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

🏆Practice variation of a function

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 $X$ variables and the height will symbolize the interval in the $Y$.

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

• 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.

Test yourself on variation of a function!

Given the following graph, determine whether function is constant

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

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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Steps for Signaling a Non-Constant Rate of Change

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

Exercise #1

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

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.

Non-uniform

Exercise #2

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

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.

Non-uniform

Exercise #3

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

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.

Non-uniform

Exercise #4

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

Step-by-Step Solution

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

Since the graph is a straight line - the rate of change is constant.

Uniform

Exercise #5

Given the following graph, determine whether function is constant