $h$ is defined by $h(x) = \frac{2}{(x-3)^2}$. Which of the following statements correctly describes the behavior of the graph of $h$ near the vertical asymptote $x = 3$?

39. The function $g$ is given by $g(x) = \frac{p(x)}{x^2 - x - 12}$, where $p(x)$ is a polynomial. The graph of $g$ has a vertical asymptote at $x = -3$ and a removable discontinuity (hole) at $x = 4$. Which of the following could be an expression for $p(x)$?

40. [Skill: 1.B | Topic: 1.6] Let $f$ be the polynomial function given by $f(x) = (2 - x)(3x^2 + 1)(x + 5)$. Which of the following statements describes the end behavior of $f$?

41. [Skill: 1B | Topic: 1.4] Let $p$ be a nonconstant polynomial function of even degree. The function $p$ has distinct real zeros at $x = -3$, $x = 1$, and $x = 4$. Which of the following statements must be true about $p$?

42. A manufacturing company plans to construct open-top storage bins from rectangular sheets of metal measuring 24 inches by 36 inches. To construct a bin, a square of side length $x$ inches is cut from each of the four corners of the sheet, and the sides are folded up. Which of the following identifies the appropriate function type to model the volume, $V(x)$, of the bin and correctly states the domain restriction for $x$ in this context?