Linear graphs.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Linear graphs

Graphs

Lesson 1 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Before We Start

Graphs

Lesson 1 of 8

Answer each one, then check.

1

Work out 3 × (−2) + 1.

2

Which axis is horizontal?

3

Write down the coordinates of the origin.

4

Rearrange y − 4 = 2x to make y the subject.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Before We Start - Answers

Graphs

Lesson 1 of 8

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

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Graphs

Lesson 1 of 8

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Tables of Values

Graphs

Lesson 1 of 8

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Table of Values for y = 2x + 1

Graphs

Lesson 1 of 8

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

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Horizontal and Vertical Lines

Graphs

Lesson 1 of 8

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Gradient and Intercept

Graphs

Lesson 1 of 8

y = 2x + 1: gradient 2, y-intercept 1.

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Gradient

Graphs

Lesson 1 of 8

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).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Gradient from Two Points

Graphs

Lesson 1 of 8

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

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

y = mx + c

Graphs

Lesson 1 of 8

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Finding the Equation from a Graph

Graphs

Lesson 1 of 8

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

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Graphs

Lesson 1 of 8

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Graphs

Lesson 1 of 8

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.

WHAT 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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Graphs

Lesson 1 of 8

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

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.

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.

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