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Quizzes > Mathematics

Rates of Change Quiz: Linear and Quadratic Functions

Quick, free linear and quadratic functions quiz-20 questions with instant results.

Editorial: Review CompletedCreated By: Betuel HimawanUpdated Aug 27, 2025
Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Paper art promoting the Slope Curve Challenge, a math practice quiz for high school students.

This quiz helps you practice rates of change in linear and quadratic functions using graphs, tables, and slope. Answer 20 quick questions with instant feedback, then deepen your skills with a derivative quiz, a calculus quiz, or a function relationships quiz. You'll learn to read change from graphs and tables and find average rate on an interval.

What is the rate of change (slope) of the linear function y = 3x - 5?
5
-3
3
-5
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For any linear function, the average rate of change over any interval equals the function's slope.
True
False
undefined
Find the slope of the line passing through the points (2, 7) and (5, 16).
-3
9
3/5
3
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A car's distance from start is given by D(t) = 60t + 120, with t in hours and D in miles. What are the units of the rate of change?
hours per mile
miles per hour
hours
miles
undefined
Lines with equal slopes are perpendicular.
False
True
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If a line has slope 4, then a line perpendicular to it has slope -1/4.
False
True
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A straight line graph indicates a constant rate of change.
False
True
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For f(x) = 2x^2 + 3x, what is the average rate of change from x = 0 to x = 2?
10
11
8
7
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Over which interval is the quadratic f(x) = (x - 2)^2 decreasing?
[2, ∞)
(-∞, 2)
(-∞, ∞)
(2, ∞)
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At the vertex of a parabola y = ax^2 + bx + c, the instantaneous rate of change is zero.
False
True
undefined
For f(x) = -x^2, the average rate of change from x = 0 to x = 2 is positive.
True
False
undefined
A sequence defined by a_n = 4n + 1 has a constant rate of change between consecutive terms.
False
True
undefined
If y = f(x) is replaced by y = f(2x) for a linear function, how does the rate of change compare to the original?
It halves
It becomes zero
It stays the same
It doubles
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A data set with constant second differences is best modeled by a linear function.
True
False
undefined
For f(x) = x^2 - 4x, find all x in [0, 4] for which the average rate of change from 0 to x equals -2.
x = 2
x = 1
x = 3
x = 4
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A position function with constant acceleration has a quadratic form, leading to a linearly changing velocity (rate of change of position).
True
False
undefined
At the vertex of y = (x + 1)^2 - 9, the average rate of change over any symmetric interval around x = -1 is zero.
True
False
undefined
For f(x) = x^2 + x, the average rate of change from x to x + 1 equals which expression?
2x + 1
x^2 + 1
x + 1
2x + 2
undefined
For f(x) = a x^2 with a > 0, the average rate of change from 0 to h is ah.
h/a
a/h
2ah
ah
undefined
Consider f(x) = -2x^2 + 8x + 1. Over which interval with x >= 0 is the average rate of change positive?
[0, 2]
[3, 5]
[5, 6]
[-1, 0]
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0

Study Outcomes

  1. Analyze the rate of change in linear functions to determine slope.
  2. Interpret how quadratic coefficients affect the curvature of graphs.
  3. Apply techniques to identify key features of both linear and quadratic graphs.
  4. Evaluate the impact of changes in function parameters on graph behavior.

Rates of Change Quiz: Linear & Quadratic Cheat Sheet

  1. Understanding Linear Functions - Linear functions look like f(x) = mx + b, where "m" tells you how steep the line is and "b" shows where it crosses the y-axis. They always graph as perfect straight lines, so predicting values is a breeze once you know m and b.
  2. Grasping Quadratic Functions - Quadratic functions follow f(x) = ax² + bx + c, giving you those signature curved shapes called parabolas. If a > 0, the curve smiles upward; if a < 0, it frowns downward. Play with different values of a, b, and c to see how the curve stretches or shifts.
  3. Constant Rate of Change in Linear Functions - In every linear function, the rate of change stays the same - that's your slope "m". No matter which points you pick, the line climbs (or falls) by m units for every 1 unit in x. This constant rate keeps graphs predictable and simple.
  4. Variable Rate of Change in Quadratic Functions - Quadratics shake things up with a rate of change that's always shifting. As x grows, the slope itself grows or shrinks, making the curve bend. This variable rate is what makes parabolas so dynamic and interesting.
  5. Calculating Slope in Linear Functions - To find a linear slope, grab two points (x₝, y₝) and (x₂, y₂) and calculate m = (y₂ - y₝)/(x₂ - x₝). This tells you exactly how fast the function is climbing or dropping. It's like measuring the steepness of a hill.
  6. Derivative of Quadratic Functions - When you take the derivative of f(x) = ax² + bx + c, you get f'(x) = 2ax + b. This formula gives the instantaneous rate of change at any x-value. It's a powerful tool for finding slopes on curves.
  7. Identifying the Vertex of a Parabola - Spot the vertex of a parabola by plugging into x = -b/(2a). That point is the peak or the valley of your curve where the rate of change hits zero. It's your parabola's very own VIP spot.
  8. Graphing Linear Functions - Graphing linear functions is as easy as 1-2-3: plot the y-intercept (0, b), use the slope "m" to find another point, then draw your line. Boom - you've got a straight-line graph in seconds. Feel free to color it in for extra fun!
  9. Graphing Quadratic Functions - For quadratic graphs, start by plotting the vertex and drawing the axis of symmetry. Then find the x-intercepts and a couple more points on either side. Sketch the smooth U-shaped curve that ties them all together.
  10. Real-World Applications - Rates of change in linear and quadratic functions pop up everywhere from physics to finance. Use these concepts to model everything from a racing car's speed to a company's profit growth. Master them, and you're ready to tackle real-world problems like a pro!
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