Function Rate of Change Practice Problems & Solutions

Master graphical analysis of function variation with step-by-step practice problems. Learn to identify constant vs variable rates and interpret slopes visually.

📚Master Rate of Change Through Graph Analysis
  • Identify constant rate of change from straight line graphs
  • Distinguish between increasing and decreasing functions using slope analysis
  • Compare steeper vs moderate slopes to determine faster or slower rates
  • Analyze four-branch graphs to understand different variation patterns
  • Determine function behavior from graphical representations without equations
  • Apply slope coefficient rules to predict function increase or decrease

Understanding Rate of change of a function represented graphically

Complete explanation with examples

Rate of Change of a Function Represented Graphically

The rate of change of a function represented graphically allows us to determine in a much more intuitive way whether it is a constant (fixed) or inconstant (not fixed) rate, and also if it is a faster (steeper slope) or slower (more moderate slope) rate.

The following graph can demonstrate the aforementioned in the best way:

Rate of Change of a Function Represented Graphically

1- Rate of Change of a Function Represented Graphically

Let's observe the graph. We will notice that it is divided into 4 different branches. Now we will analyze each of the branches:

  • Branch 1: the graph rises (increasing function) at a constant rate (straight line).
  • Branch 2: The graph falls (decreasing function) at a constant rate (straight line).
  • Branch 3: the graph rises (increasing function) at a constant rate (straight line) and more quickly than branch 1 (the slope is steeper).
  • Branch 4: The graph falls (decreasing function) at a constant rate (straight line) and more slowly than branch 2 (the slope is more moderate).
Detailed explanation

Practice Rate of change of a function represented graphically

Test your knowledge with 9 quizzes

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

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Examples with solutions for Rate of change of a function represented graphically

Step-by-step solutions included
Exercise #1

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

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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 #2

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

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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

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 #4

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

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Step-by-Step Solution

The problem requires us to determine whether the rate of change in a given graph is uniform.

A uniform rate of change corresponds to a constant slope, which is characteristic of a linear graph. First, we'll examine the graphical representation.

Upon observing the graph, we see that it displays a straight horizontal line. A horizontal line on a graph indicates that for any two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), the difference in yy-values is zero, i.e., y2−y1=0y_2 - y_1 = 0. This implies that the slope, given by the formula y2−y1x2−x1 \frac{y_2 - y_1}{x_2 - x_1} , is zero and remains constant as we move along the line.

Because the line is horizontal and does not change its slope throughout, the rate of change is indeed uniform across the entire graph.

Therefore, the rate of change is uniform.

Answer:

Uniform

Video Solution
Exercise #5

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

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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

Frequently Asked Questions

Everything you need to know about Rate of change of a function represented graphically

How do you find the rate of change of a function from a graph?

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To find the rate of change from a graph, examine the slope of the line. A straight line indicates constant rate of change, while the steepness shows the speed of change. Rising lines have positive rates (increasing function), falling lines have negative rates (decreasing function).

What does a constant rate of change look like on a graph?

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A constant rate of change appears as a perfectly straight line on a graph. The line can be rising (positive slope), falling (negative slope), or horizontal (zero slope), but it must be straight throughout to maintain constant variation.

How can you tell if a function is increasing or decreasing from its graph?

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Look at the direction of the line: 1) Increasing function - line rises from left to right (positive slope), 2) Decreasing function - line falls from left to right (negative slope), 3) Constant function - horizontal line (zero slope).

What is the difference between steep and moderate slopes in function graphs?

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Steep slopes indicate faster rates of change, while moderate slopes show slower rates. A steeper rising line means the function increases more rapidly, and a steeper falling line means it decreases more quickly than lines with gentler slopes.

Can you determine the rate of change without the function equation?

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Yes, you can analyze rate of change purely from the graph. Observe whether the line is straight (constant rate) or curved (variable rate), its direction (positive/negative), and its steepness (fast/slow rate).

How do you analyze multi-branch function graphs for rate of change?

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Examine each branch separately: identify if each section is increasing/decreasing, compare the steepness between branches, and note whether each segment shows constant (straight) or variable (curved) rate of change.

What does slope coefficient tell you about function variation?

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The slope coefficient (coefficient of X) determines function behavior: positive coefficient creates an increasing function with rising graph, negative coefficient creates a decreasing function with falling graph. The absolute value indicates the rate's magnitude.

Why do curved lines indicate inconsistent rate of change?

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Curved lines show inconsistent rates because the slope constantly changes along the curve. Unlike straight lines with fixed slopes, curves have varying steepness at different points, meaning the rate of change is not constant throughout the function.

More Rate of change of a function represented graphically Questions

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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 Rate of change of a function represented graphically first, then advance to these related topics that build on your fraction skills

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