Math 2 Quiz: Comparing Quadratic Functions
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Comparing Quadratic FunctionsQuestion 1 of 10

Two quadratic functions p(x)=ax2+bx+cp(x) = ax^2 + bx + c and q(x)=dx2+ex+fq(x) = dx^2 + ex + f satisfy the following: p(1)=3p(1) = 3, p(3)=7p(3) = 7, p(5)=15p(5) = 15, and q(1)=8q(1) = 8, q(3)=2q(3) = 2, q(5)=4q(5) = 4. Over the interval [1,5][1, 5], which statement is most accurate?

Function pp has a consistently increasing rate of change while qq has a consistently decreasing rate
Function qq has a greater average rate of change than pp over this interval despite local variations
Both functions have the same average rate of change, but their instantaneous rates differ significantly
Function pp has a greater average rate of change than qq, with pp accelerating and qq decelerating
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Math 2 Quiz

Math 2 Quiz: Comparing Quadratic Functions

Practice Comparing Quadratic Functions in Math 2 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 Comparing Quadratic Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

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

Two quadratic functions p(x)=ax2+bx+cp(x) = ax^2 + bx + c and q(x)=dx2+ex+fq(x) = dx^2 + ex + f satisfy the following: p(1)=3p(1) = 3, p(3)=7p(3) = 7, p(5)=15p(5) = 15, and q(1)=8q(1) = 8, q(3)=2q(3) = 2, q(5)=4q(5) = 4. Over the interval [1,5][1, 5], which statement is most accurate?

  1. Function pp has a consistently increasing rate of change while qq has a consistently decreasing rate
  2. Function qq has a greater average rate of change than pp over this interval despite local variations
  3. Both functions have the same average rate of change, but their instantaneous rates differ significantly
  4. Function pp has a greater average rate of change than qq, with pp accelerating and qq decelerating (correct answer)
Explanation: For p(x): average rate over [1,5] = (15-3)/(5-1) = 3. The differences are: p(3)-p(1) = 4, p(5)-p(3) = 8, showing increasing rate (acceleration). For q(x): average rate over [1,5] = (4-8)/(5-1) = -1. The differences are: q(3)-q(1) = -6, q(5)-q(3) = 2, showing the rate is becoming less negative (deceleration in the negative direction). Since 3 > -1, function p has greater average rate of change. Function p shows acceleration (increasing positive rate), while q shows deceleration (rate changing from negative to positive). Choice A is wrong about q having consistently decreasing rate. Choice B is wrong because p has greater average rate. Choice C is wrong because average rates differ (3 vs -1).

Question 2

Two quadratic functions u(x)=−x2+6x−5u(x) = -x^2 + 6x - 5 and v(x)=2x2−8x+11v(x) = 2x^2 - 8x + 11 are compared on the interval [1,4][1, 4]. Which function has the greater average rate of change, and why?

  1. Function uu because it increases on this entire interval while vv decreases
  2. Function vv because its rate of change is 22 compared to uu's rate of 11 (correct answer)
  3. Function uu because both functions have the same rate of change
  4. Function vv because it has a larger leading coefficient magnitude
Explanation: Calculate the average rates of change on [1,4]. For u(x): u(1) = -1 + 6 - 5 = 0, u(4) = -16 + 24 - 5 = 3. Average rate = (3-0)/(4-1) = 1. For v(x): v(1) = 2 - 8 + 11 = 5, v(4) = 32 - 32 + 11 = 11. Average rate = (11-5)/(4-1) = 2. Since 2 > 1, function v has the greater average rate of change.

Question 3

Consider the quadratic functions F(x)=−2x2+16x−30F(x) = -2x^2 + 16x - 30 and G(x)=x2−6x+13G(x) = x^2 - 6x + 13. On the interval [3,6][3, 6], which function has the greater average rate of change?

  1. Function FF because it has a larger coefficient for the xx term
  2. Function GG because its average rate of change is 33 (correct answer)
  3. Function FF because its average rate of change is −6-6
  4. Function GG because it increases while FF decreases on this interval
Explanation: Calculate average rates on [3,6]. For F(x): F(3) = -18 + 48 - 30 = 0, F(6) = -72 + 96 - 30 = -6. Average rate = (-6-0)/(6-3) = -2. For G(x): G(3) = 9 - 18 + 13 = 4, G(6) = 36 - 36 + 13 = 13. Average rate = (13-4)/(6-3) = 3. Since 3 > -2, function G has the greater average rate of change. F decreases on this interval (vertex at x = 4), while G increases throughout (vertex at x = 3).

Question 4

Function p(x)=ax2+bx+cp(x) = ax^2 + bx + c passes through points (−1,8)(-1, 8), (0,3)(0, 3), and (2,7)(2, 7). Function q(x)=2x2−x+1q(x) = 2x^2 - x + 1. On the interval [1,3][1, 3], which function grows faster?

  1. Function pp grows faster because it has a y-intercept of 3
  2. Function qq grows faster because its leading coefficient is positive
  3. Function pp grows faster because its average rate of change is 4
  4. Function qq grows faster because its average rate of change is 7 (correct answer)
Explanation: First, find p(x) using the three points. From (0,3): c = 3. Using (-1,8) and (2,7): a - b + 3 = 8 and 4a + 2b + 3 = 7, giving a - b = 5 and 4a + 2b = 4, so 2a + b = 2. Solving: a = 1, b = 0. Thus p(x) = x² + 3. On [1,3]: p(1) = 4, p(3) = 12, average rate = (12-4)/(3-1) = 4. For q(x): q(1) = 2(1) - 1 + 1 = 2, q(3) = 2(9) - 3 + 1 = 16, average rate = (16-2)/(3-1) = 7. Since 7 > 4, function q grows faster.

Question 5

Function m(x)=0.5x2+3x−1m(x) = 0.5x^2 + 3x - 1 and function n(x)n(x) has the property that n(0)=5n(0) = 5, n(2)=1n(2) = 1, and n(4)=5n(4) = 5. Over the interval [0,4][0, 4], which statement about their growth patterns is correct?

  1. Function mm grows faster because it has a positive leading coefficient
  2. Function nn grows faster because its average rate of change is zero
  3. Function mm grows faster because its total change is 20 units (correct answer)
  4. Function nn grows faster because it decreases then increases symmetrically
Explanation: For m(x): m(0) = -1, m(4) = 0.5(16) + 3(4) - 1 = 8 + 12 - 1 = 19. Total change = 19 - (-1) = 20 units. For n(x): n(0) = 5, n(4) = 5. Total change = 0 units. Average rate for m: 20/4 = 5. Average rate for n: 0/4 = 0. Function m grows faster with a positive average rate of change compared to n's zero average rate.

Question 6

Function r(x)=x2−2x+4r(x) = x^2 - 2x + 4 and function s(x)=−0.5x2+4x−3s(x) = -0.5x^2 + 4x - 3 are analyzed on the interval [0,2][0, 2]. Based on their average rates of change, which conclusion is most accurate?

  1. Function rr grows faster with an average rate of change of 00
  2. Function ss grows faster with an average rate of change of 33 (correct answer)
  3. Function rr grows faster because it has positive leading coefficient
  4. Function ss grows faster because its vertex is within the interval
Explanation: Calculate average rates on [0,2]. For r(x): r(0) = 4, r(2) = 4 - 4 + 4 = 4. Average rate = (4-4)/(2-0) = 0. For s(x): s(0) = -3, s(2) = -0.5(4) + 4(2) - 3 = -2 + 8 - 3 = 3. Average rate = (3-(-3))/(2-0) = 3. Since 3 > 0, function s grows faster with an average rate of change of 3.

Question 7

Consider the quadratic functions h(x)=2x2−8x+11h(x) = 2x^2 - 8x + 11 and j(x)=−x2+6x−4j(x) = -x^2 + 6x - 4. Over the interval [2,4][2, 4], which statement about their comparative behavior is most accurate?

  1. Both functions are increasing, with hh increasing faster due to its larger leading coefficient magnitude
  2. Function hh is increasing while jj is decreasing, making direct rate comparison challenging but meaningful (correct answer)
  3. Function jj is increasing while hh is decreasing, with jj having the greater rate of positive change
  4. Both functions are decreasing, with jj decreasing faster due to its negative leading coefficient
Explanation: For h(x): vertex at x = 2, so h is increasing for x > 2. Over [2,4]: h(2) = 8-16+11 = 3, h(4) = 32-32+11 = 11, so h increases by 8. For j(x): vertex at x = 3, so j increases for x < 3 and decreases for x > 3. j(2) = -4+12-4 = 4, j(3) = -9+18-4 = 5, j(4) = -16+24-4 = 4. So j increases from 2 to 3, then decreases from 3 to 4, with net change = 0. Since h is consistently increasing while j increases then decreases over [2,4], choice B correctly identifies this behavior. The comparison is meaningful because we can compare average rates (8/2 = 4 for h vs 0/2 = 0 for j). Choice A is wrong because j isn't consistently increasing. Choice C has the directions reversed. Choice D is wrong because h isn't decreasing.

Question 8

Two quadratic functions f(x)=2x2−4x+1f(x) = 2x^2 - 4x + 1 and g(x)=−x2+4x−1g(x) = -x^2 + 4x - 1 are analyzed over the interval [0,3][0, 3]. Which statement best characterizes how their growth patterns compare?

  1. Both functions start increasing, but ff accelerates upward while gg reaches maximum then decelerates downward
  2. Function ff decreases then accelerates upward, while gg increases then gradually decreases to negative values
  3. Function gg increases then decreases symmetrically, while ff decreases then increases with greater final slope (correct answer)
  4. Both functions exhibit similar initial behavior but diverge significantly in their rates during the latter portion
Explanation: For f(x): vertex at x = 1, f'(x) = 4x - 4. f decreases for x < 1, increases for x > 1. f(0) = 1, f(1) = -1, f(3) = 7. For g(x): vertex at x = 2, g'(x) = -2x + 4. g increases for x < 2, decreases for x > 2. g(0) = -1, g(2) = 3, g(3) = 2. Over [0,3]: f decreases then increases (1 to -1 to 7), while g increases then decreases (-1 to 3 to 2). Function g's behavior is more symmetric around its vertex at x = 2. Function f has steeper final slope: f'(3) = 8, while g'(3) = -2. Choice C correctly describes both behaviors and notes f's greater final slope magnitude.

Question 9

Function A(x)=3x2−12x+15A(x) = 3x^2 - 12x + 15 and function B(x)=−x2+8x−12B(x) = -x^2 + 8x - 12 are compared on interval [2,5][2, 5]. Which function exhibits faster growth over this interval?

  1. Function AA because its average rate of change is 99 (correct answer)
  2. Function BB because its average rate of change is −1-1
  3. Function AA because it increases throughout the interval
  4. Function BB because its vertex creates optimal growth conditions
Explanation: Calculate average rates on [2,5]. For A(x): A(2) = 12 - 24 + 15 = 3, A(5) = 75 - 60 + 15 = 30. Average rate = (30-3)/(5-2) = 9. For B(x): B(2) = -4 + 16 - 12 = 0, B(5) = -25 + 40 - 12 = 3. Average rate = (3-0)/(5-2) = 1. Function A grows much faster with rate 9 vs B's rate of 1. The vertex of A is at x = 2 and it increases from there. The vertex of B is at x = 4, so B increases then decreases on [2,5].

Question 10

Two quadratic functions are given: h(x)=x2−4x+7h(x) = x^2 - 4x + 7 and k(x)=3(x−1)2+2k(x) = 3(x-1)^2 + 2. Over which interval does function kk have a greater average rate of change than function hh?

  1. [0,1][0, 1] because kk has a larger leading coefficient
  2. [1,3][1, 3] because kk increases more rapidly away from its vertex (correct answer)
  3. [2,4][2, 4] because both functions are increasing but kk has steeper growth
  4. [−1,1][-1, 1] because hh is decreasing while kk remains relatively constant
Explanation: First, find the vertices: h(x) has vertex at x = 2, k(x) has vertex at x = 1. For [1,3]: h(1) = 4, h(3) = 4, so average rate = 0. k(1) = 2, k(3) = 14, so average rate = 6. For [0,1]: h(0) = 7, h(1) = 4, rate = -3. k(0) = 5, k(1) = 2, rate = -3. For [2,4]: h(2) = 3, h(4) = 7, rate = 2. k(2) = 5, k(4) = 29, rate = 12. For [-1,1]: h(-1) = 12, h(1) = 4, rate = -4. k(-1) = 14, k(1) = 2, rate = -6. Only on [1,3] does k have a greater (more positive) average rate of change.