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# 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.

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.

## 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).

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.