Year 10: Gradient and line transformations

The gradient of a line tells you how steep it is and whether it rises or falls from left to right. In Australian Year 10 Mathematics, you also start looking at how lines shift when constants in their equations change. Those shifts are line transformations.

y = m x + c

, the gradient is

m

and the vertical shift is controlled by

c

Reading and calculating gradient

Gradient can be found as $rise / run$ between two points. If the equation is already written as $y = m x + c$ , the gradient is simply the coefficient of $x$ .

Example 1: Find the gradient of the line through

(1, 2)

and

(5, 10)

Rise = $10 - 2 = 8$ .
Run = $5 - 1 = 4$ .
Gradient = $8 / 4 = 2$ .

Example 2: State the gradient of

y = - 3 x + 7

The coefficient of $x$ is $- 3$ , so the gradient is $- 3$ .

Understanding line transformations

If the gradient stays the same but the y-intercept changes, the line moves up or down and stays parallel to the original. If the gradient changes, the steepness or direction changes as well.

Example 3: Compare

y = 2 x + 1

and

y = 2 x - 4

Both lines have gradient $2$ , so they are parallel.
The second line has a smaller y-intercept, so it is shifted downward.

Using gradient and intercept together

In many Year 10 questions, you combine the gradient with the y-intercept to describe or sketch a line. Once you know how steep it is and where it crosses the y-axis, you already know a lot about the graph.

Example 4: Describe the graph of

y = - 1 / 2 x + 3

It falls from left to right because the gradient is negative.
It is not steep because the gradient magnitude is only $1 / 2$ .
It crosses the y-axis at $(0, 3)$ .

Common mistakes

Mixing up rise and run when calculating gradient.
Confusing the gradient $m$ with the y-intercept $c$ .
Thinking a vertical shift changes the gradient.
Forgetting that parallel lines have the same gradient.

Study routine