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

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

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

So, two important properties are: 

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

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

 

Written Examples

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

 

Question 1

$$y=3x

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

Do:  When $$x=0, $$y=3×0 = $$0 

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

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

Question 2

$$y=4x−7

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

Do: When $$x=0

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

When $$y=0

$$0	$$=	$$4x−7
$$7	$$=	$$4x
$$74​	$$=	$$x

So the $$x intercept is $$74​

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

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

 

Question 3

$$2y−5x−10=0

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

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

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

So the $$y intercept is $$5

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

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

 

So the $$x intercept is $$−2

 

 

Let's have a look at some worked solutions.

Question 4

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

 

Question 5

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

 

Question 6

A line has equation $$−6x+2y−12=0.

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

 

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

 

c) Use these two points to graph the line.