M4.10.4 Chapter 4

Exponential graphs

What does the graph of y = 2ˣ look like? It starts small. How fast does it rise?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A A straight line, going up by 2 at each step.
You picked A. Read on. At the end, the chapter comes back to your guess.
B A curve that rises slowly, then levels off.
You picked B. Read on. At the end, the chapter comes back to your guess.
C A curve that rises slowly at first, then faster and faster.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Draw y = 2ˣ from a table of values
Tell a growth curve from a decay curve

Some things double again and again. A patch of algae on a pond covers 1 m² today. It doubles each day, so it covers 2 m² tomorrow, then 4 m², then 8 m². After x days the area is x 2s multiplied together.

We write that as y = 2ˣ. The x is the number of days. It stands in the power, not as a number to multiply by. A rule with the letter in the power is called an exponential function. The number 2 is its base.

The table for y = 2ˣ. Each y value is double the one before.

For x = 0 the table gives 1. Any number except 0, to the power 0, is 1. For negative x, use 2⁻¹ = 1/2 and 2⁻² = 1/4. These are the decimals 0.5 and 0.25.

Worked example

y = 2ˣ. Find y when x = 5, and when x = −2.

When x = 5, multiply five 2s: 2 × 2 × 2 × 2 × 2 = 32.
When x = −2, use the rule for a negative power: 2⁻² = 1/2². First 2² = 4.
So 2⁻² = 1/4 = 0.25.

Answer: When x = 5, y = 32. When x = −2, y = 0.25.

Draw it first

The graph of y = 2ˣ

1 Plot the points of the table. The first ones are very close to the x axis.
2 Join them with one smooth curve. It rises slowly, then faster and faster.

Look at what the graph does. It crosses the y axis at (0, 1), because 2⁰ = 1. It is always above the x axis, because a power of 2 is never 0 or negative. On the left it gets closer and closer to the x axis, which is an asymptote.

Worked example

The algae covers 2ˣ m² after x days. After how many days is the area 64 m²?

We want 2ˣ = 64. Try whole numbers for x.
Try x = 5: 2 × 2 × 2 × 2 × 2 = 32. This is too small.
Try x = 6: 2 × 32 = 64.
So x = 6. The area is 64 m² after 6 days.

Answer: x = 6

Now take a base between 0 and 1, such as 0.5. In y = 0.5ˣ the value is halved each time x goes up by 1. The values 1, 0.5, 0.25, 0.125 fall towards 0. A sample of material whose mass halves every hour follows this rule.

The growth curve y = 2ˣ and the decay curve y = 0.5ˣ. Both pass through A at (0, 1).

A graph y = aˣ with a above 1 is a growth curve. It rises as x increases. With a between 0 and 1 it is a decay curve. It falls as x increases. The base a must be positive. If a = 1, then y = 1 for every x, a flat line that neither grows nor decays.

Worked example

Is y = 0.9ˣ a growth curve or a decay curve?

Work out y for x = 0, 1 and 2. Each step multiplies by 0.9.
At x = 0, y = 1.
At x = 1, y = 1 × 0.9 = 0.9.
At x = 2, y = 0.9 × 0.9 = 0.81.
The values 1, 0.9, 0.81 fall. So y = 0.9ˣ is a decay curve.

Answer: It is a decay curve, because the values 1, 0.9, 0.81 fall.

See it three ways

With things

A patch of algae covers 1 m² on day 0. It doubles each day. On day 1 it covers 2 m², on day 2 it covers 4 m², and on day 3 it covers 8 m².

As a picture

In words

The area doubles each day, so it climbs faster each day. Each day adds more than the day before.

In symbols

y = 2ˣ, so y = 2 when x = 1, y = 4 when x = 2, and y = 8 when x = 3.

Try it

Fold a sheet of paper

You need: a sheet of paper.

Fold the sheet in half. Count the layers.
Fold it in half again, and count the layers. Do this up to 4 folds.
Write the number of folds and the number of layers in a table.

What you will see: The layers double with each fold: 1, 2, 4, 8, 16. After x folds there are 2ˣ layers.

If you cannot try it, this is what happens.

A wrong idea many people have

y = 2ˣ is the same as y = 2x.

Why it feels right: Both have a 2 and an x, and both go up as x gets bigger.

The test that breaks it: Take x = 5. Then 2x = 10, but 2ˣ = 32. Take x = 10. Then 2x = 20, but 2ˣ = 1 024.

What is true: In 2x the x multiplies 2. In 2ˣ the x is a power, so x 2s are multiplied together. The graph of 2x is a straight line, and the graph of 2ˣ curves up ever more steeply.

A wrong idea many people have

A decay curve reaches zero in the end.

Why it feels right: The values fall and fall, so it seems they must run out.

The test that breaks it: Take y = 0.5ˣ. At x = 10 the value is about 0.00098, which is small but above zero. Halving a positive number never gives zero.

What is true: A decay curve gets as close to the x axis as you like, but it never reaches it. The x axis is an asymptote.

Key words

exponential: a rule with the letter in the power, like y = 2ˣ.
growth: an exponential rule y = aˣ with the base a above 1. The values rise as x rises.
decay: an exponential rule y = aˣ with the base a between 0 and 1. The values fall as x rises.

Back to your guess

The answer is C: A curve that rises slowly at first, then faster and faster.

In y = 2ˣ the x is a power, so y doubles each time x goes up by 1. The values 1, 2, 4, 8, 16 climb faster and faster. The graph is a steep curve that never levels off. For negative x it comes very close to the x axis but never touches it.

Keep it

Copy this into your exercise book.

y = aˣ passes through (0, 1) and stays above the x axis. Base above 1: growth, rising faster. Base between 0 and 1: decay, falling towards the axis.

Draw: One graph with a rising curve and a falling curve crossing at (0, 1).