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

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?

–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 #3

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 #4

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

–3–3–3–2–2–2–1–1–1111222333444–1–1–1111222333000

Step-by-Step Solution

To determine if the rate of change in the given graph is uniform, we need to analyze the graph and check if it is a straight line.

Step 1: Check for linearity - The most direct way to determine if the graph has a uniform rate of change is by inspecting it for linearity, which means the graph forms a straight line.

Step 2: Analyze the path - The given SVG code and description imply a straight diagonal line, suggesting a constant slope.

For a linear function, the slope m=y2−y1x2−x1 m = \frac{y_2 - y_1}{x_2 - x_1} is constant throughout. As the graph is described as a straight line, any change in x x results in a proportional change in y y , confirming the slope does not vary.

Consequently, the graph displays a uniform rate of change. Therefore, the solution to this problem is uniform.

Answer:

Uniform

Video Solution
Exercise #5

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

111222333444555666111222333000

Step-by-Step Solution

To solve this problem, let's analyze the graph of the line:

  • Step 1: Identify two points on the line. For simplicity, let's choose the intercept at x=1 x = 1 and y=3 y = 3 , and another at x=6 x = 6 and y=0 y = 0 (assuming these are easily readable points).
  • Step 2: Calculate the slope using the formula y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}.
  • Step 3: Substituting in our chosen points, the slope is 0−36−1=−35\frac{0 - 3}{6 - 1} = \frac{-3}{5}.
  • Step 4: Since the graph is a straight line and the slope is constant, the rate of change is uniform.

Therefore, the graph shows a constant or uniform rate of change.

The solution to the problem is thus Uniform.

Since the correct answer is shown in the multiple-choice option "Uniform", we conclude it matches the analysis result.

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

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