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Unit 2 · Exponential & Logarithmic Functions Flashcards Cheat Sheet Essentials Visual Review MC Practice FRQ Practice

AP Precalculus Unit 2 Visual Review

A topic-by-topic visual walkthrough of Unit 2: Exponential and Logarithmic Functions — sequences, exponential and logarithmic functions, composition, inverses, and semi-log plots.

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TOPIC 2.1 Arithmetic & Geometric Sequences Arithmetic: add a common difference d aₙ = a₁ + (n − 1)d Linear-style: same amount added each step. 3, 7, 11, 15… d = 4 a₅ = 3 + 4·4 = 19 Graph of terms is linear. Geometric: multiply by a common ratio r gₙ = g₁ · r⁽ⁿ⁻¹⁾ Exponential-style: same factor each step. 2, 6, 18, 54… r = 3 g₅ = 2 · 3⁴ = 162 Graph of terms is exponential. Sequences are functions of the term number n Arithmetic ↔ linear (constant difference); geometric ↔ exponential (constant ratio). Both are discrete. Arithmetic adds d (aₙ=a₁+(n−1)d); geometric multiplies by r (gₙ=g₁·rⁿ⁻¹). The Review Hub · AP Precalculus Unit 2 TOPIC 2.2 Change in Linear & Exponential Functions Linear: constant ADDED rate f(x) = mx + b Over equal x-intervals, y changes by equal DIFFERENCES. 5, 8, 11, 14 (+3 each) Use for steady, additive growth. Exponential: constant MULTIPLIED rate f(x) = a · bˣ Over equal x-intervals, y changes by equal RATIOS (proportional). 5, 10, 20, 40 (×2 each) Use for percent growth/decay. Differences vs. ratios Equal successive DIFFERENCES → linear. Equal successive RATIOS → exponential. Check a table both ways. Linear grows by equal differences; exponential grows by equal ratios. The Review Hub · AP Precalculus Unit 2 TOPIC 2.3 Exponential Functions b>1 growth 0<b<1 decay HA y=0 f(x) = a · bˣ, b > 0, b ≠ 1 a = initial value (y-intercept) b = growth factor (b>1 grow, 0<b<1 decay) Key features Domain: all reals. Range: y > 0 (if a>0). Horizontal asymptote y = 0. Always concave up (a>0); never crosses y=0. f(x) = a·bˣ: a is the initial value, b the growth factor; asymptote y = 0. The Review Hub · AP Precalculus Unit 2 TOPIC 2.4 Exponential Function Manipulation // exponent rules bᵐ · bⁿ = bᵐ⁺ⁿ bᵐ / bⁿ = bᵐ⁻ⁿ (bᵐ)ⁿ = bᵐⁿ b⁰ = 1 b⁻ⁿ = 1/bⁿ b^(1/n) = ⁿ√b A shift can rescale A horizontal shift of an exponential is equivalent to a VERTICAL stretch. 2⁽ˣ⁺³⁾ = 2³ · 2ˣ = 8·2ˣ So a and a horizontal shift are interchangeable. Natural base e e ≈ 2.718. Any base can be written as e: bˣ = e^(x·ln b) Continuous growth model: A = P·eʳᵗ. Apply exponent rules; a horizontal shift equals a vertical stretch (2ˣ⁺³ = 8·2ˣ). The Review Hub · AP Precalculus Unit 2 TOPIC 2.5 Exponential Context & Data Modeling A(t) = a(1 + r)ᵗ growth (r>0) or decay (−1<r<0) Worked example $500 grows 6% per year: A(t) = 500(1.06)ᵗ After 10 years: A(10) = 500(1.06)¹⁰ ≈ $895 Decay: half-life uses factor (1/2)^(t/h). Fit a model to data Use exponential regression to get a·bᵗ from a data set. Percent rate r = b − 1. Interpret a (start) and b (per-period multiplier) in the context. Model percent change with A(t) = a(1 + r)ᵗ; r > 0 grows, r < 0 decays. The Review Hub · AP Precalculus Unit 2 TOPIC 2.6 Competing Function Model Validation Exponential eventually dominates For large x, an increasing exponential GROWS FASTER than any polynomial. 2ˣ overtakes x¹⁰⁰ eventually Order (large x): logarithmic < linear < polynomial < exponential. Validate with residuals Compare competing models by their RESIDUALS (actual − predicted). The better model has smaller residuals with NO leftover pattern. A curved residual plot signals a wrong model. Match behavior AND context Pick the model whose long-run behavior and shape best fit the data and make sense for the real situation. Exponential beats polynomial for large x; validate model choice with residuals. The Review Hub · AP Precalculus Unit 2 TOPIC 2.7 Composition of Functions (f ∘ g)(x) = f( g(x) ) — inner g first, then f Worked example f(x)=x², g(x)=x+1 (f∘g)(3) = f(g(3)) = f(4) = 16 (g∘f)(3) = g(f(3)) = g(9) = 10 Order matters: f∘g ≠ g∘f in general. (f∘g)(x)=(x+1)², (g∘f)(x)=x²+1 Domain of a composite x must be in the domain of g, AND g(x) must be in the domain of f. Build complex functions by composing simpler ones — and decompose to analyze. Check for values that break the inner or outer. (f∘g)(x) = f(g(x)) — evaluate the inner function first; order matters. The Review Hub · AP Precalculus Unit 2 TOPIC 2.8 Inverse Functions f(f⁻¹(x)) = x and f⁻¹(f(x)) = x — inverses undo each other Find an inverse 1. Write y = f(x). 2. Swap x and y. 3. Solve for y = f⁻¹(x). f(x)=2x+3 → f⁻¹(x)=(x−3)/2 Inputs and outputs SWAP: domain and range trade places. Graph & existence The graph of f⁻¹ is f REFLECTED over the line y = x. f has an inverse only if it is ONE-TO-ONE (passes the horizontal line test). Else restrict the domain to make it invertible. ↔Inverses swap x and y and reflect over y = x; f must be one-to-one. The Review Hub · AP Precalculus Unit 2 TOPIC 2.9 Logarithmic Expressions log_b(y) = x ⟺ bˣ = y A log answers "what exponent?" log_b(y) is the power you raise b to in order to get y. log₂(8) = 3 since 2³ = 8 log₁₀(1000) = 3 ln x = log_e x (natural log, base e). Special values log_b(1) = 0 log_b(b) = 1 log_b(bˣ) = x b^(log_b x) = x The argument must be POSITIVE: you can't take the log of 0 or a negative. Domain of log_b(x) is x > 0. log_b(y) = x ⟺ bˣ = y — a logarithm is an exponent; the argument must be > 0. The Review Hub · AP Precalculus Unit 2 TOPIC 2.10 Inverses of Exponential Functions y=x bˣ log_b x Log is the inverse of exp y = log_b(x) is the inverse of y = bˣ. Their graphs are reflections over y = x. Domain/range swap between the two. Features flip bˣ: domain all reals, range y>0, HA y=0. log_b x: domain x>0, range all reals, VERTICAL asymptote x = 0. y = log_b x is the inverse of y = bˣ — reflections over y = x; domain/range swap. The Review Hub · AP Precalculus Unit 2 TOPIC 2.11 Logarithmic Functions Key features of y = log_b(x) Domain x > 0; range all real numbers. Vertical asymptote at x = 0. x-intercept at (1, 0) since log_b(1)=0. Increasing & concave down for b > 1. Grows without bound, but ever more slowly. Slow growth A logarithm increases without bound but SLOWER than any positive power of x. Equal RATIO changes in x give equal DIFFERENCE changes in y. log₂: 2→4→8 gives 1→2→3 Transformations apply the same way a·log_b(x − h) + k shifts, stretches, and reflects the graph just like any function (Topic 1.12). y = log_b x has domain x>0, VA at x=0, passes (1,0), and grows slowly. The Review Hub · AP Precalculus Unit 2 TOPIC 2.12 Logarithmic Function Manipulation // the log laws (b assumed valid) PRODUCT: log_b(MN) = log_b M + log_b N QUOTIENT: log_b(M/N) = log_b M − log_b N POWER: log_b(Mᵖ) = p · log_b M Change of base log_b(x) = ln x / ln b Lets you compute any base on a calculator. Expand or condense Use the laws to rewrite a log expression. log(x²y) = 2log x + log y Reverse to condense into a single log. Log laws: product→sum, quotient→difference, power→coefficient; change base = ln x / ln b. The Review Hub · AP Precalculus Unit 2 TOPIC 2.13 Exp & Log Equations and Inequalities Solve exponential: take a log 3ˣ = 20 ln(3ˣ) = ln 20 x·ln 3 = ln 20 x = ln 20 / ln 3 ≈ 2.73 Log both sides, then use the power law. Solve logarithmic: exponentiate log₂(x) = 5 x = 2⁵ = 32 Check for EXTRANEOUS solutions — the argument must stay positive. Reject any x that makes a log undefined. Inequalities: mind the direction Because exp and log (b>1) are increasing, the inequality direction is preserved when you apply them. Log both sides to solve exponentials; exponentiate to solve logs — check for extraneous roots. The Review Hub · AP Precalculus Unit 2 TOPIC 2.14 Logarithmic Context & Data Modeling Logs model wide-range data A logarithmic function fits data that rises quickly then levels off. Good for diminishing returns and quantities spanning many magnitudes. y = a + b·ln x (logarithmic regression) Log scales Real scales use logarithms to compress huge ranges into readable numbers: • Richter (earthquakes) • decibels (sound), pH (acidity) Each +1 unit is a ×10 change in the quantity. Interpret the parameters In y = a + b·ln x, b sets how fast y rises per multiplicative change in x; use the model to predict and compare. Logs model rapid-then-leveling data and power log scales (Richter, decibels, pH). The Review Hub · AP Precalculus Unit 2 TOPIC 2.15 Semi-log Plots y-axis is log-scaled 1101001k exp data → straight line Log the y-axis A semi-log plot scales the y-axis logarithmically (x-axis stays linear). EXPONENTIAL data becomes a STRAIGHT LINE. Read the line y = a·bˣ → log y = log a + (log b)x. SLOPE = log b (growth factor). INTERCEPT = log a (initial value). On a semi-log plot, exponential data becomes a straight line (slope = log b). The Review Hub · AP Precalculus Unit 2
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How to use the visual review

Spend 30 seconds per slide before clicking next. Look at the graph, then ask yourself: "Could I sketch this from memory and explain what its algebra is doing?"

Use the fullscreen button () on desktop for the best experience. Use arrow keys to navigate. Tap "Show all slides" to jump around.

This is great for review the night before the exam — fast, visual, and covers every graph feature you need to recognize in Unit 2.