Year 9: Linear intercepts

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What are linear intercepts

In year 9, you work a lot with straight line graphs, also called linear graphs. A straight line on a coordinate plane usually crosses the axes. The points where the line crosses the axes are called intercepts.

X intercept: Where the line crosses the x axis (the horizontal axis).
Y intercept: Where the line crosses the y axis (the vertical axis).

We always write intercepts as ordered pairs, like $(3, 0)$ or $(0, - 5)$ .

Reminder about the coordinate plane

The coordinate plane is made of:

The x axis: Horizontal line.
The y axis: Vertical line.
The origin $(0, 0)$ , where the two axes cross.

On the x axis, all points have a y value of $0$ . On the y axis, all points have an x value of $0$ .

Important: This is the key idea:

For the x intercept, $y = 0$ .
For the y intercept, $x = 0$ .

Finding intercepts from a graph

If you already have the graph of a line, finding the intercepts is simple:

Look where the line crosses the x axis. That point is the x intercept.
Look where the line crosses the y axis. That point is the y intercept.

Example 1: Reading intercepts from a graph

Imagine a line that crosses:

The x axis at $(4, 0)$ .
The y axis at $(0, 2)$ .

Then:

The x intercept is (4, 0).
The y intercept is (0, 2).

Example 2: Line through the origin

Suppose a line passes exactly through the origin $(0, 0)$ . It might also cross the axes only at that point.

In that case:

The x intercept is (0, 0).
The y intercept is also (0, 0).

Finding intercepts from an equation

Most of the time in algebra you are given an equation of a line, like $y = 2 x + 3$ . You can find intercepts without drawing the graph.

Finding the y intercept

Remember: On the y axis, $x = 0$ .
Substitute $x = 0$ into the equation.
Solve for $y$ .

Example 3: Y intercept of y = 2x + 3

Equation: $y = 2 x + 3$ .

Set $x = 0$ :

$y = 2 (0) + 3 = 3$

So the y intercept is (0, 3).

Finding the x intercept

Remember: On the x axis, $y = 0$ .
Substitute $y = 0$ into the equation.
Solve for $x$ .

Example 4: X intercept of y = 2x + 3

Equation: $y = 2 x + 3$ .

Set $y = 0$ :

$0 = 2 x + 3$

Solve for x:

$2 x = - 3$
$x = - 3 \div 2 = - 1.5$

So the x intercept is (−1.5, 0).

More worked examples with equations

Example 5: Y = −3x + 6

Y intercept (set x = 0):

$y = - 3 (0) + 6 = 6$

Y intercept is (0, 6).

X intercept (set y = 0):

$0 = - 3 x + 6$

$- 3 x = - 6$
$x = (- 6) \div (- 3) = 2$

X intercept is (2, 0).

Example 6: Y = 0.5x − 4

Y intercept (set x = 0):

$y = 0.5 (0) - 4 = - 4$

Y intercept is (0, −4).

X intercept (set y = 0):

$0 = 0.5 x - 4$

$0.5 x = 4$
$x = 4 \div 0.5 = 8$

X intercept is (8, 0).

Example 7: 2x + 4y = 12

This equation is not in the form y = mx + c, but we can still find intercepts easily.

Y intercept (set x = 0):

$2 (0) + 4 y = 12$
$4 y = 12$
$y = 3$

Y intercept is (0, 3).

X intercept (set y = 0):

$2 x + 4 (0) = 12$
$2 x = 12$
$x = 6$

X intercept is (6, 0).

Special cases horizontal and vertical lines

Example 8: Horizontal line y = −2

This line has the same y value, −2, everywhere.

To find the y intercept, set x = 0. The equation is still y = −2, so the y intercept is (0, −2).
For an x intercept we would need y = 0, but y is always −2, so it never reaches 0.

So there is no x intercept for this line.

Example 9: Vertical line x = 4

This line has the same x value, 4, everywhere.

To find the x intercept, set y = 0. The equation x = 4 is still true, so the x intercept is (4, 0).
For a y intercept we would need x = 0, but x is always 4.

So there is no y intercept for this line.

Summary:

Horizontal line y = k (k not zero): Has a y intercept but no x intercept.
Vertical line x = k (k not zero): Has an x intercept but no y intercept.

Using intercepts to sketch a line quickly

Intercepts are very useful for drawing a straight line without making a big table of values.

Find the x intercept.
Find the y intercept.
Plot both points on the grid.
Draw a straight line through the two points.

Example 10: Sketching 3x + 2y = 12

Step 1: Find the y intercept (x = 0)

$3 (0) + 2 y = 12$
$2 y = 12$
$y = 6$

Y intercept is (0, 6).

Step 2: Find the x intercept (y = 0)

$3 x + 2 (0) = 12$
$3 x = 12$
$x = 4$

X intercept is (4, 0).

Step 3: Plot and join

Plot the points (0, 6) and (4, 0), then draw a straight line through them. That is the graph of $3 x + 2 y = 12$ .

Intercept form of a straight line

Sometimes a line is written in a special “intercept form”:

$x / a + y / b = 1$

In this case:

The x intercept is $(a, 0)$ .
The y intercept is $(0, b)$ .

Example 11: X/5 + y/3 = 1

Comparing to x/a + y/b = 1, we have a = 5 and b = 3.

X intercept is (5, 0).
Y intercept is (0, 3).

Example 12: X/2 + y/−4 = 1

Here a = 2 and b = −4.

X intercept is (2, 0).
Y intercept is (0, −4).

Worded problems with intercepts

Example 13: Phone plan

A phone company charges a connection fee of $20 and then $5 per week. The total cost after x weeks is given by:

$y = 5 x + 20$

Here y is the total cost in dollars.

Y intercept

Set x = 0: $y = 5 (0) + 20 = 20$

Y intercept is (0, 20), which means at week 0 the cost is $20. This is the connection fee.

X intercept

To get an x intercept we would set y = 0. But the total cost can never be 0 dollars if you already pay $20 at the start and then extra each week. So there is no x intercept in this situation.

Example 14: Tank of water

A tank holds 100 litres of water and is being drained at 4 litres per minute. The amount of water left after t minutes is:

$V = 100 - 4 t$

Here V is the volume in litres.

Y intercept (starting amount)

Set t = 0: $V = 100 - 4 (0) = 100$

So the y intercept is (0, 100). At time 0 there are 100 litres.

X intercept (when the tank is empty)

Set V = 0: $0 = 100 - 4 t$
$4 t = 100$
$t = 25$

The x intercept is (25, 0). After 25 minutes the tank is empty.

Summary

The x intercept is where a line crosses the x axis, so y = 0.
The y intercept is where a line crosses the y axis, so x = 0.
To find intercepts from an equation, substitute x = 0 or y = 0 and solve.
Intercepts make it quick to sketch straight lines.
Horizontal and vertical lines behave differently with intercepts.
In real life problems, intercepts often have clear meanings, like starting values or times when something runs out.