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Study GuideYear 9Level 2 · Standard

Free Year 9 Linear Graphs: Gradient and Intercept

Explains the y = mx + c form of a straight line for Year 9: what the gradient and y-intercept mean, how to calculate the gradient from two points, how to read m and c from an equation (including ones that need rearranging), how to find the equation of a line from a point and a gradient, and how to sketch a line quickly.

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Year
Year 9
Subject
Mathematics
Topic
Linear Relationships & Graphs
Difficulty
Level 2 · Standard
Estimated time
20 minutes
Curriculum
Australian Curriculum
Answers
Not applicable
Format
PDF (A4) + print

Students will practise

  • understanding gradient as rise over run and the y-intercept as the value of y when x = 0
  • calculating the gradient of a line through two points
  • finding the equation of a line from its gradient and a point, or from two points
  • sketching a line from its equation using the intercept and gradient

Curriculum: Australian Curriculum. We show specific outcome codes only where they have been verified against the official curriculum document.

What's next?

Completed: Linear Graphs: Gradient and Intercept

  1. 1Year 9 Factorising and Solving Quadratics Worksheet — Level 1
Study guide

How to use this study guide

  1. Read it together first, pausing at each worked example to try the step before reading the answer.
  2. Attempt the "Check yourself" questions at the end without looking back.
  3. Then practise with a worksheet from the pathway above and finish with the topic test.

Common questions

Who is this study guide for?

Year 9 students (typically ages 14–15) working on linear relationships & graphs. It is pitched at level 2 · standard.

Are the answers included?

This is a study guide, so there is no separate answer sheet; the 'Check yourself' questions include answers.

How long does it take?

About 20 minutes. Short, regular sessions work best: two or three a week beats one long one.

Do I need to sign up to download?

No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.

What should we do next?

Try Year 9 Factorising and Solving Quadratics Worksheet — Level 1.

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Year 9 · Mathematics · Linear Relationships & Graphs

Linear Graphs: Gradient and Intercept

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What you need to know

Every straight line can be written as y = mx + c. The number m is the gradient: how steep the line is, measured as rise ÷ run (how far up for each step across). The number c is the y-intercept: where the line crosses the y-axis, which is the value of y when x = 0.

To find the gradient from two points, subtract the y-values and divide by the difference in the x-values, in the same order: m = (y₂ − y₁) ÷ (x₂ − x₁). A positive gradient rises from left to right, a negative gradient falls, a zero gradient is a horizontal line, and a vertical line has no defined gradient.

Find the gradient of the line through (1, 3) and (4, 9)

  1. Rise: 9 − 3 = 6.
  2. Run: 4 − 1 = 3.
  3. m = 6 ÷ 3 = 2. For every 1 step right, the line goes up 2.

State the gradient and y-intercept of y = −3x + 5, then of 2y = 6x − 8

  1. y = −3x + 5 is already in the form y = mx + c: m = −3, c = 5.
  2. 2y = 6x − 8 is not. Divide every term by 2 to make y the subject: y = 3x − 4.
  3. Now read off: m = 3, c = −4.

Find the equation of the line with gradient 3 that passes through (2, 7)

  1. Start with y = 3x + c.
  2. Substitute the point: 7 = 3 × 2 + c, so 7 = 6 + c and c = 1.
  3. Equation: y = 3x + 1. Check: when x = 2, y = 6 + 1 = 7.

From two points. Find the gradient first, then substitute one of the points to find c. For the line through (0, −2) and (4, 6): m = (6 − (−2)) ÷ (4 − 0) = 8 ÷ 4 = 2. The point (0, −2) is on the y-axis, so c = −2 straight away, and the equation is y = 2x − 2. Check with (4, 6): 2 × 4 − 2 = 6.

Sketch y = x − 1

  1. Plot the y-intercept: (0, −1).
  2. Use the gradient as rise 1, run 2: from (0, −1) move 2 right and 1 up to (2, 0). Plot it.
  3. Rule a straight line through both points and extend it in both directions.
  4. Check with the x-intercept: set y = 0, so 0 = x − 1, giving x = 2. The line crosses the x-axis at (2, 0), which matches.
GradientWhat the line looks likeExample
PositiveRises from left to righty = 2x + 1
NegativeFalls from left to righty = −x + 4
ZeroHorizontaly = 3
UndefinedVerticalx = −2
Common mistake: reading m and c from an equation that is not in the form y = mx + c. In 2y = 6x − 8 the gradient is 3, not 6, and in y = 5 − 2x the gradient is −2, not 5. Rearrange first, then read the number in front of x.
Common mistake: subtracting the coordinates in different orders on the top and bottom. For (2, 5) and (6, 3), m = (3 − 5) ÷ (6 − 2) = −2 ÷ 4 = −. Whichever point you put first on the top must also be first on the bottom.

Check yourself

  1. State the gradient and y-intercept of y = 4x − 7.
  2. Find the gradient of the line through (0, 2) and (3, 11).
  3. Find the gradient of the line through (2, 5) and (6, 3).
  4. Rearrange 3y = 9x + 6 into the form y = mx + c and state m and c.
  5. Find the equation of the line with gradient −2 passing through (0, 4).
  6. Find the x-intercept of y = 2x − 8.

Answers: 1. m = 4, c = −7 2. 3 3. − 4. y = 3x + 2, so m = 3, c = 2 5. y = −2x + 4 6. x = 4 (set y = 0)

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Last reviewed
1 October 2026
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