Algebra 2 / Precalculus · Step-by-step guide

Exponential Functions

Graph and model exponential growth and decay, identify rates, and solve practical examples.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

An exponential function has the variable in the exponent, commonly f(x)=abx\displaystyle f(x)=ab^x. The initial value is a, and b is the growth factor: b > 1 gives growth while 0 < b < 1 gives decay.

f(x)=abx\displaystyle f(x)=ab^x
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Factor versus rate

For growth rate r, use b = 1 + r; for decay, use b = 1 − r, with r written as a decimal.

2

Read the initial value

At x = 0, b0=1\displaystyle b^0=1, so f(0) = a.

3

Recognize the asymptote

Without a vertical shift, the graph approaches y = 0 but does not cross it when a is nonzero.

02

See the structure

A picture makes the relationships easier to remember.

The curve passes through its initial value and rises increasingly quickly for a growth factor greater than 1.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Growth

A $500 account grows 6% annually for 3 years.

  1. Use A=500(1.06)3\displaystyle A=500(1.06)^3.
  2. The factor 1.06 represents keeping 100% plus 6%.
  3. Evaluate and round currency at the end.
Answer$595.51
Example 2Decay

A value of 240 decreases 15% per period for 2 periods.

  1. Use factor 1−0.15=0.85\displaystyle 1-0.15=0.85.
  2. A=240(0.85)2\displaystyle A=240(0.85)^2.
  3. Do not subtract 15 twice as a fixed amount.
Answer173.4
Example 3Build a rule

The initial value is 8 and values triple each step.

  1. Initial value gives a = 8.
  2. Tripling gives b = 3.
  3. Insert into abx\displaystyle ab^x.
Answerf(x)=8⋅3x\displaystyle f(x)=8\cdot3^x
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Using the percent rate itself as the growth factor.

  • Treating exponential change as equal additive change.

  • Confusing initial value with the base.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Exponential Functions

Practice growth, decay, tables, graphs, and models.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

For f(x)=3(2)x\displaystyle f(x)=3(2)^x, find f(4)\displaystyle f(4).

Show hint

Substitute 4 for x.

Show worked solution
  1. f(4)=3(24)\displaystyle f(4)=3(2^4).
  2. 24=16\displaystyle 2^4=16.
  3. 3(16)=48\displaystyle 3(16)=48.

Answer: 48\displaystyle 48

2

Identify the initial value and growth factor of y=7(1.4)x\displaystyle y=7(1.4)^x.

Show hint

Compare with abx\displaystyle ab^x.

Show worked solution
  1. The coefficient a is 7.
  2. The base b is 1.4, which is greater than 1.

Answer: Initial value 7; growth factor 1.4.

3

A $500 account grows 6% yearly. Write a model.

Show hint

Growth factor is 1 plus the rate.

Show worked solution
  1. Initial value a=500\displaystyle a=500.
  2. b=1+0.06=1.06\displaystyle b=1+0.06=1.06.

Answer: A(t)=500(1.06)t\displaystyle A(t)=500(1.06)^t

4

A 200 mg dose decays by 25% each hour. Find the amount after 3 hours.

Show hint

The decay factor is 0.75.

Show worked solution
  1. A=200(0.75)3\displaystyle A=200(0.75)^3.
  2. 0.753=0.421875\displaystyle 0.75^3=0.421875.
  3. Multiply by 200.

Answer: 84.375 mg.

5

Solve 2x=32\displaystyle 2^x=32.

Show hint

Write both sides with base 2.

Show worked solution
  1. 32=25\displaystyle 32=2^5.
  2. Equal bases have equal exponents.

Answer: x=5\displaystyle x=5

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

An exponential function has the variable in the exponent, commonly f(x)=abx\displaystyle f(x)=ab^x. The initial value is a, and b is the growth factor: b > 1 gives growth while 0 < b < 1 gives decay.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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