Lesson

Here is a quick recap of what we know about straight lines on the coordinate plane:

• The gradient is a measure of the steepness of a line
• An increasing line has a positive gradient
• A decreasing line has a negative gradient
• Gradient can be determined using $\frac{\text{rise }}{\text{run }}$rise run
• A horizontal line has a gradient of zero
• The gradient of a vertical line is undefined
• The $x$x-intercept is the point where the line crosses the $x$x-axis
• The $y$y-intercept is the point where the line crosses the $y$y-axis

## Gradient-intercept form of a straight line

Any straight line on the coordinate plane is defined entirely by its gradient and its $y$y-intercept. These two features are all we need to write an equation for the line.

The variable $m$m is used to represent the gradient, and the variable $c$c is used for the $y$y-intercept.

We can represent the equation of any straight line, except vertical lines, using what is known as the gradient-intercept form of a straight line.

Gradient-intercept form of a straight line

$y=mx+c$y=mx+c

To represent a particular line on the coordinate plane, with it's own unique gradient and $y$y-intercept, we simply replace $m$m and $c$c with their corresponding values.

We can use the applet below to see the effect of varying $m$m and $c$c on both the line and its equation.

 Created with Geogebra

As can be seen from the applet, we can make the following statements about the gradient.

The value of the gradient, $m$m, relates to the line as follows:

• A negative gradient ($m<0$m<0) means the line is decreasing
• A positive gradient ($m>0$m>0) means the line is increasing
• A zero gradient ($m=0$m=0) means the line is horizontal
• The higher the value of $m$m, the steeper the line.

Coefficients and constant terms

In algebra, any number written immediately in front of a variable, is called a coefficient. For example, in the term $3x$3x, the coefficient of $x$x is $3$3. Any number by itself is known as a constant term.

In the gradient-intercept form of a line, $y=mx+c$y=mx+c, the gradient, $m$m, is the coefficient of $x$x, and the $y$y-intercept, $c$c, is a constant term.

## Using the gradient and $y$y-intercept to write the equation of a line

If we know the gradient, $m$m, and $y$y-intercept, $c$c, of a line, we can substitute these values into $y=mx+c$y=mx+c to write the equation of the line.

#### Worked example

##### Example 1

Write the equation of a line that has a gradient of $\frac{3}{4}$34 and a $y$y-intercept of $-2$2.

Solution

Substitute $m=\frac{3}{4}$m=34 and $c=-2$c=2 into $y=mx+c$y=mx+c

 $y$y $=$= $mx+c$mx+c $y$y $=$= $\frac{3}{4}x+\left(-2\right)$34​x+(−2) $y$y $=$= $\frac{3}{4}x-2$34​x−2

#### Practice questions

##### Question 1

Write down the equation of a line which has a gradient of $-3$3 and crosses the $y$y-axis at $1$1.

##### Question 2

State the gradient and $y$y-intercept of the equation $y=-9x-3$y=9x3.

1.  Gradient $\editable{}$ $y$y-intercept $\editable{}$

## Use the gradient and $y$y-intercept to graph a line

To graph a line from an equation, we use the gradient, $m$m, and the $y$y-intercept, $c$c.

We begin by locating the $y$y-intercept as a point on the $y$y-axis.

From this point, we can use the gradient to draw the correct slope of the line, as outlined in the three examples below:

#### Worked example

##### Example 2

Graph the line that has a gradient of $4$4 and a $y$y-intercept of $-1$1.

Solution

By comparing the equation of the line with $y=mx+c$y=mx+c, we see that the gradient is $4$4 and the y-intercept is $-1$1. The gradient ($4=\frac{4}{1}$4=41) tells us that for a 'run' of $1$1, we have a 'rise' of $4$4.

We can now create the graph on a coordinate plane, in a series of steps:

• Locate the $y$y-intercept at $-1$1 on the $y$y-axis, and mark it with a point
• From the $y$y-intercept, we use the value for the gradient to move $1$1 unit to the right and then $4$4 units up. This gives us the location of a second point.
• Draw a straight line between the two points, and extend the line beyond the points, across the entire coordinate plane.

#### Example 3

Graph the line with equation $y=-\frac{2}{3}x$y=23x.

Solution

By comparing the equation of the line with $y=mx+c$y=mx+c, we see that the gradient is $-\frac{2}{3}$23 and the y-intercept is $0$0, meaning the line passes through the origin. The gradient tells us that for a 'run' of $3$3, we have a 'rise' of $-2$2.

We can now create the graph on a coordinate plane, in a series of steps:

• Locate the $y$y-intercept at the origin, $\left(0,0\right)$(0,0), and mark it with a point
• From the $y$y-intercept, we use the value for the gradient to move $3$3 units to the right and then $2$2 units down. This gives us the location of a second point.
• Draw a straight line between the two points, and extend the line beyond the points, across the entire coordinate plane.

#### Practice Question

##### Question 3

Sketch a graph of the linear equation $y=4x+3$y=4x+3.

### Equation of a horizontal line

A horizontal line has a gradient of zero ($m=0$m=0), so the equation of the line becomes:

$y=c$y=c

where $c$c is the $y$y-intercept of the line.

Here are two examples of horizontal lines:

Horizontal lines $y=2$y=2 and $y=-3$y=3

### Equation of a vertical line

A vertical line has an undefined gradient, so we can't use the gradient-intercept form to write its equation. Instead we define a vertical line as having the equation:

$x=b$x=b

where $b$b is the $x$x-intercept of the line.

Here are two examples of vertical lines:

Vertical lines $x=-1$x=1 and $x=4$x=4

#### Practice question

##### Question 4

Write down the equation of the graphed line.