# Graphs of Linear Functions

## Key Questions

• For a linear function of the form

$f \left(x\right) = a x + b$,

$a$ is the slope, and $b$ is the $y$-intercept.

I hope that this was helpful.

See below

#### Explanation:

$f \left(x\right) = x : \forall x \in \mathbb{R}$

Let's think for a moment about what this means.
"$f$ is function of $x$ that is equal to the value $x$ for all real numbers $x$"

The only way this is possible is if $f \left(x\right)$ is a straight line through the origin with a slope of $1$.

In slope/intercept form: $y = 1 x + 0$

We can visualise $f \left(x\right)$ from the graph below.

graph{x [-10, 10, -5, 5]}

• The easiest way (In my opinion) to graph a linear function is to enter two points, and connect them.

For example, the function:
$f \left(x\right) = 5 x + 3$
First you choose two $x$ value. I will choose $0$ and $1$. Then you enter them in the function one by one:
$f \left(0\right) = 5 \cdot 0 + 3 = 3$
=> There is a point $\left(0 , 3\right)$, that's part of the function.

$f \left(1\right) = 5 \cdot 1 + 3 = 8$
=> There is a point $\left(1 , 8\right)$, that's part of the function.

Since any line can be represented by two points, you can graph a linear function (line) by connecting the two points.

I hope this helped.