# Linear Function

🏆Practice linear function

A linear function, as it is called, is an algebraic expression that represents the graph of a straight line.

When we talk about functions, it's important to highlight that the graphs of functions are represented in an axis system where there is a horizontal axis $X$ and a vertical axis $Y$.

Linear functions can be expressed by the expressions $y = mx$ or $y = mx + b$, where m represents the slope of the line while $b$ (when it exists) represents the y-intercept.

To plot a linear function, all we need are $2$ points. If the linear function is given, you can substitute a value for $X$ and obtain the corresponding $Y$ value.

## Test yourself on linear function!

Look at the linear function represented in the diagram.

When is the function positive?

## Linear Function

A linear function is from the family of functions $Y=mx+b$ and represents a straight line.

### $m$

We mark the slope of the function: positive or negative.

$m>0$ : upward sloping line

$m<0$  : downward sloping line

$m=0$ : a line parallel to the $X$ axis.

### $b$

Marks the point where the function intersects the $Y$ axis.

Important points:

For every value of $X$, the function will return a value of $Y$.

The linear function is a straight line that is either ascending, descending, or parallel to the $X$ axis, but never parallel to the $Y$ axis.

Let's observe the line from left to right.

### Different Representations of Linear Functions

$Y=mx+b$ : standard representation

$Y=mx$ : (when $b=0$, the line cuts the $Y$ axis at the origin, when $Y=0$)

$Y=b$ (when $m=0$, the slope of the line is $0$ and therefore parallel to the $X$ axis)

Pay attention:

Sometimes, you'll find an equation that is not arranged and looks like this $mx-y=b$

This is also a linear function. Simply isolate $Y$ and see how it comes to the standard representation.

Sometimes, $X$ will be divided by some number $C$:

$Y=\frac{mx}{c}+b$

This equation also represents a linear function.

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### When is a function not a linear function?

• When $X$ is raised to a power: $Y=X^2$
• When there is a square root for: $Y=\sqrt{X}​$
• When there is no $Y: X=b$

### Graphical Representation of a Function that Depicts a Straight-Line Relationship

The linear function will appear as a straight line that is ascending, descending, or parallel to the $X$ axis but never parallel to the $Y$ axis.

For each value of $X$, we will obtain only one value of $Y$ and vice versa.

We observe the line from left to right to see whether it ascends or descends.

The concept of slope in the function $y=mx$

$M$ represents the slope of the function for us and determines whether the straight line goes up, goes down, or is parallel to the $X$ axis.

### The linear function$y=mx+b$

The linear function represents a straight line where $M$ represents the slope and $b$ represents the y-intercept of the line with the $Y$ axis.

Do you know what the answer is?

### Finding the Equation of a Straight Line

We can find the equation of a straight line in 5 ways:

• Using slope and a point
• With the help of 2 points the line passes through.
• Using parallel lines.
• With the help of perpendicular lines.
• With the help of a graph.

### Positivity and Negativity of a Function

The function is positive when it is above the $X$ axis when $Y>0$

Find the values of $X$ for which the function takes on positive Y values.

The function is negative when it is below the $X$ axis as $Y>0$

Find the values of $X$ for which the function takes on negative $Y$ values.

### Representation of Phenomena Using Linear Functions

A linear function describes the relationship between $X$ and $Y$.

Therefore, we can represent different phenomena with the help of a linear function.

We will understand what graph represents each situation and draw the correct conclusions.

### Inequality

We can solve inequalities between linear functions in two ways:

Using the given equations and with the help of graphical representations.

Do you think you will be able to solve it?

### Let's illustrate this with an example.

Given the function: $y = 2x + 1$

We are asked to graph it on the coordinate system.

As we have discussed, to do this we need two points, which we will place in the function's expression. Choose any two points we like, it doesn't matter.

Now we will plot the two points on the coordinate system and connect them. This is actually a graph of the function for $y=2x+1$.

## Examples and Exercises with Solutions for Linear Functions

### Exercise #1

Look at the function shown in the figure.

When is the function positive?

### Step-by-Step Solution

The function we see is a decreasing function,

Because as X increases, the value of Y decreases, creating the slope of the function.

We know that this function intersects the X-axis at the point x=-4

Therefore, we can understand that up to -4, the values of Y are greater than 0, and after -4, the values of Y are less than zero.

Therefore, the function will be positive only when

X < -4

-4 > x

### Exercise #2

Solve the following inequality:

5x+8<9

### Step-by-Step Solution

This is an inequality problem. The inequality is actually an exercise we solve in a completely normal way, except in the case that we multiply or divide by negative.

Let's start by moving the sections:

5X+8<9

5X<9-8

5X<1

We divide by 5:

X<1/5

And this is the solution!

x<\frac{1}{5}

### Exercise #3

Solve the inequality:

5-3x>-10

### Step-by-Step Solution

Inequality equations will be solved like a regular equation, except for one rule:

If we multiply the entire equation by a negative, we will reverse the inequality sign.

We start by moving the sections, so that one side has the variables and the other does not:

-3x>-10-5

-3x>-15

Divide by 3

-x>-5

Divide by negative 1 (to get rid of the negative) and remember to reverse the sign of the equation.

x<5

5 > x

### Exercise #4

What is the solution to the following inequality?

$10x-4≤-3x-8$

### Step-by-Step Solution

In the exercise, we have an inequality equation.

We treat the inequality as an equation with the sign -=,

And we only refer to it if we need to multiply or divide by 0.

$10x-4 ≤ -3x-8$

We start by organizing the sections:

$10x+3x-4 ≤ -8$

$13x-4 ≤ -8$

$13x ≤ -4$

Divide by 13 to isolate the X

$x≤-\frac{4}{13}$

Answer A is with different data and therefore was rejected.

Answer C shows a case where X is greater than$-\frac{4}{13}$, although we know it is small, so it is rejected.

Answer D shows a case (according to the white circle) where X is not equal to$-\frac{4}{13}$, and only smaller than it. We know it must be large and equal, so this answer is rejected.

Therefore, answer B is the correct one!

### Exercise #5

Given the function of the graph.

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

### Step-by-Step Solution

When we are asked what the domains of positivity of the function are, we are actually being asked at what values of X the function is positive: it is above the X-axis.

At what values of X does the function obtain positive Y values?

In the given graph, we observe that the function is above the X-axis before the point X=7, and below the line after this point. That is, the function is positive when X>7 and negative when X<7,

And this is the solution!