Gradient and intercepts | Year 11

01 / LEARN

Understand the idea.

Gradient measures a rate

For a non-vertical line, gradient is vertical change divided by horizontal change. Subtract coordinates in the same point order in numerator and denominator. Its sign records whether output rises or falls as the input increases.

A horizontal line has zero gradient; a vertical line has undefined gradient because its horizontal change is zero. Calculate from coordinates, since unequal axis scales can distort apparent steepness. In a model, gradient has units of output per unit of input.

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

Intersections with axes

An intercept is a point where the graph meets an axis. On the horizontal axis $y = 0$ ; on the vertical axis $x = 0$ . Substitute these conditions separately and report the resulting coordinate pairs.

In $y = m x + c$ , the gradient is $m$ and the vertical-axis intercept is $(0, c)$ . A line may lack one intercept: for example, $y = 3$ never reaches the horizontal axis. Do not assume every line crosses both axes once.

The straight line y=2x+1 on a coordinate grid from -5 to 5 on each axis, with equal physical scale per x and y unit. The visible segment runs from (-3,-5) to (2,5).

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Two points

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

Worked solution

1
Calculate changes in a consistent order.
$\begin{aligned} Δ x & = 4 - (- 2) = 6 \\ Δ y & = - 7 - 5 = - 12 \end{aligned}$
Both differences describe movement from the first point to the second.
2
Divide vertical change by horizontal change.
$m = \frac{Δ y}{Δ x} = \frac{- 12}{6} = - 2$
Six units to the right correspond to twelve units downward.
3
Interpret the rate.
$Δ x = 1 ⟹ Δ y = - 2$
Along this line, each unit increase in input decreases the output by two.

Answer

$m = - 2$

EXAMPLE 02

Both intercepts

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

Worked solution

1
Set the vertical coordinate to zero.
$\begin{aligned} 3 x + 2 (0) & = 12 \\ x & = 4 \end{aligned}$
Every point on the horizontal axis has zero y-coordinate.
2
State the horizontal-axis intercept.
$(4, 0)$
An intercept is a coordinate pair identifying the point on the axis.
3
Set the horizontal coordinate to zero.
$\begin{aligned} 3 (0) + 2 y & = 12 \\ y & = 6 \end{aligned}$
Every point on the vertical axis has zero x-coordinate.
4
State the vertical-axis intercept.
$(0, 6)$
The calculated output belongs to the input zero.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

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

Comparing $y = - 4 x + 7$ with $y = m x + c$ gives $m = - 4$ and $c = 7$ . The vertical-axis intercept is the point $(0, 7)$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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
$\frac{10 - 2}{5 - 1} = \frac{8}{4} = 2$ .
$2$
Find the gradient through $(- 3, 6)$ and $(1, - 2)$ .
2 marks
$\frac{- 2 - 6}{1 + 3} = \frac{- 8}{4} = - 2$ .
$- 2$
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 $\frac{1 - 5}{2} = - 2$ , so $y = - 2 x + 5$ .
Set $y = 0$ to get $x = \frac{5}{2}$ .
$(2.5, 0)$ .

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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 ↗

In this lesson

Understand the idea.

Gradient measures a rate

Intersections with axes

See the method in action.

Two points

Calculate changes in a consistent order.

Divide vertical change by horizontal change.

Interpret the rate.

Both intercepts

Set the vertical coordinate to zero.

State the horizontal-axis intercept.

Set the horizontal coordinate to zero.

State the vertical-axis intercept.

Try it yourself.

Check your understanding

Keep practising