ACCUPLACER Advanced Algebra & Functions Quiz: Interpreting Function Graphs
19 questions · exam conditions
0:00
Interpreting Function GraphsQuestion 1 of 19

The graph of y=f(x)y = f(x) has x-intercepts at (−3,0)(-3, 0) and (5,0)(5, 0). Which of the following transformations results in a function whose graph has x-intercepts at (−1,0)(-1, 0) and (7,0)(7, 0)?

g(x)=f(x+2)g(x) = f(x + 2)
g(x)=f(x−2)g(x) = f(x - 2)
g(x)=f(x)+2g(x) = f(x) + 2
g(x)=2f(x)g(x) = 2f(x)
← Back to quizzes

ACCUPLACER Advanced Algebra & Functions Quiz

ACCUPLACER Advanced Algebra & Functions Quiz: Interpreting Function Graphs

Practice Interpreting Function Graphs in ACCUPLACER Advanced Algebra & Functions with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Interpreting Function Graphs, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Advanced Algebra & Functions.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The graph of y=f(x)y = f(x) has x-intercepts at (−3,0)(-3, 0) and (5,0)(5, 0). Which of the following transformations results in a function whose graph has x-intercepts at (−1,0)(-1, 0) and (7,0)(7, 0)?

  1. g(x)=f(x+2)g(x) = f(x + 2)
  2. g(x)=f(x−2)g(x) = f(x - 2) (correct answer)
  3. g(x)=f(x)+2g(x) = f(x) + 2
  4. g(x)=2f(x)g(x) = 2f(x)
Explanation: The original x-intercepts are -3 and 5. The new x-intercepts are -1 and 7. The change from -3 to -1 is an addition of 2. The change from 5 to 7 is also an addition of 2. This means every point on the graph has been shifted 2 units to the right. A horizontal shift of 2 units to the right is represented by the transformation f(x−2)f(x - 2).

Question 2

The rate of change of a function f(x)f(x) is positive on the interval (−∞,1)(-\infty, 1) and negative on the interval (1,∞)(1, \infty). Which statement accurately describes the graph of f(x)f(x)?

  1. The graph of f(x)f(x) has a local minimum at x=1x = 1.
  2. The graph of f(x)f(x) has a local maximum at x=1x = 1. (correct answer)
  3. The graph of f(x)f(x) is positive on (−∞,1)(-\infty, 1).
  4. The graph of f(x)f(x) has an x-intercept at x=1x = 1.
Explanation: A positive rate of change means the function is increasing. A negative rate of change means the function is decreasing. The function f(x)f(x) is increasing up to x=1x=1 and then decreasing after x=1x=1. This behavior defines a local maximum at x=1x=1. We do not have enough information to determine if the function is positive or negative, or where its intercepts are.

Question 3

The graph of a function f(x)f(x) has a domain of [−4,6][-4, 6] and a range of [−2,10][-2, 10]. Which of the following points CANNOT be on the graph of f(x)f(x)?

  1. (6,11)(6, 11) (correct answer)
  2. (−4,−2)(-4, -2)
  3. (5,9)(5, 9)
  4. (0,0)(0, 0)
Explanation: The domain specifies the possible x-values, and the range specifies the possible y-values. For a point (x,y)(x, y) to be on the graph, its x-coordinate must be in the interval [−4,6][-4, 6] and its y-coordinate must be in the interval [−2,10][-2, 10]. The point (6,11)(6, 11) has an x-coordinate of 6, which is in the domain, but its y-coordinate of 11 is outside the range [−2,10][-2, 10]. Therefore, this point cannot be on the graph.

Question 4

The graph of a function f(x)f(x) is periodic with a period of 6. Given that f(2)=5f(2) = 5, which of the following must be true?

  1. f(8)=5f(8) = 5 (correct answer)
  2. f(5)=2f(5) = 2
  3. f(6)=5f(6) = 5
  4. f(12)=10f(12) = 10
Explanation: A function with period pp satisfies f(x)=f(x+np)f(x) = f(x + np) for any integer nn. Here, the period is p=6p=6. We are given f(2)=5f(2) = 5. We can find the value of the function at x=8x=8 by noting that 8=2+68 = 2 + 6. Thus, f(8)=f(2+6)=f(2)=5f(8) = f(2 + 6) = f(2) = 5.

Question 5

The graph of a function h(x)h(x) is symmetric with respect to the origin. If the point (a,−b)(a, -b) is on the graph of h(x)h(x), which of the following points must also be on the graph?

  1. (a,b)(a, b)
  2. (−a,−b)(-a, -b)
  3. (−a,b)(-a, b) (correct answer)
  4. (b,−a)(b, -a)
Explanation: Symmetry with respect to the origin means the function is an odd function, which satisfies the property h(−x)=−h(x)h(-x) = -h(x). We are given that the point (a,−b)(a, -b) is on the graph, meaning h(a)=−bh(a) = -b. To find the y-coordinate corresponding to x=−ax = -a, we use the property: h(−a)=−h(a)h(-a) = -h(a). Substituting h(a)=−bh(a) = -b, we get h(−a)=−(−b)=bh(-a) = -(-b) = b. Therefore, the point (−a,b)(-a, b) must be on the graph.

Question 6

The graph of a function f(x)f(x) has a y-intercept at (0,5)(0, 5). The average rate of change of f(x)f(x) over the interval [0,4][0, 4] is −12-\frac{1}{2}. If g(x)=f(x)+3g(x) = f(x) + 3, what is the value of g(4)g(4)?

  1. 1
  2. 3
  3. 6 (correct answer)
  4. 8
Explanation: First, find f(4)f(4) using the average rate of change formula: f(4)−f(0)4−0=−12\frac{f(4) - f(0)}{4 - 0} = -\frac{1}{2}. We are given f(0)=5f(0) = 5, so f(4)−54=−12\frac{f(4) - 5}{4} = -\frac{1}{2}. Multiplying both sides by 4 gives f(4)−5=−2f(4) - 5 = -2, so f(4)=3f(4) = 3. Next, find g(4)g(4) using the definition g(x)=f(x)+3g(x) = f(x) + 3. So, g(4)=f(4)+3=3+3=6g(4) = f(4) + 3 = 3 + 3 = 6.

Question 7

The graph of a continuous function h(x)h(x) has exactly two x-intercepts, one at x=ax = a and one at x=bx = b, where a<ba < b. The function has a single local maximum at x=cx = c and no local minimum. Which of the following statements must be true?

  1. h(c)<0h(c) < 0
  2. a<c<ba < c < b (correct answer)
  3. The function is increasing on the interval (b,∞)(b, \infty).
  4. The range of the function is (−∞,0](-\infty, 0].
Explanation: A continuous function with two x-intercepts and a single local maximum must be shaped like a downward-opening parabola. The local maximum (the vertex) must occur between the two x-intercepts. Therefore, a<c<ba < c < b. The maximum value, h(c)h(c), must be positive for the graph to cross the x-axis. The function is decreasing after the maximum, so it must be decreasing on (b,∞)(b, \infty). The range is (−∞,h(c)](-\infty, h(c)] where h(c)>0h(c) > 0.

Question 8

The graph of the function g(x)g(x) has a vertical asymptote at x=−2x = -2. As xx approaches -2 from the right, g(x)g(x) approaches +∞+\infty. As xx approaches -2 from the left, g(x)g(x) approaches −∞-\infty. Which of the following statements is true for values of xx in the interval (−3,−1)(-3, -1)?

  1. The function g(x)g(x) is always positive.
  2. The function g(x)g(x) is always increasing.
  3. The function g(x)g(x) has both positive and negative values. (correct answer)
  4. The function g(x)g(x) has a maximum value but no minimum value.
Explanation: The interval (−3,−1)(-3, -1) contains the vertical asymptote at x=−2x=-2. On the sub-interval (−3,−2)(-3, -2), g(x)g(x) approaches −∞-\infty, so it takes on negative values. On the sub-interval (−2,−1)(-2, -1), g(x)g(x) approaches +∞+\infty, so it takes on positive values. Therefore, over the entire interval (−3,−1)(-3, -1), the function has both positive and negative values.

Question 9

The graph of f(x)f(x) is increasing for all real numbers and passes through the origin. The graph of g(x)g(x) is decreasing for all real numbers and has a y-intercept at (0,3)(0, 3). What can be determined about the value of (g∘f)(2)(g \circ f)(2)?

  1. (g∘f)(2)>3(g \circ f)(2) > 3
  2. (g∘f)(2)<3(g \circ f)(2) < 3 (correct answer)
  3. (g∘f)(2)=3(g \circ f)(2) = 3
  4. (g∘f)(2)=0(g \circ f)(2) = 0
Explanation: First, evaluate the inner function, f(2)f(2). Since f(x)f(x) is increasing and f(0)=0f(0)=0, for any x>0x>0, f(x)>f(0)f(x)>f(0). Thus, f(2)>0f(2) > 0. Let k=f(2)k = f(2), where kk is a positive number. Now, evaluate the outer function, g(k)g(k). Since g(x)g(x) is decreasing and g(0)=3g(0)=3, for any positive input kk, g(k)<g(0)g(k) < g(0). Therefore, g(k)<3g(k) < 3, which means (g∘f)(2)<3(g \circ f)(2) < 3.

Question 10

The graph of a continuous function f(x)f(x) on the domain [−10,10][-10, 10] has exactly one local maximum at (−4,15)(-4, 15) and one local minimum at (5,−2)(5, -2). The values at the endpoints are f(−10)=−8f(-10) = -8 and f(10)=7f(10) = 7. What is the absolute maximum value of the function on its domain?

  1. 15 (correct answer)
  2. 10
  3. 7
  4. The maximum cannot be determined.
Explanation: For a continuous function on a closed interval, the absolute maximum must occur at either a local maximum or at an endpoint. The candidates for the absolute maximum value are the local maximum value, f(−4)=15f(-4) = 15, and the values at the endpoints, f(−10)=−8f(-10) = -8 and f(10)=7f(10) = 7. Comparing these values {−8,7,15}\{-8, 7, 15\}, the largest is 15.

Question 11

The graph of a one-to-one function f(x)f(x) passes through the point (3,7)(3, 7) and is strictly increasing for all real numbers. Let f−1(x)f^{-1}(x) be the inverse function of f(x)f(x). Which statement about the graph of f−1(x)f^{-1}(x) must be true?

  1. It passes through (7,3)(7, 3) and is strictly decreasing.
  2. It passes through (3,7)(3, 7) and is strictly increasing.
  3. It passes through (−3,−7)(-3, -7) and is strictly decreasing.
  4. It passes through (7,3)(7, 3) and is strictly increasing. (correct answer)
Explanation: The graph of an inverse function f−1(x)f^{-1}(x) is a reflection of the graph of f(x)f(x) across the line y=xy=x. This means two things: First, if a point (a,b)(a, b) is on the graph of f(x)f(x), then the point (b,a)(b, a) is on the graph of f−1(x)f^{-1}(x). Since (3,7)(3, 7) is on the graph of f(x)f(x), (7,3)(7, 3) must be on the graph of f−1(x)f^{-1}(x). Second, if f(x)f(x) is strictly increasing, its inverse f−1(x)f^{-1}(x) must also be strictly increasing.

Question 12

The graph of f(x)f(x) is a smooth curve. The slope of the graph is increasing on the interval (−∞,2)(-\infty, 2) and the slope of the graph is decreasing on the interval (2,∞)(2, \infty). Which of the following is a correct description of the graph at x=2x=2?

  1. The graph has a local maximum.
  2. The graph has a local minimum.
  3. The graph has a point of inflection. (correct answer)
  4. The graph has a vertical asymptote.
Explanation: The 'slope of the graph' refers to the function's rate of change. When the slope is increasing, the graph is concave up. When the slope is decreasing, the graph is concave down. The point where the concavity changes is called a point of inflection. Since the slope's behavior changes at x=2x=2, the graph has a point of inflection there.

Question 13

The graph of a function f(x)f(x) consists of two line segments. The first segment connects (−4,2)(-4, 2) to (0,0)(0, 0). The second segment connects (0,0)(0, 0) to (2,6)(2, 6). What is the value of f(−2)+f(1)f(-2) + f(1)?

  1. 2
  2. 4 (correct answer)
  3. 5
  4. 7
Explanation: First, find the equation for the line segment from (−4,2)(-4, 2) to (0,0)(0, 0). The slope is 0−20−(−4)=−24=−12\frac{0-2}{0-(-4)} = -\frac{2}{4} = -\frac{1}{2}. The equation is y=−12xy = -\frac{1}{2}x. So, f(−2)=−12(−2)=1f(-2) = -\frac{1}{2}(-2) = 1. Second, find the equation for the line segment from (0,0)(0, 0) to (2,6)(2, 6). The slope is 6−02−0=3\frac{6-0}{2-0} = 3. The equation is y=3xy = 3x. So, f(1)=3(1)=3f(1) = 3(1) = 3. The sum is f(−2)+f(1)=1+3=4f(-2) + f(1) = 1 + 3 = 4.

Question 14

The graph of a function h(x)h(x) has a horizontal asymptote at y=4y = 4. The graph does not intersect its horizontal asymptote. The function is increasing for all xx in its domain. Which of the following could be the range of h(x)h(x)?

  1. (−∞,∞)(-\infty, \infty)
  2. (−∞,4](-\infty, 4]
  3. [4,∞)[4, \infty)
  4. (4,∞)(4, \infty) (correct answer)
Explanation: An always-increasing function must either approach its horizontal asymptote from below as x→∞x \to \infty or approach it from above as x→−∞x \to -\infty. Since the graph never intersects y=4y=4, the value 4 is not in the range. Case 1: The graph approaches y=4y=4 from below. Since it's always increasing, the range would be (−∞,4)(-\infty, 4). Case 2: The graph approaches y=4y=4 from above. Since it's always increasing, it must go to +∞+\infty as x→∞x \to \infty, so the range would be (4,∞)(4, \infty). Of the choices provided, only (4,∞)(4, \infty) is a possibility.

Question 15

The graph of a polynomial function f(x)f(x) has two local extrema: a local maximum at x=−1x = -1 and a local minimum at x=3x = 3. For which of the following conditions will the equation f(x)=kf(x) = k have exactly three distinct real solutions?

  1. k>f(−1)k > f(-1)
  2. k<f(3)k < f(3)
  3. k=f(−1)k = f(-1) or k=f(3)k = f(3)
  4. f(3)<k<f(−1)f(3) < k < f(-1) (correct answer)
Explanation: The solutions to f(x)=kf(x) = k correspond to the intersection points of the graph of f(x)f(x) and the horizontal line y=ky=k. For a polynomial with a local maximum and a local minimum, a horizontal line will intersect the graph at three distinct points only if its y-value, kk, is strictly between the y-value of the local minimum and the y-value of the local maximum. Therefore, kk must be greater than f(3)f(3) and less than f(−1)f(-1).

Question 16

The graph of a polynomial function g(x)g(x) has a relative maximum at (−4,3)(-4, 3) and a relative minimum at (2,−5)(2, -5). The function has no other local extrema. Which of the following statements must be true about the end behavior of the graph of g(x)g(x)?

  1. As x→∞x \to \infty, g(x)→∞g(x) \to \infty, and as x→−∞x \to -\infty, g(x)→∞g(x) \to \infty.
  2. As x→∞x \to \infty, g(x)→−∞g(x) \to -\infty, and as x→−∞x \to -\infty, g(x)→−∞g(x) \to -\infty.
  3. As x→∞x \to \infty, g(x)→∞g(x) \to \infty, and as x→−∞x \to -\infty, g(x)→−∞g(x) \to -\infty. (correct answer)
  4. As x→∞x \to \infty, g(x)→−∞g(x) \to -\infty, and as x→−∞x \to -\infty, g(x)→∞g(x) \to \infty.
Explanation: A polynomial with exactly one relative maximum followed by one relative minimum (as x increases) must be an odd-degree polynomial with a positive leading coefficient. The graph falls from the left, rises to a maximum, falls to a minimum, then rises to the right. This corresponds to the end behavior: as x→−∞x \to -\infty, g(x)→−∞g(x) \to -\infty, and as x→∞x \to \infty, g(x)→∞g(x) \to \infty.

Question 17

The graph of a function y=f(x)y=f(x) is decreasing on the interval [−2,1][-2, 1] and increasing on the interval [1,4][1, 4]. Which of the following statements must be true?

  1. f(−1)<f(0)f(-1) < f(0)
  2. f(2)<f(3)f(2) < f(3) (correct answer)
  3. f(0)<f(2)f(0) < f(2)
  4. f(1)f(1) is the absolute maximum value on [−2,4][-2, 4].
Explanation: By definition, a function is increasing on an interval if for any two points aa and bb in the interval with a<ba < b, we have f(a)<f(b)f(a) < f(b). Since the function is increasing on [1,4][1, 4], and 22 and 33 are in this interval with 2<32 < 3, it must be true that f(2)<f(3)f(2) < f(3). Choice A is incorrect because on a decreasing interval, −1<0-1 < 0 implies f(−1)>f(0)f(-1) > f(0).

Question 18

The graph of a function f(x)f(x) is continuous and has a range of [−5,3][-5, 3]. What is the range of the function g(x)=∣f(x)∣g(x) = |f(x)|?

  1. [−5,3][-5, 3]
  2. [0,3][0, 3]
  3. [3,5][3, 5]
  4. [0,5][0, 5] (correct answer)
Explanation: The function f(x)f(x) produces output values from -5 to 3. The function g(x)g(x) takes the absolute value of these outputs. The absolute value of any number is non-negative, so the minimum value of the range of g(x)g(x) is 0. The maximum value in the original range is 3, and ∣3∣=3|3| = 3. The minimum value in the original range is -5, and ∣−5∣=5|-5| = 5. The largest possible output from g(x)g(x) is 5. Therefore, the range of g(x)g(x) is [0,5][0, 5].

Question 19

The graph of a function f(x)f(x) is symmetric with respect to the y-axis and contains the point (−3,5)(-3, 5). The graph is decreasing on the interval (0,∞)(0, \infty). Which statement must be true?

  1. The graph is increasing on the interval (−∞,0)(-\infty, 0). (correct answer)
  2. The graph contains the point (−3,−5)(-3, -5).
  3. The function has a local minimum at x=0x = 0.
  4. The y-intercept is negative.
Explanation: Symmetry with respect to the y-axis means the function is even. The behavior on the left side of the y-axis is a mirror image of the behavior on the right. Since the function is decreasing for x>0x > 0 (moving away from the y-axis), it must be increasing for x<0x < 0 (moving towards the y-axis). This also implies a local maximum, not a minimum, at x=0x=0.