**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.**

In what interval is the function increasing?

Purple line: \( x=0.6 \)

**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

In what domain does the function increase?

Question 2

In what domain is the function negative?

Question 3

In what domain is the function increasing?

**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

In what domain does the function increase?

Question 2

Determine which domain corresponds to the described function:

The function represents the amount of water in a pool while it is being filled.

Question 3

Determine which domain corresponds to the function described below:

The function represents the height of a child from birth to first grade.

**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

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.

Question 2

Which domain corresponds to the described function:

The function represents the velocity of a stone after being dropped from a great height as a function of time.

Question 3

Determine which domain corresponds to the described function:

The function describes a person's energy level throughout the day.

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.

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

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).

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.

Choose the graph that best represents the following:

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

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.

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$(-1)$.

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.

Decreasing

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

Each number is divided by $(-1)$.

The function is:

$f(x)=\frac{x}{-1}$

Let's start by assuming that x equals 0:

$f(0)=\frac{0}{-1}=0$

Now let's assume that x equals 1:

$f(1)=\frac{1}{-1}=-1$

Now let's assume that x equals 2:

$f(-1)=\frac{-1}{-1}=1$

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

We see that we got a decreasing function.

Decreasing

Related Subjects

- Inequalities
- Inequalities with Absolute Value
- Algebraic Method
- Factorization: Common factor extraction
- The Extended Distributive Property
- Coordinate System
- Ordered pair
- Graphs
- Reading Graphs
- Value Table
- Discrete graph
- Continuous Graph
- Domain of a Function
- Indefinite integral
- Inputing Values into a Function
- Rate of Change of a Function
- Variation of a Function
- Rate of change represented with steps in the graph of the function
- Rate of change of a function represented graphically
- Constant Rate of Change
- Variable Rate of Change
- Rate of Change of a Function Represented by a Table of Values
- Ways to represent a function
- Representing a Function Verbally and with Tables
- Graphical Representation of a Function
- Algebraic Representation of a Function
- Notation of a Function
- Absolute Value
- Absolute Value Inequalities
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- Slope in the Function y=mx
- The Linear Function y=mx+b
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- Representation of Phenomena Using Linear Functions