# 2. Graphs of Linear Functions

It is very important for many math topics to know how to quickly sketch straight lines. When we use math to **model** real-world problems, it is worthwhile to have a sense of how straight lines "work" and what they look like.

We met this topic before in The Straight Line. The following section serves as a reminder for you.

## a. Slope-Intercept Form of a Straight Line: `y = mx + c`

If the slope (also known as gradient) of a line is *m*, and the *y*-intercept is *c*, then the equation of the line is written:

`y = mx + c`

### Example 1

The line `y = 2x + 6` has slope `m = 6/3 = 2` and `y`-intercept `c = 6`.

Graph of the linear equation `y = 2x+6`.

## b. Intercept Form of a Straight Line: `ax + by = c`

Often a straight line is written in the form *ax *+* by *=* c*. One way we can sketch this is by finding the *x*- and *y*-intercepts and then joining those intercepts.

### Example 2

Sketch the line 3*x* + 2*y* = 6.

Answer

The *x*-intercept (that is, when `y = 0`) is:

3

x= 6

This gives:

x= 2.

The *y*-intercept (that is, when `x = 0`) is:

2

y= 6

This gives:

y= 3.

Joining the intercepts `(2, 0)` and `(0, 3)` gives the graph of the straight line 3*x* + 2*y* = 6:

Graph of the linear equation `3x + 2y = 6`.

## Slope of a Line

The slope (or gradient) of a straight line is given by:

`m=text(vertical rise)/text(horizontal run)`

We can also write the slope of the straight line passing through the points (*x*_{1},* y*_{1}) and (*x*_{2},* y*_{2}) as:

`m=(y_2-y_1)/(x_2-x_1)`

Using this expression for slope, we can derive the following.

## c. Point-slope Form of a Straight Line: `y − y_1= m(x − x_1)`

If a line passes through the point (*x*_{1},* y*_{1}) and has slope *m*, then the equation of the line is given by:

`y − y_1= m(x − x_1)`

### Example 3

Find the equation of the line with slope `−3`, and which passes through `(2, −4)`.

Answer

Here,

`m = −3`,`x_1= 2`, and

`y_1= −4`.

So using the formula `y − y_1= m(x − x_1)`, the equation is

y− (−4) = −3(x− 2)

y+ 4 =−3x+ 6

y=−3x+ 2

The graph is:

Graph of the linear equation `y = -3x+2`.