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Algebra 1 Unit 5: Exponents & exponential functions

Exponents are repeated multiplication. Exponential functions use a variable exponent and model things that grow or shrink by a constant percent.

Covers: Exponent rules, zero and negative exponents, exponential growth and decay, comparing linear vs exponential

Study guide

Exponent rules simplify repeated multiplication. When multiplying powers with the same base, add exponents; when dividing, subtract; when raising a power to a power, multiply. A product or quotient raised to a power distributes the exponent to each factor: (2x)³ = 8x³. Any nonzero number to the zero power is 1, and a negative exponent means a reciprocal: x⁻ⁿ = 1/xⁿ, so 2⁻³ = 1/8.

Exponential functions have the form y = a·bˣ, where a is the starting value and b is the growth factor. If b > 1 the function grows; if 0 < b < 1 it decays. For percent change, the growth factor is 1 + r (6% growth → 1.06) and the decay factor is 1 − r (6% decay → 0.94). Examples include compound interest, population growth, and the value of a car that loses a percent of its worth each year.

The big comparison is linear vs exponential. Linear functions add the same amount each step, so their tables have constant differences. Exponential functions multiply by the same factor each step, so their tables have constant ratios. An exponential growth function eventually outgrows any linear function, even if it starts smaller.

An arithmetic sequence adds the same common difference d each time, so its nth term is aₙ = a₁ + (n − 1)d. It behaves like a linear function whose inputs are whole numbers. A geometric sequence multiplies by the same common ratio r each time, so aₙ = a₁·rⁿ⁻¹. It behaves like an exponential function whose inputs are whole numbers.

A recursive formula gives the first term and a rule that builds each term from the one before, such as a₁ = 4 and aₙ = aₙ₋₁ + 3 (arithmetic) or aₙ = 2·aₙ₋₁ (geometric). Generate terms one at a time, in order.

Key ideas

• Multiply like bases: add exponents. Divide: subtract. Power of a power: multiply.
• a⁰ = 1 and a⁻ⁿ = 1/aⁿ.
• y = a·bˣ: a is the starting value, b the growth factor.
• b > 1 is growth; 0 < b < 1 is decay.
• Linear adds the same amount each step; exponential multiplies by the same factor.

Formulas and rules

Common mistakes

• Multiplying exponents when you should add them.
• Thinking a negative exponent makes a number negative.

Words to know

Product of powers

xᵃ · xᵇ = xᵃ⁺ᵇ

Power of a power

(xᵃ)ᵇ = xᵃᵇ

Negative exponent

x⁻ⁿ = 1/xⁿ (2⁻³ = 1/8)

Growth factor

1 + r (6% growth → × 1.06)

Linear vs exponential change

Linear adds a constant; exponential multiplies by a constant.

Zero exponent

a⁰ = 1 for a ≠ 0.

Quotient of powers

aᵐ/aⁿ = aᵐ⁻ⁿ

Power of a product

(ab)ⁿ = aⁿbⁿ

Exponential function

y = a·bˣ: multiplies by b each step.

Exponential growth

y = a(1 + r)ᵗ with b > 1.

Exponential decay

y = a(1 − r)ᵗ with 0 < b < 1.

Constant ratio

Exponential data multiply by the same factor each step.

Sample practice questions

31 more questions like these, from level 1 to 5, are in the practice set.