Grade by Grade
B. Exponential Functions · Exponential Functions

The Exponential Function f(x) = aˣ

B1.1About 70 minutes
Learning goals
  • I can make a table of values and graph y = aˣ with and without technology.
  • I can explain why y = aˣ is a function and use the notation f(x) = aˣ.
  • I can explain why the base must be positive and not equal to 1.
  • I can tell an exponential function from a power function such as y = x³.

Words to know

exponential function
A function of the form f(x) = aˣ, where the base a is a positive constant other than 1 and the exponent x is the variable.
base
The number being multiplied by itself. In f(x) = 2ˣ the base is 2.
exponent
The number that tells how many factors of the base to use. In an exponential function the exponent is the variable.
function
A relation in which every input (x-value) has exactly one output (y-value).
vertical line test
A graph is a function if no vertical line crosses it more than once.
mapping diagram
A picture that joins each input to its output with an arrow.

A rule where the variable climbs into the exponent

A streamer posts a video. In the first hour, 2 people share it. In the next hour, each of them gets 2 more people to share it, and so on. Counting the streamer at the start, the number of new sharers goes 1, 2, 4, 8, 16, … Every hour the number doubles.

If x is the number of hours, the number of new sharers is 2ˣ. Look at where the x is: not in the base, but up in the exponent. That small change makes a completely new kind of function, the exponential function. It describes money growing in a bank, populations, cooling cups of hot chocolate, medicine leaving the body and much more. This whole unit is about it.

Remember from Grades 9 and 10

In Grade 9 you learned the exponent laws and found that a⁰ = 1 and a⁻ⁿ = 1/aⁿ. In Grade 10 you compared the graphs of y = x² and y = 2ˣ and saw that a negative exponent gives a reciprocal, never a negative number: 2⁻³ = 1/8. In this unit you will go much further: exponents that are fractions, transformations of exponential graphs, and real-world models.

Building y = 2ˣ from a table

Start with a table of values. Use integers first.

  • 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1
  • 2⁻¹ = 1/2, 2⁻² = 1/4, 2⁻³ = 1/8

Each time x goes up by 1, y is multiplied by 2. Each time x goes down by 1, y is divided by 2. The values get closer and closer to 0 on the left, but they never reach 0 and never become negative.

What about x-values between the integers? A calculator says 2^0.5 ≈ 1.414 and 2^1.5 ≈ 2.828. (On most calculators you press 2, then the ^ or xʸ key, then 0.5.) Every real number x gives a value, so we join the points with a smooth curve, not separate dots. In the next two lessons you will find out exactly what a fractional exponent like 0.5 means.

xy−4−3−2−101234−10123456789(−2, 1/4)(−1, 1/2)(0, 1)(1, 2)(2, 4)(3, 8)y = 2ˣ
The graph of y = 2ˣ. It passes through (0, 1), climbs faster and faster to the right, and flattens toward the x-axis on the left without ever touching it.

Why y = aˣ is a function

A relation is a function if every input has exactly one output. For y = 2ˣ, pick any x. There is only one number 2ˣ, so there is only one y. On the graph, this means any vertical line crosses the curve once. The graph passes the vertical line test.

A mapping diagram shows the same idea: each input has exactly one arrow leaving it.

So we can use function notation: f(x) = aˣ. Read f(3) as "f of 3". It means "the output when the input is 3". For f(x) = 2ˣ, f(3) = 2³ = 8.

x (input)2ˣ (output)−21/4−11/201122438
A mapping diagram for f(x) = 2ˣ. Every input has exactly one arrow, so the relation is a function. (Different inputs also give different outputs, which will matter in Grade 12.)

Why the base must be positive and not 1

The definition says a > 0 and a ≠ 1. Here is why.

  • a = 1: 1ˣ = 1 for every x. The graph is the horizontal line y = 1. That is a constant (linear) function, not an exponential one.
  • a = 0: 0ˣ is 0 when x > 0, but 0⁰ and 0⁻¹ = 1/0 are not defined. Too many holes.
  • a < 0: (−2)ˣ jumps between positive and negative: (−2)² = 4, (−2)³ = −8. Worse, (−2)^0.5 would be the square root of −2, which is not a real number. The "graph" would be scattered points, not a curve.

With a > 0 and a ≠ 1, aˣ is defined for every real x, is always positive, and makes a smooth curve.

Bases between 0 and 1

If 0 < a < 1, the function decreases. For f(x) = (1/2)ˣ: f(0) = 1, f(1) = 1/2, f(2) = 1/4, and f(−2) = (1/2)⁻² = 2² = 4. The graph of y = (1/2)ˣ is the mirror image of y = 2ˣ in the y-axis.

xy−4−3−2−101234−10123456789(0, 1)(2, 4)(−2, 4)y = 2ˣy = (1/2)ˣ
y = 2ˣ (solid) increases. y = (1/2)ˣ (dashed) decreases. They are reflections of each other in the y-axis, and both pass through (0, 1).
Worked example

Example 1: Table of values and graph of y = 3ˣ

Make a table of values for y = 3ˣ for x = −2, −1, 0, 1, 2. Describe the graph and explain why it is a function.

  1. x = −2: 3⁻² = 1/3² = 1/9. x = −1: 3⁻¹ = 1/3. x = 0: 3⁰ = 1.
  2. x = 1: 3¹ = 3. x = 2: 3² = 9.
  3. Plot (−2, 1/9), (−1, 1/3), (0, 1), (1, 3), (2, 9) and join them with a smooth curve. Each step right multiplies y by 3, so the curve rises more steeply than y = 2ˣ for x > 0.
  4. Every x gives exactly one value of 3ˣ, so no vertical line can hit the curve twice: it is a function.

Table: 1/9, 1/3, 1, 3, 9. The curve passes through (0, 1), increases, and approaches the x-axis on the left. It is a function because each x has one y.

Worked example

Example 2: Using function notation

For f(x) = (1/2)ˣ, find f(3), f(0), f(−2) and f(−5).

  1. f(3) = (1/2)³ = 1/8.
  2. f(0) = (1/2)⁰ = 1. (Any non-zero base to the power 0 is 1.)
  3. f(−2) = (1/2)⁻² = (2/1)² = 4. A negative exponent flips the fraction.
  4. f(−5) = (1/2)⁻⁵ = 2⁵ = 32.

f(3) = 1/8, f(0) = 1, f(−2) = 4, f(−5) = 32.

Worked example

Example 3: Which of these is an exponential function?

Decide which are exponential functions of the form f(x) = aˣ: (a) y = 5ˣ (b) y = x⁵ (c) y = 1ˣ (d) y = (0.3)ˣ (e) y = (−4)ˣ.

  1. (a) Base 5 > 0, 5 ≠ 1, variable in the exponent: yes.
  2. (b) The variable is in the base and the exponent is fixed. This is a power function (a polynomial), not exponential.
  3. (c) 1ˣ = 1 always: a horizontal line. Not exponential (a = 1 is excluded).
  4. (d) Base 0.3 is between 0 and 1: yes, a decreasing exponential function.
  5. (e) Negative base: not allowed. For example (−4)^0.5 is not a real number.

Only (a) and (d) are exponential functions.

Just for fun

Why did the exponential function never feel lonely?

Because it always had a strong base to support it, and things only got better from there.

Hands-on activity

Graph-paper investigation: two curves and a ruler

You need: graph paper (or squared paper from a notebook), a pencil, a ruler and a calculator.

  1. Make a table for y = 2ˣ and for y = (1/2)ˣ using x = −3, −2, −1, 0, 1, 2, 3. Use the calculator to add x = 0.5 and x = −0.5 (type 2 ^ 0.5 and 0.5 ^ 0.5).
  2. Draw axes with 1 square = 1 unit across and 1 square = 1 unit up (y from 0 to 9). Plot both tables and join each set of points with a smooth curve.
  3. Hold the ruler straight up and down (vertical) and slide it slowly across the page. Does it ever touch either curve in two places?
  4. Fold the paper along the y-axis and hold it up to a window or lamp. What do you notice about the two curves?

What to look for: 2^0.5 ≈ 1.41 and 0.5^0.5 ≈ 0.71. The ruler always touches each curve exactly once, so both are functions. When folded on the y-axis the curves lie on top of each other: y = (1/2)ˣ is the reflection of y = 2ˣ in the y-axis. Both pass through (0, 1).

Your turn

1

Make a table of values for f(x) = 4ˣ for x = −2, −1, 0, 1, 2.

2

For g(x) = (1/3)ˣ, find g(2), g(0) and g(−3).

3

Is y = x³ an exponential function? Explain.

4

Why is the base a = 1 not allowed in f(x) = aˣ?

5

Explain, using the vertical line test, why y = 5ˣ is a function.

6

Use a calculator to find 3^0.5 and 3^1.5 to three decimal places. Between which two integers does each lie?

7

Which point is on the graph of every function y = aˣ? Why?

More practice: warm-up

1

Evaluate 2⁵.

2

Evaluate 2⁻⁴.

3

Evaluate (1/5)⁻².

4

For f(x) = 10ˣ, find f(3).

5

Does y = 6ˣ increase or decrease as x increases?

More practice: on your own

1

For h(x) = 2ˣ, find h(−1) + h(1).

2

Copy and complete the table for y = 0.1ˣ: x = −2, −1, 0, 1, 2.

3

A relation is given by y = (−3)ˣ. Find y when x = 2 and x = 3, and explain why it is not used as an exponential function.

4

On the graph of y = 2ˣ, between which two consecutive integers is x when y = 20? Explain.

5

Describe how the graphs of y = 3ˣ and y = (1/3)ˣ are related. Give one point on each that shows it.

Challenge

1

For f(x) = aˣ, f(2) = 49. Find a and then f(−1).

2

Show that for f(x) = 2ˣ, f(x + 1) = 2f(x). What does this say about the table of values?

3

Explain why no graph of the form y = aˣ (a > 0, a ≠ 1) can ever cross the x-axis.

Check your understanding

Quiz

1.

Which equation defines an exponential function?

2.

If f(x) = 3ˣ, what is f(−2)?

3.

Every graph y = aˣ passes through which point?

4.

Why is y = 4ˣ a function?

5.

Which base makes y = aˣ a decreasing function?

6.

The graph of y = (1/2)ˣ is the reflection of y = 2ˣ in the:

For parents and teachers

Big idea: Expectation B1.1 asks students to graph y = aˣ (a > 0, a ≠ 1) with and without technology, to name it f(x) = aˣ, and to explain why it is a function.

How to help: Ask your teen to explain the difference between x² and 2ˣ in their own words (the variable moves from the base to the exponent). Have them make a quick table for 2ˣ with negative x-values, saying "divide by 2" each step to the left.

Common difficulties:

  • Thinking a negative exponent makes a negative number: 2⁻³ is 1/8, not −8.
  • Joining plotted points with straight segments. Exponential graphs are smooth curves.
  • Drawing the left side of the curve touching or crossing the x-axis. It only gets closer.
  • Forgetting why the base must be positive and not equal to 1. Ask "what goes wrong with 1ˣ? with (−2)ˣ?"
  • Calculator entry: for 2^0.5 use the ^ or xʸ key, not the × key.