Mathematical Architects · Algebra I · Foundations

Get Ready: Module 4 — Investigating Growth & Decay

Everyone starts somewhere. Before we raise the building, we pour the floor — the few skills that make growth & decay click. Go at your own pace; there's no clock here.

Read this first. If the module felt fast or fuzzy, that's not on you — it just means a foundation block is still being set. This page rebuilds the five skills you lean on most in Module 4, one small step at a time. Try the worked example, then the practice, and reveal the answer only when you're ready. You don't have to be perfect. You just have to start.
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Skills to build first

Five blocks, in order. Each one shows what it is, why Module 4 needs it, a fully worked example, and practice you can check yourself.

1

Exponents & Powers (repeated multiplication)

What it is: an exponent is a shortcut for multiplying the same number over and over. \(b^{x}\) means "multiply \(b\) by itself \(x\) times." Why Module 4 needs it: growth and decay are built on \(y = a\,b^{x}\). The exponent \(x\) is how many times the multiplier \(b\) gets applied — so if exponents feel shaky, the whole curve feels like magic instead of multiplication.

Worked example

Evaluate \(2^{4}\).

  1. Write what it means. \(2^{4}\) is four 2's multiplied together: \(2 \times 2 \times 2 \times 2\).
  2. Multiply two at a time. \(2 \times 2 = 4\), then \(4 \times 2 = 8\), then \(8 \times 2 = 16\).
  3. Answer. \(2^{4} = 16\). (Notice each step doubles — that's exactly what a base of 2 does in growth.)

Try it

a) Evaluate \(3^{3}\).    b) Evaluate \(\left(\tfrac{1}{2}\right)^{3}\).

Show answer

a) \(3^{3} = 3 \times 3 \times 3 = 9 \times 3 = 27\).

b) \(\left(\tfrac{1}{2}\right)^{3} = \tfrac{1}{2}\times\tfrac{1}{2}\times\tfrac{1}{2} = \tfrac{1}{8}\). A base between 0 and 1 keeps shrinking — that's decay.

2

Percents & Percent Change

What it is: a percent is just "out of 100," and \(25\%\) of a number means \(0.25\) times that number. Why Module 4 needs it: real growth and decay are described in percents — "grows 8% a year," "loses 30% each hour." You'll turn those percents into the multiplier \(b\): a \(+8\%\) increase makes \(b = 1.08\); a \(30\%\) loss makes \(b = 0.70\).

Worked example

A $50 item grows by 8%. What is the new amount, and what multiplier does that match?

  1. Turn the percent into a decimal. \(8\% = \dfrac{8}{100} = 0.08\).
  2. Find the increase. \(0.08 \times 50 = 4\). The item goes up by $4.
  3. Add it back on. \(50 + 4 = 54\). New amount is $54.
  4. See the multiplier. Going up 8% is the same as multiplying by \(1 + 0.08 = 1.08\): check \(1.08 \times 50 = 54\). So \(b = 1.08\).

Try it

a) What is \(20\%\) of \(80\)?    b) A population drops by \(15\%\). What multiplier \(b\) matches that?

Show answer

a) \(20\% = 0.20\), and \(0.20 \times 80 = 16\).

b) A 15% drop keeps \(100\% - 15\% = 85\%\), so \(b = 0.85\). (Subtract for loss, add for gain.)

3

Multiplication & Division Fluency

What it is: being comfortable multiplying and dividing whole numbers, decimals, and simple fractions — including by numbers less than 1. Why Module 4 needs it: every step of a growth or decay table is one more multiplication by \(b\). Building the table for \(y = a\,b^{x}\) means multiplying again and again, so speed and confidence here keep your focus on the pattern, not the arithmetic.

Worked example

Start at 100 and multiply by 1.5 three times (build the first rows of a growth table).

  1. Step 1. \(100 \times 1.5 = 150\).
  2. Step 2. Multiply the result again: \(150 \times 1.5 = 225\).
  3. Step 3. Once more: \(225 \times 1.5 = 337.5\).
  4. Notice. You always multiply the previous answer, never the original 100. That repeated multiply is the engine of exponential change.

Try it

a) \(64 \div 2 \div 2\).    b) Start at 80 and multiply by \(0.5\) twice.

Show answer

a) \(64 \div 2 = 32\), then \(32 \div 2 = 16\). (Dividing by 2 is the same as multiplying by \(0.5\) — that's halving, i.e. decay.)

b) \(80 \times 0.5 = 40\), then \(40 \times 0.5 = 20\). Each step keeps half.

4

Evaluating Expressions

What it is: plugging a number in for a letter and simplifying, following order of operations (exponents before multiplying, multiplying before adding). Why Module 4 needs it: to find any point on a growth/decay graph you substitute a value of \(x\) into \(y = a\,b^{x}\) and simplify. Getting the order right — exponent first, then multiply by \(a\) — is the difference between the right curve and a wrong one.

Worked example

Evaluate \(y = 5 \cdot 2^{x}\) when \(x = 3\).

  1. Substitute. Replace \(x\) with 3: \(y = 5 \cdot 2^{3}\).
  2. Exponent first. Order of operations says do the power before multiplying: \(2^{3} = 8\).
  3. Then multiply. \(y = 5 \cdot 8 = 40\).
  4. Answer. \(y = 40\). If you had multiplied \(5 \cdot 2\) first you'd get the wrong curve — the exponent always goes first.

Try it

a) Evaluate \(y = 3 \cdot 4^{x}\) when \(x = 2\).    b) Evaluate \(y = 100 \cdot (0.5)^{x}\) when \(x = 2\).

Show answer

a) \(4^{2} = 16\), then \(3 \cdot 16 = 48\). So \(y = 48\).

b) \((0.5)^{2} = 0.25\), then \(100 \cdot 0.25 = 25\). So \(y = 25\). (After 2 halvings, a quarter remains.)

5

Function Notation \(f(x)\) Basics

What it is: \(f(x)\) is just a name for a rule, read "\(f\) of \(x\)." \(f(3)\) means "run the rule with \(x = 3\)." The number in the parentheses is the input; the result is the output. Why Module 4 needs it: exponential models are written as functions like \(f(x) = a\,b^{x}\), and questions ask for \(f(0)\), \(f(1)\), \(f(5)\). Reading that notation correctly tells you exactly which number to plug in.

Worked example

If \(f(x) = 4 \cdot 3^{x}\), find \(f(0)\) and \(f(2)\).

  1. Read the notation. \(f(0)\) means use \(x = 0\); \(f(2)\) means use \(x = 2\).
  2. Find \(f(0)\). \(f(0) = 4 \cdot 3^{0}\). Anything to the power 0 equals 1, so \(3^{0} = 1\) and \(f(0) = 4 \cdot 1 = 4\). (That's the starting value \(a\)!)
  3. Find \(f(2)\). \(f(2) = 4 \cdot 3^{2} = 4 \cdot 9 = 36\).

Try it

For \(f(x) = 6 \cdot 2^{x}\):   a) find \(f(0)\);   b) find \(f(3)\).

Show answer

a) \(f(0) = 6 \cdot 2^{0} = 6 \cdot 1 = 6\) — the initial value.

b) \(f(3) = 6 \cdot 2^{3} = 6 \cdot 8 = 48\).

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Quick Readiness Check

Six short questions across the five skills. No grade, no timer — just a temperature check.

  1. Evaluate \(2^{5}\).
    Show answer
    \(2^{5} = 32\) (five 2's: \(2\cdot2\cdot2\cdot2\cdot2\)).
  2. What is \(30\%\) of \(60\)?
    Show answer
    \(0.30 \times 60 = 18\).
  3. A quantity grows by \(12\%\). What multiplier \(b\) is that?
    Show answer
    \(b = 1 + 0.12 = 1.12\).
  4. Start at 200 and multiply by \(0.5\) twice. What do you get?
    Show answer
    \(200 \to 100 \to 50\).
  5. Evaluate \(y = 7 \cdot 2^{x}\) when \(x = 3\).
    Show answer
    \(2^{3} = 8\), then \(7 \cdot 8 = 56\).
  6. If \(f(x) = 5 \cdot 3^{x}\), what is \(f(0)\)?
    Show answer
    \(f(0) = 5 \cdot 3^{0} = 5 \cdot 1 = 5\).

If these feel comfortable, you're ready for the module. If one or two felt rough, scroll back to that skill block and run the practice again — that's exactly what it's for.

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Practice more (free)

Want extra reps on any block? These free sites let you drill until it clicks. (External links open in a new tab.)

1 · Exponents & Powers

2 · Percents & Percent Change

3 · Multiplication & Division

4 · Evaluating Expressions

5 · Function Notation \(f(x)\)

When you're ready →

Floor poured? Beautiful. Step up into the module — the Visual Lab lets you see these skills turn into a curve.

Go to Module 4 overview Open the Visual Lab Read the Syllabus

Module and topic structure follow the TEA Bluebonnet Learning — Secondary Mathematics open curriculum (Edition 1, adapted from Carnegie Learning), licensed CC BY-NC 4.0. Khan Academy and IXL are independent third parties; links are provided for free supplemental practice.