Year 9 Maths The Thai Guy

Lesson 8.4 · 20 min

Linear Functions in Two Variables

Reading the gradient and intercept from , and finding the gradient of the line through two points.

In this lesson you will learn to…

  • Identify the gradient and -intercept from
  • Find the gradient of a line from a graph using rise over run
  • Find the gradient of the line joining two points from their coordinates
  • Explain why a gradient is positive or negative

Why this matters

Straight-line graphs model anything that changes at a steady rate — a taxi fare, a phone contract, water filling a tank. The gradient is the rate (dollars per km, litres per minute) and the intercept is the starting value.

Rise over run

Pick any two points on the line and draw the right-angled triangle between them:

  • the rise is how far up (or down) you go,
  • the run is how far across you go.

gradient=riserun\text{gradient} = \frac{\text{rise}}{\text{run}}

Using coordinates, with points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2):

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Keep the order the same. If you start with the second point on top, start with the second point on the bottom too. Mixing them gives the wrong sign.

This lesson connects directly to Sequences, Functions and Graphs (where you drew these lines) and to the next lesson, where two lines meeting gives the solution of a pair of simultaneous equations.

Key rules and formulas

  • The explicit form

    is the gradient (how steep) and is the -intercept (where the line crosses the -axis). has gradient 2 and crosses at .

  • Gradient

    It does not matter which two points on the line you use — you always get the same gradient.

  • Positive and negative

    Going left to right, a line that goes up has a positive gradient; one that goes down has a negative gradient. A horizontal line has gradient 0.
Try it

Gradient and intercept explorer

Equation
y = 1x + 0
Crosses the y-axis at
(0, 0)
For every 1 across
1 up

Set and to see the line from the first worked example. Then try a negative gradient — the line slopes down from left to right.

Worked examples

Worked example 1

From a graph

A line passes through A(0, 1) and B(2, 5). Find its equation.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

From two coordinates

Find the gradient of the line joining and .

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

A negative gradient

Find the gradient of the line joining and .

Try it yourself first, then reveal the solution one step at a time.

Practice: linear functions

8 questions

Eight questions on gradients and intercepts. Gradients may be fractions — type them like 3/4.
  1. Question 1 of 8Complete the tableFoundation
    Write down the gradient and -intercept of each line.
    LineGradient, my-intercept, c
    y = 2x + 5
    y = −3x + 1
    y = x − 4
    y = 7 − 2x
  2. Question 2 of 8Type the answer
    Find the gradient of the line joining and .
  3. Question 3 of 8Type the answer
    Find the gradient of the line joining and .
  4. Question 4 of 8Explore and match
    Set the sliders so the line passes through and .

    Move the controls until you match the question, then press Check.

    Equation
    y = 1x + 0
    Crosses the y-axis at
    (0, 0)
    For every 1 across
    1 up
  5. Question 5 of 8Click the diagram
    Click the -intercept of this line.

    Click the right part of the diagram. You can also tab to each part and press Enter.

  6. Question 6 of 8Choose all that apply
    Select every line with a negative gradient.

    Select every correct answer — there may be more than one.

  7. Question 7 of 8Algebraic answer
    A line has gradient 3 and passes through . Write its equation. (Type )

    Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.

  8. Question 8 of 8True or false
    Using the points and on the line gives a different gradient from using and .