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

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

In what interval is the function increasing?

Purple line: \( x=0.6 \)

Let's assume we have two elements $X$, which we will call $X1$ and $X2$, where the following is true: X1<X2, meaning, 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 decreasing when:** **$X2>X1$**** and also** **$Y2<Y1$****.**

The function can be decreasing in intervals or 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 decreasing area of the function

$y=(x+1)+1$

Solution

$a$ coefficient of $x^2$

Therefore $0<a$

is the minimum point

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

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

**Answer**

$x<-1$

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 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**

$x<-4$

**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 $x$

**Answer**

For all $x$

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**

Given the function in the diagram

What are the areas of positivity and negativity of the function?

**Solution**

Let's remember that a function is positive when it is above the axis: $x$ and the function is negative when it is below the axis $x$

Given that the point of intersection with the axis: $x$ is $\left(3.5,0\right)$

When $x>3.5$ it is below: $x$

When $x<3.5$ it is above: $x$

Therefore, the function is positive when $x<3.5$ and negative when $x>3.5$

**Answer**

Positive when $x<3.5$

Negative when $x>3.5$

**Assignment**

Find the increasing and decreasing area of the function

$f(x)=-2x^2+10$

**Solution**

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

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

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

according to the data we know

$a=-2,b=0,c=10$

We replace the data in the formula

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

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

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

$x=0$

Then we know that: $x=0$ and we replace it in the function and find that $y$

$y=10$

**Answer**

$0<x$ Decreasing

$x<0$ Increasing

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
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- Rate of Change of a Function
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- Variable Rate of Change
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- 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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- Representation of Phenomena Using Linear Functions