Week 13 — Exponential Functions

Welcome to your Week 13 Start Here page.

Why this matters

Exponential functions are how a saver compares two bank offers and how a city planner projects a town's growth. Each one is a starting amount multiplied by the same factor again and again.

In plain language

Some things change by adding the same amount each time, and some change by multiplying by the same number each time. This week is about the second kind. A function that multiplies is called an exponential function (a rule where the input sits up in the exponent, so every step multiplies the output by the same number). It describes savings that earn interest, a town that grows by a percent each year, a phone that loses value, and a sample that shrinks by half on a schedule. Once you can read the starting amount and the multiplier from the formula, you can predict any of them and check your answer on a graph.

• Already covered: Week 1 set the exponent rules, including zero and negative exponents; Week 3 named functions, domain, and range; Week 4 read a line's constant rate of change; Week 10 met horizontal asymptotes; Week 12 used rational exponents.
• This week: the exponential function f(x) = a · bˣ, its graph, growth and decay, percent-rate models, compound interest, and the natural base e.
• How they connect: a line adds its slope at every step, and an exponential multiplies by its base, so Week 4's table and this week's table sit side by side.
• Where we're going next: Week 14, Logarithmic Functions, runs this week's rule backwards, so you will later be able to ask which exponent gives a certain output.

Focus

What happens when a quantity multiplies by the same factor at every step?

By the end of this module, you should be able to:

• [ ] tell a constant ratio from a constant difference in a table
• [ ] evaluate a · bˣ at positive, zero, and negative inputs, exponent first
• [ ] call a function growth or decay from its base and read its graph
• [ ] build a percent-rate model and compute compound interest

To-do

Plan your week

1. Read Chapter 13 (about 45 minutes), then the lecture and slides.
2. Work the lecture tutorial (60–90 minutes) and the practice set (about 20 minutes).
3. Do the Model/Build Studio (about 90 minutes), the discussion, and the assignment.
4. Take the quiz.

This week's checkpoint: evaluate 3 · 2ˣ at x = −2, and say whether the function shows growth or decay.

You've got this

Take it step by step, and ask your instructor early.

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