Year 9 Maths The Thai Guy

Lesson 12.3 · 28 min

Linear Functions

Straight-line graphs, what the gradient and the intercept mean, and drawing a line from its equation.

In this lesson you will learn to…

  • Draw the graph of a linear function from a table of values
  • Identify the gradient and the y-intercept from an equation
  • Write the equation of a line from its graph
  • Decide whether a point lies on a given line

Why straight lines matter

Any relationship where something changes by the same amount every time draws a straight line. A taxi that charges $3 to start and $2 per kilometre; a phone plan with a monthly fee plus a rate per gigabyte; a tank draining at a steady rate. All of them are y=mx+cy = mx + c in disguise.

What mm and cc actually do

Use the explorer above while you read this — it is much faster than taking it on trust.

cc moves the line up and down. Change cc and the line slides vertically without tilting. It is the value of yy when x=0x = 0, which is exactly where the line meets the vertical axis.

mm tilts the line.

mmWhat the line does
Large positive, e.g. 55Rises steeply left to right
Small positive, e.g. 0.50.5Rises gently
ZeroFlat — a horizontal line y=cy = c
Negative, e.g. −3-3Falls left to right

Drawing a line from its equation

  1. Make a table with a handful of xx values, including 00 and some negatives.
  2. Substitute each one into the equation. Take your time with negatives.
  3. Plot the points.
  4. Join them with a ruler and extend the line to the edges of the grid.
  5. Check it looks right — does it cross the yy-axis at cc? Does its steepness match mm?

If one point is out of line with the others, you have made an arithmetic slip, not a discovery. Go back and check that value.

Finding the equation from a graph

Read off cc first — it is free, you just look at where the line crosses the yy-axis.

Then find mm by picking two points where the line passes exactly through grid corners and counting:

m=how far uphow far acrossm = \frac{\text{how far up}}{\text{how far across}}

Counting between two corners avoids estimating, which is where errors creep in. If the line goes down as you move right, mm is negative — write the minus sign immediately, before you forget.

The quickest check of all. Once you have an equation, test it on a point you did not use to find it. If it works, you are right; if it does not, you have the gradient or the intercept wrong.

Key rules and formulas

  • The equation of a straight line

    is the gradient — how steep the line is. is the y-intercept — where the line crosses the -axis.

  • Gradient

    Pick two points on the line, ideally where it crosses grid corners, and count across then up. Going down as you move right means a negative gradient.

  • Reading the intercept

    The -intercept is the value of when . On a graph it is simply where the line crosses the vertical axis — no calculation needed.

  • Is a point on the line?

    Substitute the coordinates into the equation. If both sides come out equal, the point is on the line. If not, it is not — no matter how close it looks.
Try it

Gradient and intercept explorer

Equation
y = 2x − 2
Crosses the y-axis at
(0, -2)
For every 1 across
2 up

Drag and watch the steepness change; drag and watch the whole line slide up and down without changing its slope. Try making negative — the line tips the other way.

Worked examples

Worked example 1

Drawing

Draw the graph of for from to .

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

Worked example 2

Finding the equation from a graph

A line passes through and . Find its equation.

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

Worked example 3

Is the point on the line?

Does the point lie on the line ?

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

Practice: linear functions

10 questions

Ten questions, including two where you drag sliders to match a line and two where you click the right part of a graph.
  1. Question 1 of 10Multiple choiceFoundation
    What is the gradient of the line ?
  2. Question 2 of 10Type the answerFoundation
    At what value of does the line cross the -axis?
  3. Question 3 of 10Complete the table
    Complete the table of values for .
  4. Question 4 of 10Explore and match
    Move the sliders so the red line matches .

    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 10Click the diagram
    On this graph, click the point where the line crosses the -axis.

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

  6. Question 6 of 10Click the diagram
    On the same graph, click the dashed step that shows the gradient — the rise for one unit across.

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

  7. Question 7 of 10Algebraic answer
    A line crosses the -axis at and passes through . Write its equation. Start your answer with y =.

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

  8. Question 8 of 10True or false
    The point lies on the line .
  9. Question 9 of 10Choose all that applyChallenge
    Select every line below that is parallel to .

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

  10. Question 10 of 10Explore and match
    Move the sliders to make a line that falls as you move right and crosses the -axis at . Use a gradient of .

    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