More linear graphs.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

More linear graphs

Graphs

Lesson 2 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before

Graphs

Lesson 2 of 8

Answer each one, then check.

1

Last lesson: what is the gradient of y = 7 − 2x?

2

Last lesson: gradient between (0, 1) and (2, 9)?

3

Solve 5 = 3 × 2 + c.

4

What is the reciprocal of 4?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before - Answers

Graphs

Lesson 2 of 8

1

Last lesson: what is the gradient of y = 7 − 2x?

−2

2

Last lesson: gradient between (0, 1) and (2, 9)?

4

3

Solve 5 = 3 × 2 + c.

c = −1

4

What is the reciprocal of 4?

¼

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Graphs

Lesson 2 of 8

1

Find the equation of a line from its gradient and one point.

2

Find the equation of a line through two points.

3

Recognise and find parallel lines.

4

Draw a line given in the form ax + by = c.

5

(Higher) Recognise and find perpendicular lines.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Parallel Lines

Graphs

Lesson 2 of 8

Parallel lines have the same gradient.

Same m

y = 3x + 1, y = 3x − 4 and y = 3x are all parallel.

Different c

Parallel lines cross the y-axis at different points.

Rearrange to compare

2y = 6x + 5 is y = 3x + 2.5, so it is parallel to y = 3x + 1.

Same line

If m and c are both the same, it is the same line.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equation from a Point and a Gradient

Graphs

Lesson 2 of 8

Write y = mx + c, then use the point to find c.

1

Gradient

Write down m - given, or from a parallel line, or worked out from two points.

2

Substitute

Put the point's x and y into y = mx + c.

3

Solve

Solve for c.

4

Write

Write the equation in full: y = mx + c with numbers.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Parallel Line through a Point

Graphs

Lesson 2 of 8

Find the equation of the line parallel to y = 3x − 4 that passes through (2, 5).

1

Parallel, so the same gradient

m = 3, so y = 3x + c

2

Substitute (2, 5)

5 = 3 × 2 + c

3

Solve

c = −1

ANSWER

y = 3x − 1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Line through Two Points

Graphs

Lesson 2 of 8

Find the equation of the line through (1, 4) and (3, 10).

1

Gradient

(10 − 4)/(3 − 1) = 6/2 = 3

2

Substitute (1, 4) into y = 3x + c

4 = 3 + c, so c = 1

3

Check with the other point

3 × 3 + 1 = 10 - correct

ANSWER

y = 3x + 1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Drawing 3x + 2y = 12

Graphs

Lesson 2 of 8

Find where 3x + 2y = 12 crosses the axes, and find its gradient.

1

Crosses the y-axis when x = 0

2y = 12, so (0, 6)

2

Crosses the x-axis when y = 0

3x = 12, so (4, 0)

3

Rearrange for the gradient

2y = −3x + 12, so y = −1.5x + 6

ANSWER

Through (0, 6) and (4, 0); gradient −1.5

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

PART TWO · HIGHER

Perpendicular Lines

Lines that meet at right angles.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Parallel and Perpendicular

Graphs

Lesson 2 of 8

Parallel: equal gradients. Perpendicular: gradients multiply to −1.

If one line has gradient m, a line perpendicular to it has gradient −1/m, the negative reciprocal. Here 2 × (−½) = −1.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Perpendicular Gradients

Graphs

Lesson 2 of 8

Flip the gradient and change its sign.

Negative reciprocal

Perpendicular to gradient m is gradient −1/m.

Examples

2 goes to −½; −3 goes to ⅓; 2/5 goes to −5/2.

The test

Two lines are perpendicular if m₁ × m₂ = −1.

Then as before

Substitute the point to find c.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Perpendicular Line through a Point

Graphs

Lesson 2 of 8

Find the equation of the line perpendicular to y = 2x + 3 that passes through (4, 1).

1

Perpendicular gradient

m = −½

2

Substitute (4, 1)

1 = −½ × 4 + c = −2 + c

3

Solve

c = 3

ANSWER

y = −½x + 3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Graphs

Lesson 2 of 8

Parallel

Lines with the same gradient that never meet.

Perpendicular

Lines that meet at a right angle.

Negative reciprocal

−1/m: the gradient of a line perpendicular to one with gradient m.

Intercept

Where a line crosses an axis.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Graphs

Lesson 2 of 8

YOUR TASK

Build a Square

12 minutes

The line y = 2x passes through (0, 0) and (1, 2). Find the equations of three more lines that make a square with it, with (0, 0) and (1, 2) as two neighbouring corners. Check your square on squared paper.

1

Parallel sides share a gradient.

2

Neighbouring sides are perpendicular.

3

Check each corner lies on two lines.

WHAT A GOOD ANSWER SHOWS

Sides through the origin and (1, 2) must be perpendicular to y = 2x: y = −½x and y = −½x + 2.5. The fourth side is parallel to y = 2x through (−2, 1): y = 2x + 5. (The square on the other side gives y = 2x − 5 instead.)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Graphs

Lesson 2 of 8

Find the equation of a line from a point and a gradient.

Find the equation of a line through two points.

Tell whether lines are parallel.

Draw a line in the form ax + by = c.

(Higher) Find a perpendicular gradient.

(Higher) Find the equation of a perpendicular line.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Parallel lines: same gradient.

Find c by substituting a point into y = mx + c.

ax + by = c: set x = 0 then y = 0 to find where it crosses the axes.

(Higher) Perpendicular gradients multiply to −1.

EXAM FOCUS

Find the equation of the line that is parallel to y = 3x − 4 and passes through (2, 5). (3 marks)

Always check your final equation by substituting the point back in. If the y doesn't come out right, you've made a slip finding c.

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