# Increasing functions

🏆Practice increasing and decreasing intervals of a function

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.

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

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, 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.

## Increasing Function Exercises

### Exercise 1

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 $x<-3$

$x<-3$

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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: $x$

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

According to the graph the function is positive

therefore $x>2$

$x>2$

### Exercise 3

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

$6

Do you know what the answer is?

### Exercise 4

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

$-3

### Exercise 5

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

$-1

## examples with solutions for increasing functions

### 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

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.

Decreasing

### Exercise #5

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 $(-1)$.

### Step-by-Step Solution

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.