Graphing Straight Lines from Intercepts

Straight lines are lines on the Cartesian Plane that extend forever in both directions.  If we ignore for a moment the special cases of horizontal and vertical lines, straight lines will cross both the $x$x-axis and the $y$y-axis, possibly at the same point (called the origin).

Here are some examples:

The word intercept in mathematics refers to a point where a line, curve or function crosses or intersects with the axes.

• We can have $x$x intercepts: where the line, curve or function crosses the $x$x axis.
• We can have $y$y intercepts: where the line, curve or function crosses the $y$y axis.

Consider what happens as you move up or down along the $y$y-axis. You eventually reach the origin ($\left(0,0\right)$(0,0)) where $y=0$y=0. Now, if you move along the $x$x-axis in either direction, the $y$y value is still $0$0.

Similarly, consider what happens as you move along the $x$x-axis. You eventually reach the origin where $x=0$x=0. Now, if you move along the $y$y-axis in either direction, the $x$x value is still $0$0. 

So, two important properties are: 

• any point on the $x$x-axis will have $y$y value of $0$0
• any point on the $y$y-axis will have $x$x value of $0$0

We can use these properties to calculate or identify $x$x and $y$y intercepts for any line, curve or function.

 

Written Examples

Find the $x$x and $y$y intercepts for the following lines.

 

Question 1

$y=3x$y=3x

Think:  The $x$x intercept occurs when $y=0$y=0.  The $y$y intercept occurs when $x=0$x=0.

Do:  When $x=0$x=0, $y=3\times0$y=3×0 = $0$0 

This means that this line passes through $\left(0,0\right)$(0,0), the origin.  The $x$x and $y$y intercept occur at the same point!

This particular form of a straight line $y=mx$y=mx always passes through the origin.  (Test it out on the applet below)  

Question 2

$y=4x-7$y=4x−7

Think:  The $x$x intercept occurs when $y=0$y=0.  The $y$y intercept occurs when $x=0$x=0.

Do: When $x=0$x=0

$y=4\times0-7$y=4×0−7 = $-7$−7  So the $y$y intercept is $-7$−7

When $y=0$y=0

$0$0	$=$=	$4x-7$4x−7
$7$7	$=$=	$4x$4x
$\frac{7}{4}$74​	$=$=	$x$x

So the $x$x intercept is $\frac{7}{4}$74​

This form of a straight line $y=mx+b$y=mx+b, always has $y$y intercept of $b$b.  

The $x$x intercept is easy to work out after that (substitute $y=0$y=0).

 

Question 3

$2y-5x-10=0$2y−5x−10=0

Think:  The $x$x intercept occurs when $y=0$y=0.  The $y$y intercept occurs when $x=0$x=0.

Do:  When $x=0$x=0, the $5x$5x term disappears.  This leaves us with:

$2y-10$2y−10	$=$=	$0$0
$2y$2y	$=$=	$10$10
$y$y	$=$=	$5$5

So the $y$y intercept is $5$5

When $y=0$y=0, the $2y$2y term disappears.  This leaves us with:

$-5x-10$−5x−10	$=$=	$0$0
$-5x$−5x	$=$=	$10$10
$x$x	$=$=	$-2$−2

 

So the $x$x intercept is $-2$−2

 

 

Let's have a look at some worked solutions.

Question 4

What is the $x$x intercept of the line $-3x+4y=-27$−3x+4y=−27?

 

Question 5

What is the $y$y intercept of the line with equation $-5x+3y=27$−5x+3y=27?

 

Question 6

A line has equation $-6x+2y-12=0$−6x+2y−12=0.

a) Find the y value at the point where $x=0$x=0.

 

b) Find the $x$x value at the point where $y=0$y=0.

 

c) Use these two points to graph the line.