Unit 2 · 8 lessons

Exponential Functions

Exponent rules, the shape and transformations of y = bˣ, and real growth/decay models — populations, half-life, depreciation.

93 practice questions
71 flashcards
1 interactive lesson

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Lessons

1Exponent LawsThe rules for combining powers — product, quotient, power-of-a-power, zero, negative, and rational exponents — all follow from what an exponent actually counts.12 questions
2Rational Exponents and RadicalsA fractional exponent is a root and a power in one: $a^{m / n} = (n a)^{m}$ . Learn to switch between radical and exponent form, evaluate powers like $(\frac{16}{81})^{- 3/4}$ by hand, and simplify expressions with fractional exponents.11 questions
3Graphs of Exponential FunctionsThe graph of $y = b^{x}$ always passes through (0,1), has a horizontal asymptote at y=0, and when $b > 1$ eventually outgrows any polynomial.11 questions
4Linear, Quadratic or Exponential?Tell the three function types apart from tables (first differences, second differences, ratios), from graphs and from equations, handle uneven $x$ -steps and messy real data, and see why an exponential always wins in the long run.12 questions
5Transformations of Exponential Functions $y = a b^{k (x - d)} + c$ stretches, reflects and slides an exponential graph exactly like any other function — with c shifting the asymptote itself.Interactive11 questions
6Finding an Exponential EquationWork backwards from information to the rule: find $a$ and $b$ in $y = a \cdot b^{x}$ from two points or a table (even with $x$ -steps of 2 or 5), handle a known asymptote with $y = a \cdot b^{x} + c$ , interpret the parameters, and fit real data with exponential regression.12 questions
7Solving Exponential EquationsWhen the unknown is in the exponent, rewrite both sides as powers of the same base and set the exponents equal. Handle fractions and roots, isolate the power, factor out a common power, spot equations with no solution, and estimate when there is no common base.12 questions
8Exponential Growth and Decay ApplicationsPopulation growth, half-life, doubling time and depreciation are all modelled with $A = A_{0} (1 + r)^{t}$ — and equations with matching bases can be solved exactly by comparing exponents.12 questions