M5.2.1 Chapter 1

Exponential functions

Can a number such as 2ˣ ever be zero or negative, and what does its graph look like between the whole numbers?

Your guess

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

Your guess

A Yes. It becomes negative when x is negative.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Yes, but only when x is 0.
You picked B. Read on. At the end, the chapter comes back to your guess.
C No. It is above zero for every x.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Sketch y = 3ˣ and y = (1/2)ˣ on one grid
Explain why aˣ is always positive

You have met 2ˣ for whole numbers x, such as 2⁵ = 32 and 2⁻² = 1/4. Now x may be any real number. The rule f(x) = aˣ is an exponential function. The number a is its base, and x is the power.

The base must be above 0 and not equal to 1. Take a = −2. Then (−2)² = 4 and (−2)³ = −8, but the power 0.5 would need the square root of −2, which is not a real number. Take a = 1. Then 1ˣ = 1 for every x, a flat line and not a new function. So we always write a > 0 and a ≠ 1.

Values of 3ˣ and 0.5ˣ. Each step right multiplies the first column by 3 and the second by 0.5.

Between the whole numbers the rule keeps the laws of powers. A power of 1/2 means a square root, so 2 to the power 0.5 is the square root of 2. A power of 3/2 means a square root, then a cube. For a power such as √2, which has endless digits, we use closer and closer decimals. The values settle on one number, so aˣ has a value for every real x.

As x gets closer to √2, which is about 1.41421, the value of 2ˣ settles at 2.6651 to 4 decimal places. So 2 to the power √2 is about 2.665.

Worked example

y = 4ˣ. Find y when x = 1.5, and when x = −0.5.

A power of 1.5 is 3/2. Take the square root of 4 first: 4 to the power 0.5 is 2.
Then cube it: 4 to the power 1.5 is 2³ = 8.
A negative power means one over the positive power: 4 to the power −0.5 is 1 ÷ 4 to the power 0.5.
So 4 to the power −0.5 is 1 ÷ 2 = 0.5.

Answer: When x = 1.5, y = 8. When x = −0.5, y = 0.5.

Draw it first

The graph of y = 3ˣ

1 Plot the points from the table. On the left they sit very close to the x axis.
2 Join them with one smooth curve. It passes through (0, 1) and rises faster and faster.

The growth curve y = 3ˣ and the decay curve y = 0.5ˣ, which is the same as y = (1/2)ˣ. Both pass through A at (0, 1).

Read the graph. With a above 1 the curve is a growth curve: it rises as x rises. With a between 0 and 1 it is a decay curve: it falls as x rises. In both cases the curve crosses the y axis at (0, 1), because a⁰ = 1. It also gets as close as you like to the x axis on one side, without touching it. A line that a curve approaches but never reaches is an asymptote. Here the asymptote is the x axis, the line y = 0.

The curves y = 2ˣ and y = 0.5ˣ are mirror images in the y axis. P is (−1, 2) and Q is (1, 2).

The two curves in the last picture are mirror images because 0.5ˣ = (2⁻¹)ˣ = 2⁻ˣ. Swapping x for −x reflects a graph in the y axis. In general y = (1/a)ˣ is the reflection of y = aˣ in the y axis. For x above 0 a bigger base gives a steeper curve. For x below 0 the order is reversed, and 3ˣ lies below 2ˣ.

Now the reason aˣ is always positive. For a whole x above 0, aˣ is a product of positive numbers. For x below 0, aˣ is one over a positive number. A fractional power is a positive root, by our definition. For the rest, here is one argument that works for every x. First, aˣ is the square of a to the power x/2, and a square is never negative. Second, aˣ × a⁻ˣ = a⁰ = 1. A product equal to 1 has no zero factor, so aˣ is not 0. A number that is not negative and not zero is above zero. So aˣ > 0 for all real x.

Worked example

Sketch y = 2ˣ − 4. Give where it crosses each axis, its asymptote and its range.

The graph of y = 2ˣ − 4 is the graph of y = 2ˣ moved down 4 units.
Cross the y axis: put x = 0. Then y = 2⁰ − 4 = 1 − 4 = −3. The point is (0, −3).
Cross the x axis: put y = 0. Then 2ˣ − 4 = 0, so 2ˣ = 4. This gives x = 2.
On the far left 2ˣ is close to 0, so y is close to −4. The asymptote is the line y = −4.
Since 2ˣ > 0, y = 2ˣ − 4 is above −4. The range is y > −4.

Answer: y-intercept (0, −3), x-intercept (2, 0), asymptote y = −4, range y > −4.

The graph of y = 2ˣ − 4 with its asymptote, the dashed line y = −4.

See it three ways

With things

A dose of 8 mg of a medicine is in the blood. Each hour half of what is there leaves. After 1 hour 4 mg is left, after 2 hours 2 mg, after 3 hours 1 mg.

As a picture

In words

The dose halves every hour, so it falls fast at first and then slowly. It gets close to 0 but is never exactly 0.

In symbols

D = 8 × 0.5ˣ, where x is the time in hours. So D = 8 when x = 0, D = 4 when x = 1, D = 2 when x = 2 and D = 1 when x = 3.

Try it

Watch 2ˣ fill the gaps

You need: a scientific calculator, paper and a pencil.

Work out 2¹, then 2 to the power 1.5, then 2² on the calculator. Use the power key.
Now work out 2 to the power 1.9 and 2 to the power 2.1.
Write the five results in order of x.

What you will see: The values climb smoothly: 2, 2.83, 3.73, 4, 4.29. There are no gaps and no jumps. The values for 1.9 and 2.1 sit just either side of 4.

If you cannot try it, this is what happens.

A wrong idea many people have

A negative power gives a negative number, so 2⁻³ = −8.

Why it feels right: The minus sign in the power looks like it should turn the answer negative.

The test that breaks it: Use 2⁻¹ = 1 ÷ 2 = 0.5, 2⁻² = 0.25 and 2⁻³ = 0.125. Each step down divides by 2, and dividing a positive number by 2 stays positive.

What is true: A negative power means one over the positive power. It is small, but always above zero. Catch the slip by asking: is this number a product, a root or a reciprocal of positive numbers?

A wrong idea many people have

The graph of a decay curve crosses the x axis somewhere on the right.

Why it feels right: The curve falls and falls, so it seems it must reach 0.

The test that breaks it: Take y = 0.5ˣ. At x = 10 it is about 0.00098, at x = 20 about 0.00000095. Each value is half of one still above 0.

What is true: The x axis is an asymptote: the curve gets as close as you like but never touches it. The range of y = aˣ is y > 0, and the equation aˣ = 0 has no solution.

A wrong idea many people have

y = 2ˣ is the same kind of curve as y = x².

Why it feels right: Both climb quickly, and both have a 2 and an x.

The test that breaks it: Compare x = 3 and x = 10. For x² you get 9 and 100. For 2ˣ you get 8 and 1 024. Now try x = −3: x² gives 9, but 2ˣ gives 0.125.

What is true: In x² the letter is the base and the power is fixed. In 2ˣ the letter is the power and the base is fixed. The curve y = x² is symmetric and touches the x axis at 0. The curve y = 2ˣ is not symmetric and stays above the x axis.

Key words

exponential function: a rule f(x) = aˣ with a above 0 and a ≠ 1, where x is the power.
base: the number a in aˣ, which is multiplied by itself.
growth: the shape of y = aˣ when a is above 1. The values rise as x rises.
decay: the shape of y = aˣ when a is between 0 and 1. The values fall as x rises.
asymptote: a line that a curve gets closer to but never reaches.

Back to your guess

The answer is C: No. It is above zero for every x.

A negative power does not make a negative number. 2⁻³ means 1 ÷ 2³ = 1/8, which is small but above zero. And 2⁰ = 1. So 2ˣ is above zero for every x, and the graph never reaches the x axis.

Keep it

Copy this into your exercise book.

y = aˣ, a > 0, a ≠ 1: passes through (0, 1), always above the x axis. a > 1 grows, 0 < a < 1 decays. The x axis is an asymptote.

Draw: A rising curve and a falling curve crossing at (0, 1), with the x axis dashed.