Rate of Change of a Function

Initially, there are three basic ways to describe the rate of change of a function:

In a graph, we can perceive if the rate of change of the function is constant or not.
If the function is a straight line, the rate of change will be constant.
If the function is not a straight line, the rate of change will not be constant.

Rate of change of a function represented by a table of values
Let's create a table of values for the function. We will place the $X$ one by one and discover the corresponding $Y$ according to the function.
Pay attention The intervals of $X$ that we choose must be fixed.
That is, $1,2,3,4$ or $3,5,7,9$ for example
We will observe at what pace the $Y$ increases in the table.
If the $Y$ increases at a constant pace, we can determine what the rate of change is.
If the $Y$ does not increase at a constant pace as the $X$ does, we can determine that the rate of change is not constant.

We can draw steps on the graph of the function that mark the intervals of the $X$ with respect to the intervals of the $Y$.
The base of the step will represent the interval in the $X$ and the height will symbolize the interval in the $Y$.

We already know that a function, in fact, represents the change in Y produced by the change in $X$.
However, have you ever thought about the rate at which this change occurs?
The rate of change of a function describes the pace at which the variables $Y$ change with respect to the change in the variables $X$.
The rate of change can be constant, fixed
or inconsistent, not fixed.
We can see the rate of change in the graphical representation, in the value table, or by drawing steps on the graph of the function.
After you read this entire page, there will be no doubt that you know everything you need to know about the rate of change of functions.

As we have noted, the rate of change of a function describes the pace at which the variables $Y$ change with respect to the change in the variables $X$. It can be said that the rate of change of a function is the slope of the function.
We will identify the slope according to the coefficient of $X$.

With a graph, we can see the rate of change of a function.
Let's see it in the following example:
Given the function $y=x+4y=x+4$
Let's illustrate the function:

The coefficient of $X$ is positive, therefore, it will be an increasing function and $b$ the independent variable is $4$, so the function will cut the $Y$ axis through $4$.
Another way to illustrate the function is to observe the points of intersection of the function with the axes:
When we place $y=0$ we will find points of intersection with the $x$ axis
and when we place $x=0$, we will find points of intersection with the $y$ axis.

Let's place $y=0$

$0=x+40=x+4$

$x=-4x=−4$

Let's set ( x=0 \)

$y=0+4y=0+4$

$y=-4y=−4$
Since the line of the function is straight in the graphical representation, we can say that the rate of change of the function is constant.
For every $X$ that we increase, the $Y$ grows at the same rate.
Furthermore, we can say that the rate of change of the function is the slope of the function, that is $1$.
In other words, how much does the $Y$ variable change when the $X$ increases by $1$? The answer is $1$.
If the slope were $2$, we would say that the rate of change of the function is $2$.
In that case, when the $X$ grew by $1$, the $Y$ would do so by $2$.

Since the function is not a straight line, it does not have a fixed slope, and its rate of change is not constant.
That is, when the value of $X$ increases by $1$, the change in $Y$ will not always be the same.

Another way to look at the rate of change is with a table of values.

The value table is a table that shows the pairs of $X$ and $Y$ for that function.
With a fairly simple calculation we can discover if the rate of change of the function is constant or not and, if it is, what it would be.

Observe the following value table that belongs to the function with which we have started:

In the table we can see that, when the $X$ increases by $1$ also the $Y$ consistently increases by $1$.

We will ask ourselves:
What happened to the $Y$ when the $X$ increased by $1$?
It increased by $1$.
Again the $X$ increased by one. What happened to the $Y$?
and so on.

Pay attention. The slope is the only thing that influences the rate of change, while the independent variable does not modify anything.
Clearly, we have obtained the same result in the graphical representation and in the value table.

Clarification:
We will ask ourselves what would happen to the $Y$ if the $X$ increases by $1$.
The Y does not have to increase by $1$. As long as it grows or decreases in a constant manner, the rate of change of the function will be constant.

What does a non-constant rate of change look like within the value table?

It can be seen that the values of $X$ always increase by $2$.
But what happens with the $Y$
Its value increases at a rate that is not constant!
In the first step the $Y$ increased by $5$
In the second the $Y$ increased by $11$
and in the third step the $Y$ increased by $2$.

Therefore, it can be said that the rate of change is not constant.

The constant or fixed rate of change describes a situation in which for a fixed interval between the variables $X$ there will also be a fixed interval between the variables $Y$.
For example, when the constant interval of $X$ is $2$ and also that of $Y$ is constant and does not vary from time to time.
If the function is represented with a straight graph, it means that the rate of change is constant.
The rate of change is the slope of the function.

The non-constant or variable rate of change describes a situation in which for a fixed interval between the variables $X$ there will also be different intervals between the variables $Y$.
For example, when the constant interval of $X$ is $2$ and that of $Y$ is not constant and varies from time to time.
If the function is not represented with a straight graph, it means that the rate of change is not constant.
Each segment of the function will have a different rate of change, a different slope.

The rate of change of a function can be seen by drawing steps on its graph.
What do we mean by steps?
We will call them this because they really look like steps, stairs, or risers.
The step will mark the interval between the $X$ variables, that is, the "jump" between the values of $X$
in relation to the interval between the $Y$ variables, that is, the "jump" in the values of $Y$.

The base of the step will represent the interval between the $X$ variables, and its height, that of the variables $Y$.

Let's take the function from the first example and draw steps on it to understand the rate of change:

$y=x+4y=x+4$

We can see that when we add $1$ to the $X$
the $Y$ will also increase by$1$.

This is another way to see if the interval is constant or not.
In the example, the constant interval is equal to $1$.

Excellent! Now you know everything you need about the rate of change of functions and also all the ways to find out if the rate is constant or not.

If you are interested in this article, you might also be interested in the following articles:

Functions for Seventh Grade

Rate of Change of a Function Represented Graphically

Rate of Change of a Function Represented by a Table of Values

Constant Rate of Change

Variable Rate of Change

Rate of Change Represented with Steps in the Function's Graph

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