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B. Exponential Functions · Exponential Functions

Graphing y = aˣ and Why It Is a Function

B1.3About 70 minutes
Learning goals
  • I can make a table of values and graph y = aˣ by hand for a > 1 and for 0 < a < 1.
  • I can explain why the base of an exponential function must satisfy a > 0 and a ≠ 1.
  • I can explain why y = aˣ is a function, define it as f(x) = aˣ, and use function notation.
  • I can describe how the graph changes as the base changes, and why y = (1/a)ˣ is the mirror image of y = aˣ.

Words to know

exponential function
A function of the form f(x) = aˣ, where the base a is a constant with a > 0 and a ≠ 1, and the variable x is in the exponent.
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.
function notation
f(x) is read "f of x" and means the output of function f for input x. If f(x) = 2ˣ, then f(3) = 8.
table of values
A list of chosen x-values and the y-values the equation gives for them.

A penny or a million dollars?

Would you rather have $1 000 000 today, or a penny today, 2 pennies tomorrow, 4 the next day, doubling every day for 30 days? On day x you get 0.01 × 2ˣ⁻¹ dollars. On day 30 alone that is 0.01 × 2²⁹ ≈ $5 368 709. The doubling plan wins easily.

Functions in which the variable is in the exponent grow (or shrink) in a way that surprises almost everyone. To understand them, we start by graphing the simplest ones, y = aˣ.

Remember from Grade 10 and the Quadratic Functions unit

In Grade 10 you compared the graphs of y = x² and y = 2ˣ. In the Quadratic Functions unit of this course you learned that a function gives exactly one output for every input, that the vertical line test checks a graph, and that f(x) notation names the output. All of these ideas carry over to exponential functions.

Graphing y = 2ˣ

Make a table of values. Include negative x-values; you know how to evaluate them now.

Notice the pattern: each time x goes up by 1, y is multiplied by 2. Because rational exponents fill in the gaps (2^(1/2) ≈ 1.41, 2^(3/2) ≈ 2.83), we join the points with a smooth curve, not line segments. (Your calculator can even evaluate irrational exponents such as 2^π ≈ 8.82, so every real x has a value.)

x−3−2−10123
y = 2ˣ1/81/41/21248
y = (1/2)ˣ84211/21/41/8
The two rows are the same numbers in the opposite order.
xy−3−2−1123123456780y = 2ˣy = (1/2)ˣboth pass through (0, 1)
y = 2ˣ (solid) rises from left to right. y = (1/2)ˣ (dashed) falls. They are mirror images in the y-axis and both pass through (0, 1).

Bases between 0 and 1

For y = (1/2)ˣ, each step of 1 to the right multiplies y by 1/2, so the graph falls. Since (1/2)ˣ = (2⁻¹)ˣ = 2⁻ˣ, the point (x, y) on y = 2ˣ matches the point (−x, y) on y = (1/2)ˣ. That is a reflection in the y-axis.

  • If a > 1, y = aˣ is increasing (exponential growth).
  • If 0 < a < 1, y = aˣ is decreasing (exponential decay).
  • The bigger a is (above 1), the faster the graph climbs for positive x, and the faster it falls toward 0 for negative x.
xy−2−1123123456780y = 3ˣy = 2ˣy = 1.5ˣy = 1ˣ = 1 (not exponential)
Three growth curves. All pass through (0, 1). For x > 0, y = 3ˣ is above y = 2ˣ, which is above y = 1.5ˣ; for x < 0 the order is reversed. With a base of 1 the "curve" is just the horizontal line y = 1.

Why a > 0 and a ≠ 1?

  • a = 1: 1ˣ = 1 for every x. That is the horizontal line y = 1, a constant function, not growth or decay.
  • a = 0: 0ˣ is 0 for x > 0 but undefined for x = 0 and for negative x (you would divide by 0).
  • a < 0: try y = (−2)ˣ. The whole-number points are 1, −2, 4, −8, 16, … jumping above and below the x-axis. Worse, (−2)^(1/2) = √(−2) is not a real number, so there is no point at x = 1/2. The points cannot be joined into a smooth curve.

So the definition of an exponential function is f(x) = aˣ with a > 0 and a ≠ 1.

01x = 0−2x = 14x = 2−8x = 316x = 4−32x = 5x = 1/2: (−2)^(1/2)= √(−2), not realy = (−2)ˣ for whole-number x
Values of y = (−2)ˣ for x = 0 to 5. They flip between positive and negative, and many values in between (like x = 1/2) are not real numbers. That is why the base must be positive.

Why y = aˣ is a function

For each input x there is exactly one value of aˣ. (For example 2³ is 8, and only 8.) So every vertical line crosses the graph exactly once, and y = aˣ passes the vertical line test. We can therefore name it with function notation:

f(x) = aˣ

If f(x) = 3ˣ, then f(2) = 9, f(0) = 1, f(−1) = 1/3 and f(1/2) = √3 ≈ 1.73.

Worked example

Example 1: Graph y = 3ˣ

Make a table of values for y = 3ˣ from x = −2 to x = 2 and describe the graph.

  1. x = −2: 3⁻² = 1/9 ≈ 0.11. x = −1: 1/3 ≈ 0.33. x = 0: 1. x = 1: 3. x = 2: 9.
  2. Each step right multiplies y by 3, so the graph rises faster and faster.
  3. On the left the y-values get closer and closer to 0 but never reach it (1/9, 1/27, 1/81, …), so the graph hugs the x-axis.
  4. Plot the points and draw a smooth curve through them.

Points (−2, 1/9), (−1, 1/3), (0, 1), (1, 3), (2, 9). An increasing curve through (0, 1) that flattens toward the x-axis on the left.

Worked example

Example 2: Function notation

f(x) = (1/4)ˣ. Find f(2), f(−1), f(1/2) and f(−3/2). Is the point (−2, 16) on the graph?

  1. f(2) = (1/4)² = 1/16.
  2. f(−1) = (1/4)⁻¹ = 4.
  3. f(1/2) = √(1/4) = 1/2.
  4. f(−3/2) = 4^(3/2) = (√4)³ = 8.
  5. f(−2) = (1/4)⁻² = 4² = 16. So yes, (−2, 16) is on the graph.

1/16, 4, 1/2, 8; yes

Worked example

Example 3: Is it an exponential function?

Which of these are exponential functions? (a) y = 5ˣ (b) y = x⁵ (c) y = 1ˣ (d) y = 0.3ˣ (e) y = (−3)ˣ

  1. Ask: is the variable in the exponent, with a constant base a > 0, a ≠ 1?
  2. (a) Yes: base 5. (b) No: the variable is the base, so it is a power function (degree 5).
  3. (c) No: base 1 gives the constant y = 1. (d) Yes: base 0.3 (decay). (e) No: negative base.

Only (a) and (d).

Just for fun

Why did y = 2ˣ never get lost on a hike?

No matter which way it walked, it always passed through the same trailhead: (0, 1).

Hands-on activity

Hand-graph a pair of mirror curves

You need: grid paper (or a ruled page where you draw your own grid), a pencil, a ruler and a calculator.

  1. Make a table for y = 1.5ˣ for x = −4, −3, −2, −1, 0, 1, 2, 3, 4. Round y to two decimal places.
  2. Make a table for y = (2/3)ˣ for the same x-values.
  3. Choose scales: x from −4 to 4, y from 0 to 6. Plot both sets of points and join each set with a smooth curve (no ruler for the curve!).
  4. Fold the paper along the y-axis and hold it up to a window. What do you notice?
  5. Explain the fold using exponent rules.

What to look for: y = 1.5ˣ gives about 0.20, 0.30, 0.44, 0.67, 1, 1.5, 2.25, 3.38, 5.06. y = (2/3)ˣ gives the same numbers in reverse order. The curves match when folded along the y-axis because 2/3 = 1.5⁻¹, so (2/3)ˣ = 1.5⁻ˣ.

Your turn

1

Complete a table for y = 4ˣ for x = −2, −1, 0, 1, 2.

2

Complete a table for y = (1/3)ˣ for x = −2, −1, 0, 1, 2.

3

Does y = 0.8ˣ increase or decrease? Explain.

4

f(x) = 2ˣ. Find f(5) and f(−5).

5

g(x) = 9ˣ. Find g(1/2) and g(−3/2).

6

Which point do y = 7ˣ and y = 0.2ˣ have in common? Why?

7

Explain why y = 2ˣ passes the vertical line test.

More practice: warm-up

1

Is y = x³ an exponential function?

2

Is y = 6ˣ increasing or decreasing?

3

f(x) = 10ˣ. Find f(3).

4

What is the y-value of y = 5ˣ at x = 0?

5

Why is y = 1ˣ not called an exponential function?

More practice: on your own

1

Write y = (1/5)ˣ using base 5 and a negative exponent. Describe how its graph is related to y = 5ˣ.

2

h(x) = (0.25)ˣ. Is the point (1/2, 0.5) on the graph? Show your work.

3

On the same axes, y = 2ˣ and y = 4ˣ. For which x-values is 4ˣ above 2ˣ? Below?

4

Give three reasons y = (−4)ˣ is not used as an exponential function.

5

f(x) = 3ˣ. Find x if f(x) = 81 and if f(x) = 1/27.

Challenge

1

The graph of y = aˣ passes through (3, 64). Find a and then f(−1/2).

2

The graph of y = aˣ passes through (−2, 25). Find a. Is the function increasing or decreasing?

3

For the penny-doubling offer in the hook, the amount on day x is A(x) = 0.01 × 2ˣ⁻¹ dollars. On which day does the daily amount first pass $1000? Show how you know.

Check your understanding

Quiz

1.

Which equation is an exponential function?

2.

Every graph of the form y = aˣ (a > 0, a ≠ 1) passes through which point?

3.

If f(x) = 8ˣ, what is f(2/3)?

4.

Which graph is the reflection of y = 6ˣ in the y-axis?

5.

Why is y = (−3)ˣ not an exponential function?

6.

Why does y = 0.7ˣ pass the vertical line test?

For parents and teachers

Big idea (B1.3): students graph exponential relations y = aˣ (a > 0, a ≠ 1) with and without technology, define the relation as the function f(x) = aˣ, and explain why it is a function.

How to help: Ask your teen to make a quick table (x = −2 to 2) before sketching any exponential graph, and to say what happens to y each time x goes up by 1 ("it doubles", "it is cut to a third"). Free graphing tools such as Desmos or GeoGebra are great for checking hand sketches.

Common difficulties:

  • Joining points with straight segments: exponential graphs are smooth curves.
  • Drawing the curve crossing or touching the x-axis: aˣ is always positive.
  • Confusing y = x² with y = 2ˣ: in an exponential function the variable is the exponent.
  • Negative x-values: 2⁻³ = 1/8, a small positive number. Review the negative-exponent lesson if needed.