Variable Rate of Change Practice Problems & Solutions

Master variable rate of change with step-by-step practice problems. Learn to identify non-constant rates, analyze graphs, and solve real-world applications.

📚Master Variable Rate of Change Through Interactive Practice
  • Identify functions with variable rates of change from graphs and tables
  • Calculate different slopes for each section of non-linear functions
  • Distinguish between constant and variable rate of change problems
  • Analyze step-wise functions and their changing rates
  • Apply variable rate concepts to real-world scenarios
  • Interpret data tables showing inconsistent interval changes

Understanding Variable Rate of Change

Complete explanation with examples

Variable Rate of Change

The meaning of a variable rate of change of a function can be seen when the variables X X change in fixed proportions and the Y Y change unevenly. 

A variable rate of change is represented by a line that is not straight, as seen in the following diagram:

B3 - Function with inconsistent rate of change

Detailed explanation

Practice Variable Rate of Change

Test your knowledge with 9 quizzes

Given a table showing points on the graph of a function, determine whether or not the rate of change is uniform.

XY1234-6-303

Examples with solutions for Variable Rate of Change

Step-by-step solutions included
Exercise #1

Look at the graph below and determine whether the function's rate of change is constant or not:

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111–3–3–3–2–2–2–1–1–1111222333444555000

Step-by-Step Solution

First we need to remember that if the function is not a straight line, its rate of change is not constant.

The rate of change is not uniform since the function is not a straight line.

Answer:

Not constant

Video Solution
Exercise #2

Given the following graph, determine whether the rate of change is uniform or not

111222333444555666777888999101010111111121212131313141414151515111222333444555666777888000

Step-by-Step Solution

Let's remember that if the function is not a straight line, its rate of change is not uniform.

Since the graph is not a straight line - the rate of change is not uniform.

Answer:

Non-uniform

Video Solution
Exercise #3

Given the following graph, determine whether the rate of change is uniform or not?

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111–3–3–3–2–2–2–1–1–1111222333444555000

Step-by-Step Solution

Let's remember that if the function is not a straight line, its rate of change is not uniform.

Since the graph is not a straight line - the rate of change is not uniform.

Answer:

Non-uniform

Video Solution
Exercise #4

Given the following graph, determine whether the rate of change is uniform or not?

–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888–3–3–3–2–2–2–1–1–1111222333444555666000

Step-by-Step Solution

Remember that if the function is a straight line, its rate of change will be constant.

Due to the fact that the graph is a straight line - the rate of change is constant.

Answer:

Uniform

Video Solution
Exercise #5

Given the following graph, determine whether the rate of change is uniform or not

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111–3–3–3–2–2–2–1–1–1111222333444555000

Step-by-Step Solution

The problem asks us to determine if the rate of change in the graph is uniform or not. To do this, we need to examine the graph closely to see whether it is linear.

If a graph is linear, it means it is a straight line, indicating a constant (uniform) rate of change. The slope of a straight line does not change, meaning that for every unit increase in x x there is a proportional and consistent change in y y .

In contrast, if a graph curves or the line is not straight, the rate of change would not be uniform. This is because a curve indicates that the amount y y changes for each unit change in x x is not constant.

By analyzing the given graph, we can see that it is a non-linear function with a visible curve. Since the line is not straight (it appears as a curved line in the graph), the rate of change of the function is not constant across its range.

Therefore, the solution to the problem is that the rate of change is non-uniform.

Consequently, the correct choice, corresponding to a non-uniform rate of change in the graph, is:

Non-uniform

Answer:

Non-uniform

Video Solution

Frequently Asked Questions

Everything you need to know about Variable Rate of Change

What is a variable rate of change in a function?

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A variable rate of change occurs when the X values change in fixed proportions but the Y values change unevenly. This creates a non-straight line graph where different sections have different slopes, meaning the rate of change varies throughout the function.

How do you identify a variable rate of change from a graph?

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Look for curves or bent lines instead of straight lines. If the graph is not a straight line, the function has a variable rate of change. Each section of the curve will have a different slope, indicating the rate is changing continuously.

What does a table with variable rate of change look like?

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In a table with variable rate of change, the X values increase by constant intervals (like 2, 4, 6, 8) but the Y values change by inconsistent amounts (like 1, 3, 7, 13). The differences between consecutive Y values are not equal.

How is variable rate of change different from constant rate?

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Constant rate of change produces straight line graphs with the same slope throughout. Variable rate of change creates curved or bent graphs where the slope changes. Constant rates have equal Y-value differences for equal X-intervals, while variable rates have unequal Y-differences.

What are step-wise functions with variable rates?

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Step-wise functions show variable rates through distinct horizontal segments at different heights. Each step represents a different rate period, and the varying heights of steps demonstrate how the rate changes over different intervals of the domain.

How do you calculate rate of change for different sections?

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For each section of a variable rate function: 1) Identify two points in that section, 2) Use the formula (y₂ - y₁)/(x₂ - x₁), 3) Calculate the slope for each section separately. Different sections will have different slope values.

What are real-world examples of variable rate of change?

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Common examples include population growth (starts slow, accelerates, then slows), car acceleration (varies with gear changes), temperature changes throughout the day, and water flow rates that change due to external factors.

Why do functions have variable rates of change?

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Functions have variable rates because many real-world relationships are not linear. Factors like acceleration, compound growth, diminishing returns, or changing conditions cause the relationship between variables to change at different rates over different intervals.

More Variable Rate of Change Questions

Click on any question to see the complete solution with step-by-step explanations

Variation of a Function

Find the Rate of Change: Solving -x+y=0 Linear Equation Find the Rate of Change in the Linear Equation: -5x+y=3 Find the Rate of Change in the Linear Equation 3-y=1/4x Find the Rate of Change in Linear Equation -8y+2x=6 Find the Rate of Change: Solving -1/4y + x = 3

Continue Your Math Journey

Master Variable Rate of Change first, then advance to these related topics that build on your fraction skills

Suggested Topics to Practice in Advance

Ways to Represent a FunctionRepresenting a Function Verbally and with TablesGraphical Representation of a FunctionAlgebraic Representation of a FunctionNotation of a Function

Topics Learned in Later Sections

Rate of Change of a FunctionVariation of a FunctionRate of change represented with steps in the graph of the functionRate of change of a function represented graphicallyConstant Rate of ChangeRate of Change of a Function Represented by a Table of ValuesFunctions for Seventh GradeIncreasing and Decreasing Intervals (Functions)Increasing functionsDecreasing functionConstant FunctionDecreasing Interval of a functionIncreasing Intervals of a functionDomain of a FunctionIndefinite integralInputing Values into a Function

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