AP Precalculus Test Review

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Question 1

x	5	6	7	8	9	10
f(x)	−17	−27	−39	−53	−69	−87

The table gives values for the function f at selected values of x. Which of the following conclusions with reason is consistent with the values in the table?

AThe graph of f is concave up because the average rate of change over consecutive equal-length intervals is increasing.BThe graph of f is concave up because the average rate of change over consecutive equal-length intervals is decreasing.CThe graph of f is concave down because the average rate of change over consecutive equal-length intervals is increasing.DThe graph of f is concave down because the average rate of change over consecutive equal-length intervals is decreasing.

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Why

The first differences are −10, −12, −14, −16, −18. Because the input intervals all have length 1, these are also the average rates of change. They are decreasing, so the slopes become more negative as x increases. Decreasing slopes mean the graph is concave down. The correct answer is the choice stating the graph is concave down because the average rate of change is decreasing.

Question 2

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The graph of the function f is shown. What is the average rate of change of f on the interval −3 ≤ x ≤ 3, if it exists?

A−1B0C1DUndefined

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Why

Average rate of change is the slope of the secant line: [f(3)−f(−3)]/[3−(−3)]. The graph gives f(−3)=3 and f(3)=3, so the numerator is 0. The average rate of change is 0.

Question 3

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A student starts to drive to school from home in a straight line and quickly realizes they forgot their backpack. The student stops the car and then drives back to their starting position at home. Which graph could model the velocity v, in miles per hour, of the car as a function of time t, in hours? (Assume driving away from home involves positive velocity.)

AGraph ABGraph BCGraph CDGraph D

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Why

The student initially drives away from home, so velocity is positive. The student stops, so velocity reaches 0. Then the student drives back toward home, so velocity is negative, and finally stops again at home. Graph A shows positive → 0 → negative → 0.

Question 4

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The graph of a polynomial function y=p(x) is shown. Which of the following expressions could define p(x)?

A(x − 3)(x + 2)B(x + 3)(x − 2)C(x − 3)²(x + 2)²D(x + 3)²(x − 2)²

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Why

The graph touches and turns at x=−3 and x=2 rather than crossing the x-axis. Both zeros therefore have even multiplicity. The matching factors are (x+3)² and (x−2)².

Question 5

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The graph of a function models the percentage of charge C in a phone battery over a period of time t, in hours. Based on the graph, which collection of statements about the function and scenario is accurate?

AThe function is sometimes decreasing and sometimes increasing. As time increases, the charge of the battery is sometimes decreasing and sometimes increasing.BThe function is sometimes decreasing and sometimes increasing. As time increases, the charge of the battery is always decreasing. The charge is decreasing at different rates at different times.CThe function is always decreasing. As time increases, the charge of the battery is always decreasing. The charge is decreasing at the same rate at all times.DThe function is always decreasing. As time increases, the charge of the battery is always decreasing. The charge is decreasing at different rates at different times.

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Why

The graph falls the entire time, so the function is always decreasing. But its steepness changes, so the battery is not losing charge at one constant rate. It is decreasing at different rates at different times.

Question 6

The polynomial function P is given by P(x)=(x−3)(x−2)(x+1), and the domain of P is −5 ≤ x ≤ 5. Which of the following is true about the graph of P?

AThe graph of P intersects the x-axis at three points, has four local extrema, and has no global extrema.BThe graph of P intersects the x-axis at three points, has two local extrema that are not also global extrema, and has two global extrema.CThe graph of P intersects the x-axis at three points, has no local extrema that are also global extrema, and has two global extrema.DThe graph of P intersects the x-axis at three points, has two local extrema that are not also global extrema, and has no global extrema.

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Why

The zeros are x=−1, 2, and 3. A cubic with three distinct real zeros has two local extrema between consecutive zeros. Because the domain is the closed interval [−5,5], absolute extrema exist; here they occur at the endpoints, while the two interior turning points are local but not global.

Question 7

A linear function P is used to model the price, in dollars, of used cars as a function of their age t, in years. It is known that P(4)=7300 and P(7)=5500. Based on this model, which of the following is true?

AFor each year that a car ages, its price decreases by approximately 25%.BFor each year that a car ages, its price decreases by approximately 75%.CFor each year that a car ages, its price decreases by approximately $600.DFor each year that a car ages, its price decreases by approximately $1800.

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Why

For a linear model, compute the constant slope: (5500−7300)/(7−4)=−1800/3=−600 dollars per year. So the price decreases by about $600 each year.

Question 8

Which of the following functions has the end behaviors lim_x→−∞ p(x)=−∞ and lim_x→∞ p(x)=∞?

Ap(x)=(x−10)(x+2)(x+3)Bp(x)=(x−10)(x+2)(x+3)(x+4)Cp(x)=(10−x)(x+2)(x+3)Dp(x)=(10−x)(x+2)(x+3)(x+4)

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Why

Left down/right up is the end behavior of an odd-degree polynomial with positive leading coefficient. The matching option is the degree-3 product (x−10)(x+2)(x+3), whose leading term is x³.

Question 9

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The graph of the function f is shown. On which of the following intervals is the graph of f increasing?

A(−∞,−2)B(−∞,−4) ∪ (0,∞)C(−2,∞)D(−4,0)

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Why

Reading left to right, the graph rises until the relative maximum near x=−4, falls until the relative minimum near x=0, and then rises again. Thus f is increasing on (−∞,−4) and (0,∞).

Question 10

The function f is given by f(x)=x³+x. Which statement is true and supports the claim that f is an odd function and not an even function?

Af(0)=−f(0)B−f(3)=f(3)Cf(−3)=f(3)Df(−3)=−f(3)

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Why

f(−x)=−x³−x=−(x³+x)=−f(x), which is exactly the definition of an odd function. Therefore f(−3)=−f(3).

Question 11

A new special attraction opened at a museum. The museum management tracked the number of people who visited the attraction each day and created a function model M for the number of people for each day d after the attraction opened. Each day they also calculated the rate of change of the number of people visiting the attraction. They created a function model R for the rate of change, in people per day, for each day d after the attraction opened. The function R is given by R(d)=1/200(−d⁴+35d³−411d²+1845d−2686.5). At which of the following values of d does the graph of y=M(d) have a point of inflection?

A8.627 onlyB3.894 and 13.728 onlyC5.911 and 11.590D3.894, 8.627, and 13.728

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Why

R is the rate of change of M, so R=M′. Points of inflection of M occur where M′ changes from increasing to decreasing or vice versa—that is, at local extrema of R. The listed extrema occur at about d=3.894, 8.627, and 13.728, so all three values are points of inflection of M.

Question 12

The polynomial function is given by p(x)=(x−3)⁷(x+2)⁵(x²+ax+9), where a is an integer. For p, which statement is true about the multiplicity of the zero x=3?

AThe multiplicity must be 7, because the exponent on the factor (x−3) is 7.BThe multiplicity can be 8, because the exponent on (x−3) is 7, and x=3 is a zero of (x²+0x+9) with multiplicity 1.CThe multiplicity can be 9, because the exponent on (x−3) is 7, and x=3 is a zero of (x²+6x+9) with multiplicity 2.DThe multiplicity can be 9, because the exponent on (x−3) is 7, and x=3 is a zero of (x²−6x+9) with multiplicity 2.

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Why

The first factor contributes multiplicity 7. If a=−6, then x²−6x+9=(x−3)², which adds multiplicity 2 for a total multiplicity of 9.

Question 13

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The graph of the function f is shown. Over which of the following intervals is f increasing?

A[−6,−4] and [2,5]B[−6,0] and [2,5]C(−4,2)D(0,2)

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Why

The graph rises from x=−6 to −4, is constant from −4 to 0, decreases from 0 to 2, and rises again from 2 to 5. So f is increasing on [−6,−4] and [2,5].

Question 14

The polynomial function p is given by p(x)=x³−10x²−4. Which of the following describes the behavior of p as the input values decrease without bound?

AThe output values decrease without bound.BThe output values increase without bound.CThe output values sometimes increase without bound and sometimes decrease without bound.DThe output values get closer to −4.

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Why

The leading term x³ controls end behavior. As x→−∞, x³→−∞, so p(x) decreases without bound.

Question 15

The polynomial function f is given by f(x)=axᵇ, where a is an integer and b is a positive integer. It is known that lim_x→−∞f(x)=−∞ and lim_x→∞f(x)=−∞. Which statement must be true?

AThe value of a must be positive, because as x increases without bound or decreases without bound, the end behaviors of f are the same.BThe value of a must be negative, because as x increases without bound or decreases without bound, the end behaviors of f are different.CThe value of b must be even, because as x increases without bound or decreases without bound, the end behaviors of f are the same.DThe value of b must be odd, because as x increases without bound or decreases without bound, the end behaviors of f are different.

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Why

Both ends go in the same direction, so the exponent b must be even. Since both ends go downward, a is negative as well, but the statement that must be true among the choices is the one saying b must be even because the end behaviors are the same.

Question 16

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The average rate of change of the function g on the interval (−1,3) is negative. Which of the following could be the graph of g?

AGraph ABGraph BCGraph CDGraph D

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Why

A negative average rate of change means [g(3)−g(−1)]/[3−(−1)]<0. The denominator is positive, so we need g(3)

Question 17

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The graph of the function f and a point labeled A on the graph are shown. Which statement about the graph of f at point A is true?

AAt point A, the rate of change of f is negative, and the graph of f is concave down.BAt point A, the rate of change of f is negative, and the graph of f is concave up.CAt point A, the rate of change of f is positive, and the graph of f is concave down.DAt point A, the rate of change of f is positive, and the graph of f is concave up.

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Why

At A the curve is rising, so the instantaneous rate of change is positive. The slope is becoming less positive as the graph approaches a local maximum, so the graph is concave down. The correct choice is the one describing a positive rate of change and a concave-down graph.

Question 18

The function h is given by h(x)=−x⁴+x³+202x². Which of the following statements is true?

Ah has a global minimum between x=−4 and x=0.Bh has a global maximum between x=−4 and x=0.Ch has a global minimum between x=0 and x=5.Dh has a global maximum between x=0 and x=5.

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Why

The leading term is −x⁴, so h(x)→−∞ as x→±∞. Therefore h has a global maximum, not a global minimum. The larger peak occurs for a critical point between x=0 and x=5, so h has a global maximum on that interval.