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 3x + 2y = 6.

Answer

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

3x = 6

This gives:

x = 2.

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

2y = 6

This gives:

y = 3.

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

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

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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 (x1, y1) and (x2, y2) 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 (x1, y1) 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`.

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