EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Linear graphs
Graphs · Lesson 1 of 8
Before We Start
Answer each one, then check.
1. Work out 3 × (−2) + 1.
−5
2. Which axis is horizontal?
The x-axis
3. Write down the coordinates of the origin.
(0, 0)
4. Rearrange y − 4 = 2x to make y the subject.
y = 2x + 4
Learning Objectives
1. Draw a straight-line graph from a table of values.
2. Recognise lines of the form x = a and y = b.
3. Work out the gradient of a line.
4. Use y = mx + c to find the gradient and y-intercept.
Tables of Values
To draw y = 2x + 1, work out y for a few values of x, plot the points and join them with a ruler.
▸ Substitute. When x = −2, y = 2 × (−2) + 1 = −3.
▸ Three points at least. Two points fix a line; a third catches a mistake.
▸ Straight line. If the points are not in a straight line, check your table.
▸ Across the grid. Draw the line through all the points, from edge to edge of the given range.
A Table of Values for y = 2x + 1
|
x |
Working |
y |
|---|---|---|
|
−2 |
2 × (−2) + 1 |
−3 |
|
−1 |
2 × (−1) + 1 |
−1 |
|
0 |
2 × 0 + 1 |
1 |
|
1 |
2 × 1 + 1 |
3 |
|
2 |
2 × 2 + 1 |
5 |
|
3 |
2 × 3 + 1 |
7 |
Horizontal and Vertical Lines
Some lines have only one letter in their equation.
▸ y = b. A horizontal line: every point on it has y-coordinate b. The x-axis is y = 0.
▸ x = a. A vertical line: every point on it has x-coordinate a. The y-axis is x = 0.
▸ y = x. A diagonal line through the origin, where both coordinates are equal.
▸ y = −x. The other diagonal through the origin.
Gradient and Intercept
|
The gradient measures steepness: how much y goes up for every 1 across. Draw a triangle under the line and work out (change in y)/(change in x) = 4/2 = 2. The y-intercept is where the line crosses the y-axis. |
y = 2x + 1: gradient 2, y-intercept 1. |
Gradient
Gradient = (change in y)/(change in x).
▸ Positive. The line goes up from left to right.
▸ Negative. The line goes down from left to right.
▸ Zero. A horizontal line has gradient 0.
▸ From two points. Between (x₁, y₁) and (x₂, y₂), gradient = (y₂ − y₁)/(x₂ − x₁).
Gradient from Two Points
|
Work out the gradient of the line through (1, 3) and (4, 12). |
1. Change in y
12 − 3 = 9
2. Change in x
4 − 1 = 3
3. Divide
9 ÷ 3 = 3
Answer: Gradient 3
y = mx + c
When the equation is written as y = mx + c, you can read the line straight off it.
▸ m. The gradient - the number in front of x.
▸ c. The y-intercept: the line crosses the y-axis at (0, c).
▸ Rearrange first. y must be on its own: 2y = 4x + 6 becomes y = 2x + 3.
▸ Order doesn't matter. y = 5 − 3x has gradient −3 and y-intercept 5.
Finding the Equation from a Graph
|
A straight line crosses the y-axis at (0, −2) and passes through (3, 4). Find its equation. |
1. y-intercept
c = −2
2. Gradient
(4 − (−2))/(3 − 0) = 6/3 = 2
3. Write y = mx + c
y = 2x − 2
Answer: y = 2x − 2
Key Terms
|
Linear Making a straight line when plotted. |
Gradient The steepness of a line: change in y divided by change in x. |
|
y-intercept Where a line crosses the y-axis. |
Coordinates A pair (x, y) giving a point's position. |
|
Origin The point (0, 0) where the axes cross. |
|
Your Task: Line Match-Up
10 minutes
|
Sort these six equations into pairs that share something - same gradient, same intercept or both: y = 3x + 2, y = 2 − x, y = 3x − 5, 2y = 6x + 4, y = −x + 7, y = 2x + 2. Then sketch each line. 1. Rearrange into y = mx + c first. 2. Write down m and c for each. 3. Group them. |
A good answer shows: y = 3x + 2 and 2y = 6x + 4 are the same line (gradient 3, intercept 2). y = 3x − 5 is parallel to them. y = 2 − x and y = −x + 7 are parallel (gradient −1). y = 2x + 2, y = 3x + 2 and y = 2 − x share intercept 2.
Can I...?
☐ Complete a table of values.
☐ Plot and draw a straight-line graph.
☐ Recognise x = a, y = b, y = x and y = −x.
☐ Find the gradient from a graph.
☐ Find the gradient between two points.
☐ Read m and c from y = mx + c.
☐ Rearrange an equation into y = mx + c.
☐ Find the equation of a line from its graph.
Summary
✓ Gradient = (change in y)/(change in x).
✓ y = mx + c: m is the gradient, c the y-intercept.
✓ x = a is vertical; y = b is horizontal.
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EXAM FOCUS The graph shows a straight line. Find the equation of the line. (3 marks) For the gradient, choose two points where the line crosses grid lines exactly, and check the sign: a line going down to the right has a negative gradient. |