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 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 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 –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 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 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 –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 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 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

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

Variation of a Function

The variation of a function means the rate at which a certain function changes. The rate of variation of a function is also called the slope.

According to the mathematical definition, the slope represents the change of the function $(Y)$ by increasing the value of $X$ by $1$ .

If the function's graph is represented by a straight line, it means that the rate of variation of the function is constant
However, if the graph is not represented by a straight line, this implies that the rate of variation of the function is not constant

Detailed explanation

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Test yourself on variation of a function!

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

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That is, there are functions, such as the linear function (which we will study in more detail later, but generally speaking, it is a function with the variable to the first power) in which the slope, or in other words, the rate of change of the function is constant, and there are other functions that may have an increasing or decreasing rate of change that is calculated separately for each value $X$ .

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

Functions for seventh grade

Graphical representation of a function

Algebraic representation of a function

Function notation

Domain of a function

Indefinite integral

Assignment of numerical value in a function

Increasing function

Decreasing function

Constant function

Intervals of increase and decrease of a function

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

Exercises on the variation of a function

Exercise 1

Assignment

$y=-5x^{2}+x$

Solution

$a$ coefficient of $x^2$

Given in the exercise: $-5$

$b$ coefficient of $x$

Given in the exercise: $1$

$c$ is a free number

Therefore it is: $0$

Answer

$a=-5,~b=1,~c=0$

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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

Exercise 2

Assignment

Given the linear function in the graph

When is the function positive?

Solution

The function is positive when it is above the axis: $x$

Pay attention that the intersection point with the axis $x$ is $\left(2,0\right)$

According to the graph, the function is positive, therefore $x\gt2$

Answer

$x\gt2$

Exercise 3

Assignment

Given the function in the graph

When is the function positive?

Solution

The intersection point with the axis : $x$ is: $\left(-4,0\right)$

First positive, then negative.

Therefore $x<-4$

Answer

$x<-4$

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

Exercise 4

Assignment

$y=-40x+40$

Solution

$a$ coefficient of $x^2$

Given in the exercise: $0$

$b$ coefficient of $x$

Given in the exercise: $-40$

$c$ is a free number

Therefore it is: $40$

Answer

$a=0,~b=-40,~c=40$

Exercise 5

Assignment

$y=-x^{2}+3x+40$

Solution

$a$ coefficient of $x^2$

Given in the exercise: $-1$

$b$ coefficient of $x$

Given in the exercise: $3$

$c$ is a free number

Therefore it is: $40$

Answer

$a=-1,~b=3,~c=40$

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 function is constant

Question 3

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

Examples with solutions for Variation of a Function

Exercise #1

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

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?

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

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

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

Exercise #5

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

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Variation of a Function

Test yourself on variation of a function!

Exercises on the variation of a function

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Examples with solutions for Variation 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