**We will say that a function is increasing when, as the value of the independent variable** **$X$**** increases, the value of the function** **$Y$**** increases.**

**We will say that a function is increasing when, as the value of the independent variable** **$X$**** increases, the value of the function** **$Y$**** increases.**

Does the function in the graph decrease throughout?

**For example**

let's assume we have two elements $X$, which we will call $X1$ and $X2$, where the following is true: $X1<X2$, that is, $X2$ is located to the right of $X1$.

- When $X1$ is placed in the domain, the value $Y1$ is obtained.
- When $X2$ is placed in the domain, the value $Y2$ is obtained.

**The function is increasing when** **$X2>X1$**** and also** **$Y2>Y1$****.**

The function can be increasing in intervals or can be continuous throughout its domain.

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

Graphical representation of a function

Algebraic representation of a function

Assignment of numerical value in a 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**

**Assignment**

Find the increasing area of the function

$y=-(x+3)^2$

**Solution**

Solve the equation using the shortcut multiplication formula

$y=-x^2-6x-9$

From this, the data we have are:

$a=-1,b=-6,c=9$

Find the vertex using the formula

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

$x=\frac{-\left(-6\right)}{2\cdot\left(-1\right)}$

$x=\frac{6}{-2}$

$x=-3$

Vertex point is $\left(-3,0\right)$

From this we know that: $a<0$

And therefore the function is maximum

The function is increasing in the area of $x<-3$

**Answer**

$x<-3$

Test your knowledge

Question 1

Is the function in the graph decreasing?

Question 2

Is the function shown in the graph below decreasing?

Question 3

Is the function in the graph decreasing?

**Assignment**

Given the linear function in the graph.

When is the function positive?

**Solution**

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

Intersection point with the axis: $x$ is $\left(2,0\right)$

According to the graph the function is positive

therefore $x>2$

**Answer**

$x>2$

**Assignment**

Find the increasing area of the function

$y=-(x-6)^2$

**Solution**

Solve the equation using the shortcut multiplication formula

$y=-x^2+12x-36$

From this, the data we have are:

$a=-1,b=12,c=36$

Find the vertex by the formula

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

$x=\frac{-12}{2\cdot\left(-1\right)}$

$x=\frac{-12}{-2}$

$x=6$

The vertex point is $\left(6,0\right)$

From this we know that: $a<0$

Therefore the function is maximum

The function is increasing in the area of $6<x$

**Answer**

$6<x$

Do you know what the answer is?

Question 1

Is the function in the graph below decreasing?

Question 2

Is the function shown in the graph below decreasing?

Question 3

In what domain does the function increase?

**Assignment**

Find the increasing area of the function

$y=-(2x+6)^2$

**Solution**

Solve the equation using the shortcut multiplication formula

$y=-4x^2-24x-36$

From this, the data we have are:

$a=-4,b=-24,c=-36$

Find the vertex using the formula

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

$x=\frac{-\left(-24\right)}{2\cdot\left(-4\right)}$

$x=\frac{24}{-8}$

$x=-3$

The vertex point $\left(-3,0\right)$

From this we know that $a<0$

Therefore, the function is maximum

The function is increasing from $-3<x$

**Answer**

$-3<x$

**Assignment**

Find the increasing area of the function

$y=(x+3)^2+2x^2$

**Solution**

Solve the equation using the shortcut multiplication formula

$y=x^2+6x+9+2x^2$

$y=3x^2+6x+9$

From this, the data we have are:

$a=3,b=6,c=9$

Find the vertex by the formula

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

$x=\frac{-6}{2\cdot3}$

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

$x=-1$

Now replace $x=-1$ in the given function

$y=3\cdot1-6+9$

$y=3-6+9$

$y=6$

The vertex point is $\left(-1,6\right)$

From this we know that: $a>0$

Therefore, the function is minimum

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

**Answer**

$-1<x$

Check your understanding

Question 1

In what domain is the function negative?

Question 2

In what domain is the function increasing?

Question 3

In what domain does the function increase?

In what domain does the function increase?

Let's remember that the function increases if the X values and Y values increase simultaneously.

On the other hand, the function decreases if the X values increase and the Y values decrease simultaneously.

In the given graph, we notice that the function increases in the domain where x > 0 meaning the Y values are increasing.

x > 0

In what domain is the function negative?

Let's remember that a function is increasing if both X values and Y values are increasing simultaneously.

A function is decreasing if X values are increasing while Y values are decreasing simultaneously.

In the graph, we can see that in the domain x > 1 the function is decreasing, meaning the Y values are decreasing.

x > 1

In what domain is the function increasing?

Let's remember that a function is increasing if both X values and Y values are increasing simultaneously.

A function is decreasing if X values are increasing while Y values are decreasing simultaneously.

In the graph shown, we can see that the function is increasing in every domain, therefore the function is increasing for all X.

Entire$x$

In what domain does the function increase?

Let's remember that the function increases if the X values and Y values increase simultaneously.

On the other hand, the function decreases if the X values increase and the Y values decrease simultaneously.

In the given graph, we notice that the function increases in the domain where x < 0 meaning the Y values are increasing.

x<0

In what interval is the function increasing?

Purple line: $x=0.6$

Let's remember that a function is increasing if both X values and Y values are increasing simultaneously.

A function is decreasing if X values are increasing while Y values are decreasing simultaneously.

In the graph, we can see that in the domain x < 0.6 the function is increasing, meaning the Y values are increasing.

x<0.6

Related Subjects

- Inequalities
- Inequalities with Absolute Value
- Algebraic Method
- Factorization: Common factor extraction
- The Extended Distributive Property
- Coordinate System
- Ordered pair
- Graphs
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- Rate of Change of a Function
- Variation of a Function
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- Representing a Function Verbally and with Tables
- Graphical Representation of a Function
- Algebraic Representation of a Function
- Notation of a Function
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- Absolute Value Inequalities
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- The Linear Function y=mx+b
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