EDEXCEL GCSE MATHS · HIGHER TIER
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Graphs · Lesson 8 of 8
Last Lesson and Before
Answer each one, then check.
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
Learning Objectives
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.
Exponential Graphs
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.
A Table of Values for y = 2 to the power x
|
x |
2^x |
y |
|---|---|---|
|
−2 |
¼ |
0.25 |
|
−1 |
½ |
0.5 |
|
0 |
1 |
1 |
|
1 |
2 |
2 |
|
2 |
4 |
4 |
|
3 |
8 |
8 |
Finding a and b
|
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
Exponential Decay
|
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
Exponentials and a Circle
|
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. |
y = 2^x and y = (½)^x both pass through (0, 1); x² + y² = 25 is a circle of radius 5. |
Circles
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).
PART TWO
Trigonometric Graphs
Sine, cosine and tangent for any angle.
The Trig Graphs
|
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. |
Sine and cosine repeat every 360°; tangent every 180°. |
Key Points of the Trig Graphs
|
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 |
None |
0°, 180°, 360° |
180° |
A Second Solution
|
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°
Key Terms
|
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. |
|
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. |
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.
Can I...?
☐ 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.
Summary
✓ 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. |