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B. Exponential Functions · Exponential Functions

Exponential Functions: Unit Review

B1.1 B1.2 B1.3 B1.4 B1.5 B1.6 B2.1 B2.2 B2.3About 40 minutes
Learning goals
  • I can simplify and evaluate expressions with integer and rational exponents and rational bases.
  • I can graph exponential functions and describe their key properties.
  • I can tell exponential functions apart from linear and quadratic functions.
  • I can build exponential models from data and descriptions and use them to solve problems.

Words to know

exponent rules
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ.
rational exponent
x^(m/n) = (ⁿ√x)ᵐ for x > 0; a negative exponent means reciprocal.
exponential function
f(x) = aˣ (or y = a(b)ˣ in models), with a positive base that is not 1.
asymptote
A line the graph approaches but never reaches; y = 0 for y = aˣ.
growth / decay factor
b = 1 + r for growth and b = 1 − r for decay, where r is the percent rate as a decimal.
half-life / doubling time
The time for an amount to halve or to double.

Let's review!

In this unit you made the exponent rules automatic, gave meaning to zero, negative and rational exponents, and evaluated expressions such as (2/3)⁻², 27^(2/3) and 1.01¹²⁰. You graphed exponential functions f(x) = aˣ, explained why they are functions, described their domain, range, intercepts and asymptotes, and told them apart from linear and quadratic functions. Finally you collected data, built models y = a(b)ˣ, recognized growth and decay with their real restrictions, and solved problems about populations, interest, half-life, depreciation, bouncing balls and cooling.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time, without looking at your notes first. Use a scientific calculator, but estimate before you press = ("Is the answer bigger or smaller than the base?"). Keep exact fractions where the question uses fractions, and round only final answers. For every question you get wrong, write down why (exponent rule? reciprocal? root vs power? ratio vs difference? growth factor 1 + r?) and go back to that lesson's worked examples. Finish with the 10-question quiz.

Powersexponent ruleszero, negative andrational exponentsFunctionsf(x) = aˣ, a > 0, a ≠ 1domain, range, asymptotevs linear and quadraticModelsdata and ratiosgrowth and decaysolving problemsLessons 1–5Lessons 6–8Lessons 9–12
The unit in three steps: powers, then exponential functions, then models of real situations.

The Exponent Rules: Multiplying, Dividing and a Power of a Power

1

Simplify, then evaluate: (2⁵ × 2³) ÷ (2²)³

2

Simplify, then evaluate: (1/3)⁵ ÷ (1/3)³

3

Is 4³ × 4² equal to 16⁵? Explain.

Zero and Negative Exponents with Fraction, Decimal and Negative Bases

1

Evaluate: (3/5)⁻²

2

Evaluate: −2⁻⁴

3

Evaluate: 5⁰ + 5⁻¹

Rational Exponents, Part 1: What Does 9^(1/2) Mean?

1

Evaluate: 121^(1/2)

2

Evaluate: (27/64)^(1/3)

3

Evaluate: 16^(−1/2)

Rational Exponents, Part 2: Powers Like 8^(2/3) and 16^(0.75)

1

Evaluate: 125^(2/3)

2

Evaluate: 9^(−3/2)

3

Evaluate: 32^0.8

Evaluating Expressions with Integer and Rational Exponents

1

Simplify, then evaluate: 5^(2/3) × 5^(4/3)

2

Evaluate: 4^(1/2) + 8^(−1/3) − (1/2)⁰

3

Evaluate 1.04¹⁵ to three decimal places and say what it means for something growing 4% per year.

xy−3−2−1123123456780growth: y = 2ˣ (solid)decay: y = (1/2)ˣ (dashed)both:domain all real xrange y > 0y-intercept (0, 1)asymptote y = 0
The two basic shapes. Growth (a > 1) rises to the right; decay (0 < a < 1) falls to the right. Both pass through (0, 1) and approach the asymptote y = 0.

Graphing y = aˣ and Why It Is a Function

1

f(x) = 4ˣ. Find f(−2) and f(3/2).

2

Why is y = (−5)ˣ not an exponential function?

3

How are the graphs of y = 3ˣ and y = (1/3)ˣ related?

Key Properties of Exponential Functions: Domain, Range, Intercepts and Asymptotes

1

State the domain, range, y-intercept and asymptote of y = 0.6ˣ.

2

State the y-intercept and asymptote of y = 2(3)ˣ − 5.

3

x: 0, 1, 2, 3; y: 7, 14, 28, 56. Write the equation.

Just for fun

Why do exponential functions make terrible secret-keepers?

You tell two friends, and everyone who hears it tells two more... In the 10th round of telling alone, 2¹⁰ = 1024 new people hear it.

Linear, Quadratic or Exponential? Comparing Three Kinds of Functions

1

Classify (equal x-steps): 4, 9, 16, 25, 36.

2

Classify (equal x-steps): 4, 6, 9, 13.5, 20.25.

3

Classify: y = 6 − 2x and y = 6(0.5)ˣ. Which one reaches 0?

Collecting Data That Can Be Modelled by an Exponential Function

1

Coin experiment data: 60, 31, 14, 8, 4. What model would you expect, and do the data fit?

2

Two-point method: y = 5 at x = 0 and y = 80 at x = 4. Find b.

3

Name one primary and one secondary source of exponential data.

Exponential Growth and Decay: Recognizing Models and Their Restrictions

1

Growth or decay, and what rate? y = 900(0.97)ᵗ

2

Write a model: 350 bacteria doubling every 5 hours.

3

T = 19 + 66(0.9)ᵗ models a cooling drink, t ≥ 0. State the range.

Solving Problems with Exponential Growth: Populations, Bacteria and Interest

1

$3000 at 5% per year compounded annually: find the amount after 7 years.

2

N = 50(3)^(t/2). When is N = 4050?

3

A population of 2000 grows 6% per year. In which year does it first pass 3000?

Solving Problems with Exponential Decay: Half-Life, Bouncing Balls, Depreciation and Cooling

1

Half-life 4 hours, start 96 mg. How much is left after 12 hours?

2

A $25 000 truck loses 18% per year. Find its value after 4 years.

3

A ball dropped from 1.8 m rebounds to 50% each time. How high is the 3rd rebound?

Check your understanding

Quiz

1.

Simplify (3⁴)² ÷ 3⁵.

2.

Evaluate (4/9)^(−1/2).

3.

Evaluate 64^(5/6).

4.

Which is the range of f(x) = 3ˣ?

5.

Which table (equal x-steps) is exponential?

6.

Which function is decreasing?

7.

A model is y = 600(0.93)ᵗ. What does it describe?

8.

A quantity doubles every 6 hours and starts at 40. Which model fits?

9.

What is the horizontal asymptote of T = 22 + 70(0.8)ᵗ?

10.

A substance has a half-life of 3 days. What fraction remains after 12 days?

For parents and teachers

How to use the results: match each kind of mistake to the lesson to revisit.

  • Adding or multiplying bases, or using the rules on sums: Lesson 1, The Exponent Rules.
  • Negative exponents giving negative answers, or fraction bases not flipped: Lesson 2, Zero and Negative Exponents.
  • Treating x^(1/2) as x ÷ 2, or calculator bracket errors: Lesson 3, Rational Exponents, Part 1.
  • Mixing up root and power in x^(m/n), or decimal exponents: Lesson 4, Rational Exponents, Part 2.
  • Order-of-operations slips in mixed expressions, or rounding too early: Lesson 5, Evaluating Expressions.
  • Joining points with straight lines, or graphs crossing the x-axis: Lesson 6, Graphing y = aˣ.
  • Wrong range (y ≥ 0 or all reals) or wrong asymptote for y = a(b)ˣ + k: Lesson 7, Key Properties.
  • Using differences instead of ratios to spot exponential tables: Lesson 8, Linear, Quadratic or Exponential?
  • Trouble building a model from data: Lesson 9, Collecting Data.
  • Using r instead of 1 + r or 1 − r, half-life exponents, or missing context restrictions: Lesson 10, Growth and Decay.
  • Difficulty finding an unknown time: Lessons 11 and 12, Solving Problems (substitution, graphs and systematic trial).

A score of 8 or more out of 10 on the quiz shows your teen is ready for the Financial Mathematics unit, which builds directly on compound growth.