# Variation of a Function

🏆Practice 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

## Test yourself on variation of a function!

Given the following graph, determine whether function is constant

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 $X$.

## 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: $0$

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

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### 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 $x\gt2$

$x\gt2$

### 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 $x<-4$

$x<-4$

Do you know what the answer is?

### 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: $40$

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

### 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: $40$

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

## Examples with solutions for Variation of a 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