Gradient and intercepts | Year 11

01 / LEARN

The idea, step by step

Gradient measures a rate

For a non-vertical line, gradient is the change in the vertical coordinate divided by the change in the horizontal coordinate. Keep the point order consistent in both differences. A positive gradient rises as x increases; a negative gradient falls. A horizontal line has gradient zero, whereas a vertical line has no defined finite gradient.

The steepness of a drawing depends on axis scales, so calculate gradient from coordinates rather than judging an angle on the page. In a practical model, its units are output units per input unit. A gradient of 3 in a distance-time model means 3 distance units per time unit.

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

Intersections with axes

An x-intercept has y=0 and a y-intercept has x=0. Substitute these conditions separately. In slope-intercept form $y = m x + c$ , the coefficient m is the gradient and c is the y-intercept value, so the y-intercept is the point (0,c).

A line need not meet both axes: a non-zero horizontal line has no x-intercept, and a vertical line away from the y-axis has no y-intercept. State a point when asked for an intercept, not just a number without identifying the axis.

A common trap

Reading the x-intercept as c in $y = m x + c$ .

Instead: c is the output at x=0, so it gives the y-intercept. For the x-intercept, solve $0 = m x + c$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Two points

Find the gradient of the line through $(- 2, 5)$ and $(4, - 7)$ .

$m = (- 7 - 5) / (4 - (- 2)) = - 12 / 6 = - 2$ .
The output decreases by 2 for every increase of 1 in x.

m = - 2

Example 2

Both intercepts

Find the intercepts of $3 x + 2 y = 12$ .

Set y=0: $3 x = 12$ , giving x=4.
Set x=0: $2 y = 12$ , giving y=6.

x-intercept (4,0); y-intercept (0,6).

Check your understanding

What are the gradient and y-intercept of $y = - 4 x + 7$ ?

Show answer

Gradient -4; y-intercept (0,7).

Compare directly with $y = m x + c$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the gradient through $(1, 2)$ and $(5, 10)$ .
2 marks
Find the gradient through $(- 3, 6)$ and $(1, - 2)$ .
2 marks
Find both intercepts of $2 x + 5 y = 20$ .
2 marks
State the gradient of y=8 and the gradient of x=-2.
2 marks
In $C = 15 + 2.4 h$ , where C is dollars and h is hours, interpret the gradient.
2 marks
A line passes through (0,5) and (2,1). Find its x-intercept. Sketch the line and label both axis intercepts.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the gradient through $(1, 2)$ and $(5, 10)$ .
2 marks
$(10 - 2) / (5 - 1) = 8 / 4 = 2$ .
$2$
Find the gradient through $(- 3, 6)$ and $(1, - 2)$ .
2 marks
$(- 2 - 6) / (1 + 3) = - 8 / 4 = - 2$ .
$- 2$
Find both intercepts of $2 x + 5 y = 20$ .
2 marks
For y=0, x=10. For x=0, y=4.
(10,0) and (0,4).
State the gradient of y=8 and the gradient of x=-2.
2 marks
A horizontal line has no vertical change. A vertical line has zero horizontal change, so the ratio is undefined.
0; undefined.
In $C = 15 + 2.4 h$ , where C is dollars and h is hours, interpret the gradient.
2 marks
Each extra hour adds 2.4 dollars.
2.40 dollars per hour.
A line passes through (0,5) and (2,1). Find its x-intercept. Sketch the line and label both axis intercepts.
3 marks
The gradient is (1-5)/2=-2, so y=-2x+5.
Set y=0 to get x=5/2.
(2.5,0).

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Topic 1: Functions and graphs. Prerequisite algebra and coordinate geometry are embedded as supporting lessons.

Official curriculum source ↗

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