Gradient/intercepts | Year 11

01 / LEARN

Understand the idea.

Gradient is change per input unit

Gradient measures output change per unit of input. For two distinct points on a non-vertical line, use $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Subtract in the same point order above and below the fraction.

The sign shows direction: positive rises as input increases, negative falls. Units follow the axes, such as dollars per kilometre. A vertical line has zero horizontal change, so this division cannot give a finite gradient.

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

Intercepts answer different questions

The $y$ -intercept is where $x = 0$ ; the $x$ -intercept is where $y = 0$ . In $y = m x + b$ , the intercept value is $b$ , and the corresponding point is $(0, b)$ .

If the equation is not already solved for $y$ , rearrange it first, dividing every term when necessary. Interpret either intercept in context only when its input is allowed by the practical domain.

y = m x + b

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Recover a line

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

Sage line y equals three x plus four is shown from x zero to eight, with marks at (0,4), (1,7), (5,19) and (8,28). The x window is zero to ten and the y window zero to forty, with the original major ticks.

Worked solution

1
Find matching coordinate changes.
$m = \frac{19 - 7}{5 - 1} = \frac{12}{4} = 3$
The output rises twelve while the input rises four.
2
Use one point to find the constant.
$7 = 3 (1) + b$
The point with coordinates one and seven lies on the required line.
3
State and verify the line.
$\begin{aligned} b & = 4 \\ y & = 3 x + 4 \\ 3 (5) + 4 & = 19 \end{aligned}$
The second point also satisfies the rule, whose gradient is three and intercept value four.

Answer

Gradient 3; y-intercept 4.

EXAMPLE 02

Read both intercepts

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

Worked solution

1
Isolate the output.
$\begin{aligned} 2 y & = 10 - 4 x \\ y & = - 2 x + 5 \end{aligned}$
Divide both the constant and the variable term by two.
2
Read the gradient and vertical intercept.
$m = - 2, (0, 5)$
The coefficient of the input is the gradient, and the constant is the output at zero.
3
Set the output to zero.
$\begin{aligned} 4 x + 2 (0) & = 10 \\ x & = \frac{10}{4} = 2.5 \end{aligned}$
A point on the horizontal axis has zero output.
4
State the other intercept.
$(2.5, 0)$
Coordinates are written as input first, then output.

Answer

Gradient −2; intercepts $(0, 5)$ and $(2.5, 0)$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Gradient −3; y-intercept 8.

Write $y = - 3 x + 8$ . Comparing with $y = m x + b$ gives gradient $- 3$ and $y$ -intercept value $8$ , at $(0, 8)$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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
$m = \frac{(17 - 5)}{(6 - 2)} = \frac{12}{4} = 3$ .
3.
Find gradient through $(1, 11)$ and $(5, 3)$ .
2 marks
$m = \frac{(3 - 11)}{(5 - 1)} = - \frac{8}{4} = - 2$ .
$- 2$ .
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 ↗

In this lesson

Understand the idea.

Gradient is change per input unit

Intercepts answer different questions

See the method in action.

Recover a line

Find matching coordinate changes.

Use one point to find the constant.

State and verify the line.

Read both intercepts

Isolate the output.

Read the gradient and vertical intercept.

Set the output to zero.

State the other intercept.

Try it yourself.

Check your understanding

Keep practising