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EDEXCEL GCSE MATHS · HIGHER TIER

More graphs

Graphs

Lesson 8 of 8

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EDEXCEL GCSE MATHS · HIGHER TIER

Last Lesson and Before

Graphs

Lesson 8 of 8

Answer each one, then check.

1

Work out 2⁻¹.

2

Work out 3 × 2⁴.

3

What is sin 30° exactly?

4

A car worth £15 000 loses 12% of its value. What multiplier gives its new value?

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EDEXCEL GCSE MATHS · HIGHER TIER

Last Lesson and Before - Answers

Graphs

Lesson 8 of 8

1

Work out 2⁻¹.

½

2

Work out 3 × 2⁴.

48

3

What is sin 30° exactly?

½

4

A car worth £15 000 loses 12% of its value. What multiplier gives its new value?

0.88

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EDEXCEL GCSE MATHS · HIGHER TIER

Learning Objectives

Graphs

Lesson 8 of 8

1

Recognise, plot and use exponential graphs y = ab^x.

2

Find the values of a and b from points on an exponential graph.

3

Recognise the equation of a circle centred at the origin.

4

Recognise and sketch the graphs of y = sin x, y = cos x and y = tan x.

5

Use trig graphs to find a second solution to an equation.

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EDEXCEL GCSE MATHS · HIGHER TIER

Exponential Graphs

Graphs

Lesson 8 of 8

In y = ab^x, the variable x is the power.

a is the y-intercept

When x = 0, b⁰ = 1, so y = a.

Growth

If b > 1, the graph rises more and more steeply: y = 2^x.

Decay

If 0 < b < 1, it falls towards zero: y = (½)^x.

Never zero

The graph gets closer to the x-axis but never touches it - the x-axis is an asymptote.

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EDEXCEL GCSE MATHS · HIGHER TIER

A Table of Values for y = 2 to the power x

Graphs

Lesson 8 of 8

x

2^x

y

−2

¼

0.25

−1

½

0.5

0

1

2

4

3

8

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EDEXCEL GCSE MATHS · HIGHER TIER

Finding a and b

Graphs

Lesson 8 of 8

The graph of y = ab^x passes through (0, 3) and (2, 12). Find a and b.

1

Use (0, 3): b⁰ = 1

3 = a × 1, so a = 3

2

Use (2, 12)

12 = 3 × b²

3

Solve

b² = 4, so b = 2 (b must be positive)

ANSWER

a = 3, b = 2, so y = 3 × 2^x

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EDEXCEL GCSE MATHS · HIGHER TIER

Exponential Decay

Graphs

Lesson 8 of 8

A car is bought for £15 000. It loses 12% of its value each year. Its value after t years is V = 15 000 × 0.88^t. Work out its value after 3 years.

1

Substitute t = 3

V = 15 000 × 0.88³

2

Calculate

0.88³ = 0.681472

3

Multiply

15 000 × 0.681472 = 10 222.08

ANSWER

£10 222.08

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EDEXCEL GCSE MATHS · HIGHER TIER

Exponentials and a Circle

Graphs

Lesson 8 of 8

y = 2^x and y = (½)^x both pass through (0, 1); x² + y² = 25 is a circle of radius 5.

Every point on a circle centred at the origin is a distance r from the origin. By Pythagoras, x² + y² = r². So x² + y² = 25 has radius √25 = 5, and (3, 4) lies on it because 9 + 16 = 25.

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EDEXCEL GCSE MATHS · HIGHER TIER

Circles

Graphs

Lesson 8 of 8

x² + y² = r² is a circle with centre (0, 0) and radius r.

Radius from the equation

x² + y² = 49 has radius 7.

Equation from the radius

Radius 6 gives x² + y² = 36.

Is a point on it?

Substitute: (4, 5) is on x² + y² = 41 because 16 + 25 = 41.

Where it crosses the axes

At (± r, 0) and (0, ± r).

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EDEXCEL GCSE MATHS · HIGHER TIER

PART TWO

Trigonometric Graphs

Sine, cosine and tangent for any angle.

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EDEXCEL GCSE MATHS · HIGHER TIER

The Trig Graphs

Graphs

Lesson 8 of 8

Sine and cosine repeat every 360°; tangent every 180°.

y = sin x and y = cos x are waves between −1 and 1. The cosine graph is the sine graph shifted 90° to the left. y = tan x has no value at 90° and 270°, where it has asymptotes.

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EDEXCEL GCSE MATHS · HIGHER TIER

Key Points of the Trig Graphs

Graphs

Lesson 8 of 8

Graph

Maximum

Minimum

Crosses the x-axis

Repeats every

y = sin x

1 at 90°

−1 at 270°

0°, 180°, 360°

360°

y = cos x

1 at 0° and 360°

−1 at 180°

90°, 270°

360°

y = tan x

None

0°, 180°, 360°

180°

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EDEXCEL GCSE MATHS · HIGHER TIER

A Second Solution

Graphs

Lesson 8 of 8

Solve sin x = 0.5 for 0° ≤ x ≤ 360°.

1

First solution from the calculator (or exact values)

x = sin ⁻¹(0.5) = 30°

2

The sine graph is symmetrical about x = 90°

180 − 30 = 150

3

Check: sin x is negative from 180° to 360°, so no more solutions

Only two

ANSWER

x = 30° or x = 150°

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EDEXCEL GCSE MATHS · HIGHER TIER

Key Terms

Graphs

Lesson 8 of 8

Exponential

A function where the variable is the power, such as y = 2^x.

Growth and decay

An exponential that increases (b > 1) or decreases (0 < b < 1).

Asymptote

A line a graph approaches but never touches.

Circle equation

x² + y² = r²: centre the origin, radius r.

Period

The interval after which a graph repeats.

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EDEXCEL GCSE MATHS · HIGHER TIER

Graphs

Lesson 8 of 8

YOUR TASK

Bacteria Model

12 minutes

A colony of bacteria starts with 500 cells and doubles every hour. (a) Write a formula for the number N after t hours. (b) Make a table for t = 0 to 5 and sketch the graph. (c) After how many whole hours does it first pass 10 000? (d) Why can't this model be right for ever?

1

Identify a and b.

2

Make the table.

3

Question the model.

WHAT A GOOD ANSWER SHOWS

(a) N = 500 × 2^t (b) 500, 1000, 2000, 4000, 8000, 16 000 (c) after 5 hours (d) The bacteria would run out of space or food; real growth levels off.

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EDEXCEL GCSE MATHS · HIGHER TIER

Can I...?

Graphs

Lesson 8 of 8

Plot an exponential graph.

Tell growth from decay.

Find a and b in y = ab^x.

Use an exponential formula in context.

Find the radius of x² + y² = r².

Check whether a point is on a circle.

Sketch the sine, cosine and tangent graphs.

Find a second solution using a trig graph.

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EDEXCEL GCSE MATHS · HIGHER TIER

Summary & Exam Focus

y = ab^x: a is the y-intercept; b > 1 growth, 0 < b < 1 decay.

x² + y² = r²: circle, centre (0, 0), radius r.

Sine and cosine: waves between −1 and 1, repeating every 360°. Tangent repeats every 180°.

Use symmetry of the graph to find a second solution.

EXAM FOCUS

The graph of y = ab^x passes through (0, 3) and (2, 12). Find the values of a and b. (3 marks)

For y = ab^x, always use the point with x = 0 first - it gives a immediately, because b⁰ = 1.

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