Gradient/intercepts | Year 11

01 / LEARN

The idea, step by step

Gradient is change per input unit

For a non-vertical line, gradient is vertical change divided by horizontal change. Use the same direction for both differences. A positive gradient means the output increases as the input increases; a negative gradient means it decreases.

The units belong to the relationship. On a cost-versus-distance graph, gradient is dollars per kilometre. A vertical line has no finite gradient because its horizontal change is zero.

m = (y_{2} - y_{1}) / (x_{2} - x_{1})

Intercepts answer different questions

In $y = m x + b$ , the constant $b$ is the value of y at $x = 0$ . This is the y-intercept. For the x-intercept, set $y = 0$ and solve instead. In context, an intercept only has a physical interpretation if the model permits that input.

An equation may need rearranging before its gradient is visible. Divide every term correctly when isolating y; then identify the coefficient of x and the constant.

y = m x + b

A common trap

Reversing the point order in only one of the gradient differences.

Instead: Subtract the coordinates in matching order in the numerator and denominator.

02 / WORKED EXAMPLES

See the thinking

Example 1

Recover a line

Find gradient and y-intercept for a line through $(1, 7)$ and $(5, 19)$ .

$m = (19 - 7) / (5 - 1) = 3$ .
Substitute $(1, 7)$ into $y = 3 x + b$ : $7 = 3 + b$ .
Thus $b = 4$ and $y = 3 x + 4$ .

Gradient 3; y-intercept 4.

Example 2

Read both intercepts

Find gradient and intercepts of $4 x + 2 y = 10$ .

Rearrange: $y = - 2 x + 5$ .
Gradient is $- 2$ and the y-intercept is $(0, 5)$ .
Set $y = 0$ : $4 x = 10$ , so the x-intercept is $(2.5, 0)$ .

Gradient −2; intercepts

(0, 5)

and

(2.5, 0)

Check your understanding

Give gradient and y-intercept of $y = 8 - 3 x$ .

Show answer

Gradient −3; y-intercept 8.

Rewrite as $y = - 3 x + 8$ and compare with $y = m x + b$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find gradient through $(2, 5)$ and $(6, 17)$ .
2 marks
Find gradient through $(1, 11)$ and $(5, 3)$ .
2 marks
Find gradient and y-intercept of $3 x + y = 12$ .
2 marks
Find both intercepts of $y = 2 x - 6$ .
2 marks
A cost graph has gradient 4.5 dollars/km and y-intercept 9 dollars. Interpret both.
2 marks
What is the gradient of $x = 4$ ?
2 marks

Read the worked solutions

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

Find gradient through $(2, 5)$ and $(6, 17)$ .
2 marks
$m = (17 - 5) / (6 - 2) = 12 / 4 = 3$ .
3.
Find gradient through $(1, 11)$ and $(5, 3)$ .
2 marks
$m = (3 - 11) / (5 - 1) = - 8 / 4 = - 2$ .
−2.
Find gradient and y-intercept of $3 x + y = 12$ .
2 marks
Rearrange to $y = - 3 x + 12$ .
Gradient −3; y-intercept 12.
Find both intercepts of $y = 2 x - 6$ .
2 marks
At $x = 0$ , $y = - 6$ .
At $y = 0$ , $2 x = 6$ .
$(0, - 6)$ and $(3, 0)$ .
A cost graph has gradient 4.5 dollars/km and y-intercept 9 dollars. Interpret both.
2 marks
The intercept is cost at zero distance.
The gradient is the charge for one extra kilometre.
A fixed fee of 9 dollars plus 4.50 dollars/km.
What is the gradient of $x = 4$ ?
2 marks
All points have zero horizontal change.
The ratio would require division by zero.
Undefined; no finite gradient.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: algebraic and graphical modelling of linear/exponential change, with interpretation and evaluation.

Official curriculum source ↗

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