Given the following graph, determine whether the rate of change is uniform or not 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 1 1 1 2 2 2 3 3 3 0 0 0

The graph does not represents a function

Given the following graph, determine whether the rate of change is uniform or not 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 12 12 12 13 13 13 14 14 14 15 15 15 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 0 0 0 <path fill="none" stroke="#1565C0" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10" stroke-opacity=".698" stroke-width="2.5" d="m72.94 884.168.751-7.27.752-7.16.752-7.047.752-6.937.752-6.828.752-6.72.752-6.612.752-6.507.752-6.4.752-6.298.752-6.193.752-6.091.752-5.99.752-5.89.752-5.79.752-5.691.752-5.594.752-5.498.752-5.401.752-5.307.751-5.213.752-5.12.752-5.029 1.504-9.784 1.504-9.426 1.504-9.077 1.504-8.734 1.504-8.398 1.504-8.068 1.504-7.746 1.504-7.431 1.504-7.122 1.503-6.82 1.504-6.523 1.504-6.234 1.504-5.952 1.504-5.674 1.504-5.404 1.504-5.14 3.008-9.511 3.007-8.526 3.008-7.587 3.008-6.693 3.008-5.844 3.008-5.039 6.015-7.827 3.008-2.87 3.008-2.225 3.008-1.62 3.008-1.05 3.007-.516 3.008-.017 3.008.447 3.008.88 6.016 2.931 6.015 4.292 6.016 5.426 6.015 6.347 6.016 7.07 6.016 7.609 6.015 7.976 6.016 8.187 6.016 8.253 6.015 8.187 6.016 8 6.015 7.707 6.016 7.313 6.016 6.836 6.015 6.282 6.016 5.662 6.015 4.987 6.016 4.265 6.016 3.507 6.015 2.718 6.016 1.91 6.016 1.09 6.015.263 6.016-.562 6.015-1.377 6.016-2.178 6.016-2.959 6.015-3.712 6.016-4.434 6.016-5.12 6.015-5.766 6.016-6.366 6.015-6.918 6.016-7.417 6.016-7.864 6.015-8.252 6.016-8.581 6.016-8.85 6.015-9.057 6.016-9.2 6.015-9.278 6.016-9.294 6.016-9.245 6.015-9.133 6.016-8.96 6.016-8.724 6.015-8.43 6.016-8.078 6.015-7.673 6.016-7.214 6.016-6.708 6.015-6.156 6.016-5.564 6.016-4.935 6.015-4.274 6.016-3.587 6.015-2.878 6.016-2.154 6.016-1.423 6.015-.687 6.016.042 6.016.76 6.015 1.456 6.016 2.125 6.015 2.757 6.016 3.342 6.016 3.873 6.015 4.338 6.016 4.728 6.016 5.032 6.015 5.239 6.016 5.339 6.015 5.319 6.016 5.167 6.016 4.871 6.015 4.419 6.016 3.796 6.016 2.99 6.015 1.985 3.008.55 3.008.219 3.008-.142 3.008-.532 3.007-.952 3.008-1.405 3.008-1.89 3.008-2.41 6.015-6.515 6.016-9.019 3.008-5.543 3.008-6.286 3.007-7.069 3.008-7.894 3.008-8.762 3.008-9.675 1.504-5.193 1.504-5.439 1.504-5.69 1.504-5.946 1.503-6.21 1.504-6.479 1.504-6.754 1.504-7.035 1.504-7.323 1.504-7.616 1.504-7.918 1.504-8.224 1.504-8.537 1.504-8.858 1.503-9.185 1.504-9.518 1.504-9.859.752-5.06.752-5.146.752-5.236.752-5.325.752-5.416.752-5.507.752-5.6.752-5.692.752-5.786.752-5.881.752-5.977.752-6.074.752-6.172.752-6.27.752-6.37.752-6.471.751-6.572.752-6.675.752-6.778.752-6.882.752-6.988.752-7.094.752-7.2.752-7.31.752-7.42.752-7.529.752-7.64.752-7.753.752-7.866.752-7.98.752-8.096.752-8.213.752-8.33.752-8.447.752-8.568.752-8.688.752-8.81.751-8.932.752-9.056.752-9.18.752-9.308.752-9.433.752-9.562.752-9.69.752-9.822.752-9.952.376-5.026.376-5.059.376-5.092" paint-order="fill stroke markers">

The graph does not represents a function

Given the following graph, determine whether the rate of change is uniform or not 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 12 12 12 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 0 0 0

The graph does not represents a function

Given the following graph, determine whether the rate of change is uniform or not 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 12 12 12 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 0 0 0

The graph does not represents a function

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 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 0 0 0

The graph does not represents a function

Given the following graph, determine whether the rate of change is uniform or not –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 –1 –1 –1 1 1 1 2 2 2 3 3 3 0 0 0

The graph does not represents a function

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 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 0 0 0

The graph does not represents a function

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 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 0 0 0

The graph does not represents a function

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 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 0 0 0

The graph does not represents a function

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 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 0 0 0

The graph does not represents a function.

Given the following graph, determine whether function is constant –9 –9 –9 –8 –8 –8 –7 –7 –7 –6 –6 –6 –5 –5 –5 –4 –4 –4 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 –4 –4 –4 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 0 0 0

The graph does not represents a function

Given the following graph, determine whether the rate of change is uniform or not? –8 –8 –8 –7 –7 –7 –6 –6 –6 –5 –5 –5 –4 –4 –4 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 0 0 0

The graph does not represent a function

Rate of Change of a Function

Different Types of Rates of Change of a Function

The rate of change of a function describes the pace of modification that the variables $Y$ experience with respect to the change in the variables $X$ .
The rate of change can be:

$b>$ Constant Rate of Change - fixed
Describes a situation in which we will see constant intervals of variables $Y$ at constant intervals of variables $X$ .
In a linear function (a straight line), the rate of change is constant and will be the slope of the function.
Non-constant Rate of Change - not fixed
Describes a situation in which we will see different intervals of variables $Y$ at constant intervals of variables $X$ .
In a function that is not linear (not a straight line), the rate of change will not be constant. Each part of the function will have a different slope.

Different Types of Rates of Change of a Function

We can see the rate of change represented in a graph, a table of values, or by creating steps or stairs in the graph of the function.

Detailed explanation

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Test yourself on variation of a function!

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

Practice more now

How can the rate of change of a function be indicated?

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

The rate of change of a function represented graphically

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.

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Test your knowledge

Question 1

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

Question 2

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

Question 3

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

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

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.

Stepwise Rate of Change in the Function's Graph

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 $Y$ .

Do you know what the answer is?

Question 1

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

Question 2

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

Question 3

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

The rate of change of a function

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.

So, what is the rate of change of a function?

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 $X$ .

Rate of change of a function represented graphically

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 $y=0$

and we will obtain

$0=x+40=x+4$

$x=-4x=−4$

Let's set ( x=0 \)

and we will obtain

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

$y=-4y=−4$
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 $2$ .

And what about the graphical representation that describes a non-constant rate of change?

Graph of the non-constant rate of change

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.

Check your understanding

Question 1

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

Question 2

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

Question 3

Look at the graph below and determine whether the function's rate of change is constant or not:

Rate of Change of a Function Represented by a Value Table

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?

Table representation of a function with a rate of change that is not constant

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.

Constant Rate of Change

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.

Do you think you will be able to solve it?

Question 1

Given the following graph, determine whether function is constant

Question 2

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

Question 3

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

Rate of change that is not constant

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.

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

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 $1$ .

Graph with steps of the constant rate of change

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

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

Examples and exercises with solutions on the rate of change of a function

Exercise #1

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

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 $x = 1$ and $y = 3$ , and another at $x = 6$ and $y = 0$ (assuming these are easily readable points).
Step 2: Calculate the slope using the formula $\frac{y_2 - y_1}{x_2 - x_1}$ .
Step 3: Substituting in our chosen points, the slope is $\frac{0 - 3}{6 - 1} = \frac{-3}{5}$ .
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

Exercise #2

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

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

Video Solution

Step-by-Step Solution

To determine if the rate of change is uniform, we need to examine the slopes of the segments in the graph.

First, let's identify the segments in the graph. The graph provided has multiple segments as follows:

Segment 1: From point $A$ to point $B$ (approximation based on graph layout)
Segment 2: From point $B$ to point $C$
Segment 3: From point $C$ to point $D$
Segment 4: From point $D$ to point $E$

Next, calculate the slope for each segment:

**Segment 1 (A to B):**

A

B

m_1 = \frac{\text{change in y}}{\text{change in x}}

**Segment 2 (B to C):**

B

C

m_2

**Segment 3 (C to D):**

C

D

m_3

**Segment 4 (D to E):**

D

E

m_4

Compare the slopes $m_1$ , $m_2$ , $m_3$ , and $m_4$ . If all the calculated slopes are the same, then the rate of change is uniform. If they differ, the rate of change is non-uniform.

Given the visual inspection of the graph and performing these calculations, you'll find that the slopes change; hence, the rate of change is not uniform.

Therefore, the solution to the problem is non-uniform.

Answer

Non-uniform

Exercise #4

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

Video Solution

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 $(x_1, y_1)$ and $(x_2, y_2)$ , the difference in $y$ -values is zero, i.e., $y_2 - y_1 = 0$ . This implies that the slope, given by the formula $\frac{y_2 - y_1}{x_2 - x_1}$ , 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

Exercise #5

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

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 $m = \frac{y_2 - y_1}{x_2 - x_1}$ is constant throughout. As the graph is described as a straight line, any change in $x$ results in a proportional change in $y$ , 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

Test your knowledge

Question 1

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

Question 2

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

Question 3

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

Start practice

Rate of Change of a Function

Different Types of Rates of Change of a Function

Test yourself on variation of a function!

How can the rate of change of a function be indicated?

The rate of change of a function represented graphically

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

Stepwise Rate of Change in the Function's Graph

The rate of change of a function

So, what is the rate of change of a function?

Rate of change of a function represented graphically

and we will obtain

and we will obtain

And what about the graphical representation that describes a non-constant rate of change?

Rate of Change of a Function Represented by a Value Table

In the table we can see that, when the XX X increases by 11 1 also the Y Y Y consistently increases by 1 1 1.

Constant Rate of Change

Rate of change that is not constant

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

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

Examples and exercises with solutions on the rate of change of a function

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Variation of a Function

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

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