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College Algebra Quiz

College Algebra Quiz: Average Rate Of Change

Practice Average Rate Of Change in College Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 2

0 of 2 answered

A function f(x)f(x)f(x) passes through the points (−3,7)(-3, 7)(−3,7) and (5,−1)(5, -1)(5,−1). If g(x)=f(x−2)+3g(x) = f(x - 2) + 3g(x)=f(x−2)+3, what is the average rate of change of g(x)g(x)g(x) over the interval [−1,7][-1, 7][−1,7]?

Select an answer to continue

What this quiz covers

This quiz focuses on Average Rate Of Change, giving you a quick way to practice the rules, question types, and explanations that matter most for College Algebra.

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

A function f(x)f(x)f(x) passes through the points (−3,7)(-3, 7)(−3,7) and (5,−1)(5, -1)(5,−1). If g(x)=f(x−2)+3g(x) = f(x - 2) + 3g(x)=f(x−2)+3, what is the average rate of change of g(x)g(x)g(x) over the interval [−1,7][-1, 7][−1,7]?

  1. −1-1−1 (correct answer)
  2. −12-\frac{1}{2}−21​
  3. 12\frac{1}{2}21​
  4. 111

Explanation: First, find the average rate of change of f(x)f(x)f(x) over [−3,5][-3, 5][−3,5]: f(5)−f(−3)5−(−3)=−1−78=−1\frac{f(5) - f(-3)}{5 - (-3)} = \frac{-1 - 7}{8} = -15−(−3)f(5)−f(−3)​=8−1−7​=−1. For g(x)=f(x−2)+3g(x) = f(x - 2) + 3g(x)=f(x−2)+3, we need g(−1)g(-1)g(−1) and g(7)g(7)g(7). Since g(−1)=f(−3)+3=7+3=10g(-1) = f(-3) + 3 = 7 + 3 = 10g(−1)=f(−3)+3=7+3=10 and g(7)=f(5)+3=−1+3=2g(7) = f(5) + 3 = -1 + 3 = 2g(7)=f(5)+3=−1+3=2, the average rate of change is 2−107−(−1)=−88=−1\frac{2 - 10}{7 - (-1)} = \frac{-8}{8} = -17−(−1)2−10​=8−8​=−1. Choice B results from incorrectly using the original interval length of 8 in the denominator twice. Choice C is the negative of the correct answer. Choice D ignores the negative sign from the calculation.

Question 2

The function k(x)k(x)k(x) has an average rate of change of −3-3−3 over the interval [2,6][2, 6][2,6] and an average rate of change of 555 over the interval [6,10][6, 10][6,10]. If k(2)=8k(2) = 8k(2)=8, what is k(8)k(8)k(8) assuming k(x)k(x)k(x) is linear on each interval?

  1. −2-2−2
  2. 222
  3. 666 (correct answer)
  4. 101010

Explanation: Since k(x)k(x)k(x) has average rate of change −3-3−3 over [2,6][2,6][2,6] and k(2)=8k(2) = 8k(2)=8, we have k(6)−k(2)6−2=−3\frac{k(6) - k(2)}{6 - 2} = -36−2k(6)−k(2)​=−3, so k(6)−84=−3\frac{k(6) - 8}{4} = -34k(6)−8​=−3, giving k(6)−8=−12k(6) - 8 = -12k(6)−8=−12, thus k(6)=−4k(6) = -4k(6)=−4. Since k(x)k(x)k(x) is linear on [6,10][6,10][6,10] with rate 555, we have k(8)=k(6)+5⋅(8−6)=−4+5⋅2=−4+10=6k(8) = k(6) + 5 \cdot (8-6) = -4 + 5 \cdot 2 = -4 + 10 = 6k(8)=k(6)+5⋅(8−6)=−4+5⋅2=−4+10=6. Choice A would result from calculation errors. Choice B incorrectly uses the rate from the first interval. Choice D is k(6)+14k(6) + 14k(6)+14, a common arithmetic error.