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B. Exponential Functions · Exponential Functions

Exponential Functions: Unit Review

B1.1 B1.2 B1.3 B1.4 B2.1 B2.2 B2.3 B2.4 B2.5 B3.1 B3.2 B3.3About 40 minutes
Learning goals
  • I can evaluate powers with rational exponents and simplify expressions with exponents.
  • I can describe, compare and transform exponential graphs.
  • I can write exponential equations from graphs, properties and data.
  • I can solve real-world growth and decay problems.

Words to know

exponential function
f(x) = bˣ, b > 0, b ≠ 1.
rational exponent
x^(m/n) = (ⁿ√x)ᵐ for x > 0.
horizontal asymptote
A line y = c that the graph approaches but never reaches.
mapping rule
(x, y) → (x/k + d, ay + c) for y = af(k(x − d)) + c.
half-life
The time for a decaying quantity to fall to half.

Let's review!

In this unit you met the exponential function f(x) = bˣ and learned what rational exponents mean, so the graph makes sense between the integers. You simplified expressions with the exponent laws, described the key properties of exponential graphs, told them apart from linear and quadratic functions, and transformed them with y = af(k(x − d)) + c. You rewrote functions with different bases, found equations from graphs and properties, collected data, and solved growth and decay problems.

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 notes first. Show your steps: for powers, say "root, then power"; for transformations, write the mapping rule before sketching. Check numeric answers with a calculator (brackets around fractional exponents!). For each mistake, write down why it happened and revisit that lesson's worked examples. Finish with the 10-question quiz.

Rational exponentsx^(m/n) = (ⁿ√x)ᵐdenominator = rootnumerator = powerProperties of y = bˣdomain ℝ, range y > 0y-intercept 1, asymptote y = 0b > 1 grows, 0 < b < 1 decaysTransformationsy = af(k(x − d)) + c(x, y) → (x/k + d, ay + c)asymptote y = cModelsgrowth a(1 + r)ᵗ, decay a(1 − r)ᵗhalf-life a(1/2)^(t/h)A = P(1 + i)ⁿ
The big ideas of the unit on one page.
Just for fun

You now know enough about exponential functions to explain why hot chocolate never gets colder than the kitchen, why a bouncing ball gets bored eventually, and why your savings grow faster the longer you leave them alone. That's a lot of power for one little exponent.

The Exponential Function f(x) = aˣ

1

For f(x) = 5ˣ, find f(−2) and f(0).

2

Why is y = (−2)ˣ not used as an exponential function?

3

Is y = x⁴ exponential? Explain.

Rational Exponents I: Powers with Exponent 1/n

1

Evaluate 81^(1/4).

2

Evaluate 125^(−1/3).

3

Write ⁵√12 as a power.

Rational Exponents II: Powers with Exponent m/n

1

Evaluate 32^(3/5).

2

Evaluate (9/16)^(−3/2).

3

Evaluate 27^(2/3) + 4^(−1/2).

Simplifying Expressions with Integer Exponents

1

Simplify (4x⁻²y³)(−3x⁵y⁻¹).

2

Simplify (2a⁻³b)⁻².

3

Evaluate (3/2)⁻³.

Simplifying Expressions with Rational Exponents

1

Simplify x^(2/3) × x^(1/6).

2

Simplify (49a⁶)^(1/2).

3

Write √x ÷ ∜x as a single power.

Key Properties of Exponential Functions

1

State the domain, range and asymptote of y = 0.4ˣ.

2

Is y = 1.8ˣ increasing or decreasing?

3

For x < 0, which is greater: 2ˣ or 6ˣ?

Exponential, Linear or Quadratic?

1

x = 0, 1, 2, 3; y = 4, 10, 25, 62.5. Classify and give the equation.

2

x = 0, 1, 2, 3; y = 1, 4, 9, 16. Classify.

3

Give one graph feature that shows y = 3ˣ is not quadratic.

Transforming Exponential Graphs I: Translations, Vertical Stretches and Reflections

1

Describe y = 2^(x + 5) − 3 as a transformation of y = 2ˣ.

2

Range of y = −4(3ˣ) + 1?

3

Image of (1, 2) on y = 2ˣ under y = 3(2ˣ) − 5?

xy−3−2−101234−6−4−20246(0, 1.5)(1, 1)(2, 0)(3, −2)asymptote y = 2
Use this graph for the last question in the next section: an exponential curve below its asymptote y = 2, through (0, 1.5), (1, 1), (2, 0) and (3, −2).

Transforming Exponential Graphs II: Horizontal Stretches, Reflections and All Four Parameters

1

Find k and d in y = 2^(4x − 12).

2

Mapping rule for y = −2(3^(0.5(x + 1))) + 4?

3

The graph above shows y = a(2^(x − d)) + c with key points (1, 1) and (2, 0). Find its equation.

Same Function, Different Bases

1

Write 16ˣ as a power of 2.

2

Solve 9ˣ = 27.

3

Show 3^(x + 2) = 9(3ˣ).

Finding the Equation of an Exponential Function

1

Asymptote y = 0, y = a(bˣ) through (0, 5) and (2, 45). Find the equation.

2

Asymptote y = 4, y-intercept 10, through (1, 7). Find y = a(bˣ) + c.

3

y = a(bˣ) through (1, 12) and (3, 108). Find a and b.

Collecting and Modelling Exponential Data

1

Coin data: 50, 24, 13, 6. Mean ratio and model?

2

Theoretical model for 90 number cubes, removing 6s?

3

Why does an experiment not match the model exactly?

Growth and Decay Models, and What Makes Sense

1

Growth or decay? Rate? y = 300(0.93)ᵗ

2

Coffee at 85°C cools in a 21°C room. Give the range of a cooling model.

3

Write a model: 50 mg, half-life 4 h.

Solving Problems with Exponential Models

1

$2500 at 3% compounded annually for 4 years?

2

A ball from 2.5 m rebounds to 80%. Height of the 3rd rebound?

3

640 mg, half-life 3 days. How much after 12 days?

Check your understanding

Quiz

1.

What is 16^(3/4)?

2.

Simplify x^(1/2) × x^(1/4).

3.

Simplify (3a⁻²)⁻².

4.

What is the range of y = 2ˣ − 7?

5.

A table with equal x-steps has y = 5, 15, 45, 135. It is:

6.

y = 3^(2x − 6) is y = 3ˣ compressed horizontally by 1/2 and translated:

7.

Which is equal to 8ˣ?

8.

An exponential graph has asymptote y = 1, y-intercept 4, and passes through (1, 7). Its equation is:

9.

Which model shows 6% decay per year?

10.

A sample of 120 mg has a half-life of 2 days. How much remains after 6 days?

For parents and teachers

Use the review to find what to revisit:

  • Confusing 2ˣ with x², or negative exponents giving negative numbers → Lesson 1 (The Exponential Function).
  • Treating 9^(1/2) as 4.5, or root/power mix-ups → Lessons 2 and 3 (Rational Exponents I and II).
  • Errors with negative exponents, coefficients or several variables → Lesson 4 (Integer Exponents).
  • Fraction arithmetic in exponents, or mixing radicals and powers → Lesson 5 (Rational Exponents in expressions).
  • Range written as y ≥ 0, or asymptote on the wrong side → Lesson 6 (Key Properties).
  • Using differences instead of ratios → Lesson 7 (Exponential, Linear or Quadratic?).
  • Translations in the wrong direction, or forgetting to move the asymptote → Lesson 8 (Transformations I).
  • Reading d without factoring, or multiplying x by k instead of dividing → Lesson 9 (Transformations II).
  • Writing 2^(3x) as 6ˣ, or common-base equations → Lesson 10 (Different Bases).
  • Using the y-intercept as a when the graph is shifted → Lesson 11 (Finding the Equation).
  • Fitting models to data, mean ratios → Lesson 12 (Collecting Data).
  • Writing 12% decay as 0.12, or ignoring restrictions on domain and range → Lesson 13 (Growth and Decay).
  • Half-life exponents, compound interest periods, "how long until" problems → Lesson 14 (Solving Problems).

Exponential functions return in this course in the Sequences and Series unit (geometric sequences) and the Financial Applications unit (compound interest and annuities), and in Grade 12 Advanced Functions (MHF4U), where logarithms solve exponential equations exactly.