# Increasing and Decreasing Intervals (Functions)

🏆Practice increasing and decreasing intervals of a function

The intervals where the function is increasing show a certain situation in which the values of $X$ and $Y$ increase together.

The intervals where the function is decreasing expose a certain situation in which the value of $X$ in a function increases while that of $Y$ decreases.

## Test yourself on increasing and decreasing intervals of a function!

In what domain does the function increase?

## What are increasing, decreasing, and constant functions

### Increasing function

If the line of the graph starts below and, as it moves to the right it goes up, that means that the function is increasing. That is, the function grows when the values of $Y$ increase as those of $X$ grow (that is, move from left to right)

Increasing Function

### Decreasing function

If the line of the graph starts at the top and, as it moves to the right it goes down, that means the function is decreasing. That is, the function decreases when the values of $Y$ go down as those of $X$ increase (that is, move from left to right)

Decreasing Function

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### Constant Function

If the line on the graph starts at a certain point on the $Y$ axis, and as it moves to the right it remains constant at the same height, that is, at the same point on the $Y$ axis, this means that it is a constant function. That is, the function is constant when the values of $Y$ keep their place and remain fixed as those of $X$ increase (that is, move from left to right)

Constant Function

## Intervals of Increase and Decrease of a Function

### Increasing Function Intervals

To identify the intervals where the function is increasing, we will look on the graph for the point where the function begins to rise.

We will mark the value on the $X$ axis. In our case, it is $-5$. Then, we will look on the $X$ axis for the point where the function stops rising. In our case, it is $7$. Therefore, the growth interval of the function will be:

$-5

We will illustrate this with a simple graph:

In the graph, it can be seen that the intervals of growth of the function are $X<-3$ (values of $X$ less than $-3$) and for the values of $X$ that are between $0$ and $3$. That is, in these intervals, the values of $X$ and $Y$ increase together.

Furthermore, it follows from the graph that the intervals of decline of the function are for the values of $X$ that are between $-3$ and $0$ and for $X>3$. That is, in these intervals, the values of $X$ increase and those of $Y$ decrease at the same time.

Exercise

Note that, in the graph, you can also see the intervals of decline of the function. Do you know what they are?

$-10

$7

Do you know what the answer is?

### Decreasing interval of the function

To identify the intervals where the function is decreasing, we will look on the graph for the point where the function starts to go down.

We will mark the value on the $X$ axis. In our case, it is $7$. Then we will look on the $X$ axis for the point where the function stops going down. In our case, it is $5$. Therefore, the interval of decrease of the function will be:

$-7

Exercise

Notice that, on the graph, you can also see the intervals of increase of the function. Do you know what they are?

$-10

$5

## Exercises with increasing and decreasing intervals of a function:

### Exercise 1

Assignment

Find the increasing area of the function

$f(x)=6x^2-12$

Solution

In the first step, let's consider that $a=6$

Therefore $a>0$ and the parabola is at a minimum

In the second step, we find $x$ of the vertex

according to the data we know that:

$a=6,b=0,c=-12$

We replace the data in the formula

$x=\frac{-b}{2\cdot a}$

$x=\frac{-0}{2\cdot6}$

$x=\frac{0}{12}$

$x=0$

Therefore

$0 Increasing

$x<0$ Decreasing

$0

### Exercise 2

Assignment

Given the function in the diagram, what is its domain of positivity?

Solution

Note that the entire function is always above the axis: $x$

Therefore, it will always be positive. Its area of positivity will be for all $x$

For all $x$

### Exercise 3

Assignment

Find the increasing area of the function

$f(x)=-4x^2-24$

Solution

In the first step, let's consider that $a=-4$

Therefore $a<0$ and the parabola is at its maximum

In the second step, we find $x$ of the vertex

according to the data we know that:

$a=-4,b=0,c=-24$

We replace the data in the formula

$x=\frac{-b}{2\cdot a}$

$x=\frac{-0}{2\cdot\left(-4\right)}$

$x=\frac{0}{-8}$

$x=0$

Therefore $x<0$ increasing area

$x<0$

Do you think you will be able to solve it?

### Exercise 4

Assignment

Find the increasing area of the function

$f(x)=2x^2$

Solution

In the first step, let's consider that $a=2$

Therefore $a>0$ and the parabola is minimum

In the second step, we find $x$ of the vertex

according to the data we know that:

$a=2,b=0,c=0$

We replace the data in the formula:

$x=\frac{-b}{2\cdot a}$

$x=\frac{0}{2\cdot2}$

$x=\frac{0}{4}$

$x=0$

Therefore, there is increase in the area $0

$0

### Exercise 5

Assignment

Find the increasing area of the function

$f(x)=-3x^2+12$

Solution

In the first step, let's consider that $a=-3$

Therefore $a<0$ and the parabola is at its maximum

In the second step, we find $x$ of the vertex

according to the data we know that:

$a=3,b=0,c=12$

We replace the data in the formula

$x=\frac{-b}{2\cdot a}$

$x=\frac{-0}{2\cdot\left(-3\right)}$

$x=\frac{0}{-6}$

$x=0$

Therefore, there is increase in the area $x<0$

$x<0$

### Exercise 6

Assignment

Find the decreasing area of the function

$y=(x+1)+1$

Solution

$a$ coefficient of $x^2$

Therefore $0

is the minimum point

The vertex of the function is $\left(-1,1\right)$

The function decreases in the area of $x<-1$

$x<-1$

### Exercise 7

Assignment

Given the function in the graph

When is the function positive?

Solution

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

Positive before, then negative.

Therefore $x<-4$

$x<-4$

Do you know what the answer is?

## examples with solutions for increasing and decreasing intervals of a function

### Exercise #1

Determine which domain corresponds to the function described below:

The function represents the amount of fuel in a car's tank according to the distance traveled by the car.

### Step-by-Step Solution

According to the definition, the amount of fuel in the car's tank will always decrease, since during the trip the car consumes fuel in order to travel.

Therefore, the domain that is suitable for this function is - always decreasing.

Always decreasing

### Exercise #2

Choose the graph that best describes the following:

The acceleration of a ball (Y) after throwing it from a building as a function of time (X).

### Step-by-Step Solution

Since acceleration is dependent on time, it will be constant.

The force of gravity on Earth is constant, meaning the velocity of Earth's gravity is constant and therefore the graph will be straight.

The graph that appears in answer B satisfies this.

### Exercise #3

Choose the graph that best represents the following:

Temperature of lukewarm water (Y) after placing in the freezer as a function of time (X).

### Step-by-Step Solution

Since the freezing point of water is below 0, the temperature of the water must drop below 0.

The graph in answer B describes a decreasing function and therefore this is the correct answer.

### Exercise #4

Is it possible to create an increasing function with the two given points?

### Step-by-Step Solution

We will connect the two points to each other, and we will see that indeed we have obtained an increasing function.

Yes

### Exercise #5

Determine whether the function is increasing, decreasing, or constant. For each function check your answers with a graph or table.

For each number, multiply by$(-1)$.

### Step-by-Step Solution

The function is:

$f(x)=(-1)x$

Let's start by assuming that x equals 0:

$f(0)=(-1)\times0=0$

Now let's assume that x equals minus 1:

$f(-1)=(-1)\times(-1)=1$

Now let's assume that x equals 1:

$f(1)=(-1)\times1=-1$

Now let's assume that x equals 2:

$f(2)=(-1)\times2=-2$

Let's plot all the points on the function graph:

We can see that the function we got is a decreasing function.