Rate of change represented with steps in the graph of the function

Rate of change represented with steps in the graph of the function - Examples, Exercises and Solutions
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Rate of change represented by steps in the function graph

We can draw stairs on the graph of the function to see the rate of change.
The base of the step will represent the interval in the X X variables and the height will symbolize the interval in the Y Y .

The step will mark the "jump" from X X in relation to the "jump" in Y Y .
The bases of the steps will always be the same since we always choose fixed intervals in X X .

  • If the heights of the steps are increasing, it means that the rate of change is increasing.
  • If the heights of the steps are decreasing, it means that the rate of change is decreasing.
  • If the heights of the steps do not change, it means that the rate of change is constant.
Rate of change represented by steps in the function graph

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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

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To use this method, you need a graph paper on which we know the length and height of each square. In this way, you can draw "steps" across the entire graph, which will help us compare the height of these steps at different points of the function.

We will demonstrate this with two different graphs drawn on graph paper:

In the first one, there is a straight line in which all the heights of the steps are equal. Consequently, we understand that this is a constant rate of change.

Steps for Signaling a Constant Rate of Change

Steps for signaling a constant rate of change

The second graph represents a function in which the heights of the steps change.

Consequently, we understand that it is a rate of change that is not constant.  

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Question 1

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

Question 2

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

111222333444555666777888999101010111111121212131313141414151515111222333444555666777888000

Question 3

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

Steps for Signaling a Non-Constant Rate of Change

image 2 Steps for signaling a non-constant rate of change

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Constant rate of change

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In the blog of Tutorela you will find a variety of articles with interesting explanations about mathematics

Examples and exercises with solutions of rate of change represented with steps on the graph of the function

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

Video Solution

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

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

Video Solution

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

Exercise #3

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

111222333444555666777888999101010111111121212131313141414151515111222333444555666777888000

Video Solution

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

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

Video Solution

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

Exercise #5

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

111222333444555666111222333000

Video Solution

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

Do you know what the answer is?
Question 1

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

111222333444555666777888999101010111111121212111222333444555666000

Question 2

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

111222333444555666111222333000

Question 3

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

–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888–3–3–3–2–2–2–1–1–1111222333444000

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