AP® Precalculus Cheat Sheets: Formulas, Flashcards, Quiz & Study Guide

Unit 2: Exponential and Logarithmic Functions

Unit 2 focuses on multiplicative change, inverse relationships, and models where growth does not happen by adding the same amount each time. A linear function adds a constant amount over equal input intervals. An exponential function multiplies by a constant factor over equal input intervals. This distinction is essential. If a quantity goes from \(100\) to \(120\) to \(144\), the increases are \(20\) and \(24\), so the change is not additive. But each value is multiplied by \(1.2\), so the pattern is exponential.

An exponential model usually has the form \( f(x)=ab^x \). The value \(a\) is the initial output when \(x=0\), assuming no horizontal shift. The base \(b\) is the growth or decay factor. If a population increases by \(8\%\) per year, the growth factor is \(1.08\), so a model may look like \( P(t)=P_0(1.08)^t \). If a value decreases by \(12\%\), the decay factor is \(0.88\), because the quantity keeps \(88\%\) of its previous value each step.

Sequences connect discrete input values to functions. Arithmetic sequences behave like linear functions because the same difference is added each time. Geometric sequences behave like exponential functions because the same ratio is multiplied each time. On the AP® exam, a sequence problem may use \(n\) as the input instead of \(x\), but the logic remains the same. Ask whether the pattern adds or multiplies.

Exponential functions have important graph characteristics. For \( f(x)=ab^x \), where \(a>0\), the function is positive for all real \(x\). The domain is all real numbers, and the range is positive outputs unless transformations shift or reflect the graph. The horizontal asymptote is usually \( y=0 \), or \( y=k \) if the function has been shifted vertically. Exponential growth eventually outpaces polynomial growth as \(x\) becomes very large, which is why exponential models are powerful but also dangerous for long-term extrapolation.

Logarithms are inverses of exponentials. The statement \( \log_b(c)=a \) means \( b^a=c \). This definition is the safest way to evaluate logs, solve equations, and check your work. For example, \( \log_2(32)=5 \) because \(2^5=32\). The domain rule is critical: the argument of a logarithm must be positive. If you solve a log equation and get a value that makes a log argument zero or negative, that value is extraneous.

The log properties are powerful but easy to misuse. Products become sums, quotients become differences, and exponents move to the front. However, there is no rule that allows \( \log_b(x+y) \) to become \( \log_b(x)+\log_b(y) \). That error is extremely common. When simplifying logs, identify whether you are working with multiplication, division, powers, or addition before choosing a rule.

Function composition and inverses often appear in Unit 2. If \( f \) and \( g \) are functions, \( f(g(x)) \) means the output of \(g\) becomes the input of \(f\). The domain of the composition must satisfy both functions: \(x\) must be allowed in \(g\), and \(g(x)\) must be allowed in \(f\). Inverse functions reverse inputs and outputs. Graphically, inverse functions reflect across the line \( y=x \). Algebraically, \( f(f^{-1}(x))=x \) and \( f^{-1}(f(x))=x \), as long as the values are in the correct domains.

Logarithmic models often appear when inputs change proportionally to produce equal output changes. For example, sound intensity, pH, and earthquake magnitude use logarithmic scales. In AP® Precalculus, you are usually not asked to memorize every scientific scale. You are asked to recognize the structure: when multiplicative changes in input correspond to additive changes in output, logarithmic thinking may be appropriate.

Unit 3: Trigonometric and Polar Functions

Unit 3 is about periodic behavior, angle measurement, circular motion, trigonometric identities, inverse trigonometric functions, and polar coordinates. Many students memorize unit circle values without understanding what they mean. A better approach is to remember that on the unit circle, the \(x\)-coordinate is cosine and the \(y\)-coordinate is sine. From that one idea, signs, symmetry, exact values, and graph behavior become easier to understand.

Radians are central in AP® Precalculus. A radian measures angle using arc length on a circle. One full rotation is \(2\pi\) radians, half a rotation is \(\pi\), and a quarter rotation is \( \frac{\pi}{2} \). Calculator mode matters because trigonometric function values differ depending on whether the calculator interprets angles as degrees or radians. Unless a problem clearly uses degrees, AP® Precalculus work should normally be done in radians.

Sine and cosine are periodic functions. Their basic period is \(2\pi\). Tangent has period \(\pi\), because its values repeat after half a full rotation. Reciprocal functions such as secant, cosecant, and cotangent are built from sine, cosine, and tangent. For example, \( \sec(\theta)=\frac{1}{\cos(\theta)} \), so secant has vertical asymptotes wherever cosine equals zero.

A sinusoidal model is commonly written as \( f(x)=A\sin(B(x-C))+D \) or \( f(x)=A\cos(B(x-C))+D \). The amplitude \( |A| \) measures distance from the midline to a maximum or minimum. The value \(D\) is the midline. The period is \( \frac{2\pi}{|B|} \). The value \(C\) is the phase shift. In context, the period might represent one day, one year, one rotation, one tide cycle, or one oscillation.

When building a sinusoidal model, start with maximum, minimum, midline, and period. If the maximum is \(18\) and the minimum is \(6\), then the amplitude is \(6\) and the midline is \(12\). If one full cycle takes \(24\) hours, then \(B=\frac{2\pi}{24}=\frac{\pi}{12}\). After that, choose sine or cosine based on where the cycle begins. Cosine is often convenient when the graph starts at a maximum or minimum. Sine is often convenient when the graph starts on the midline.

Trigonometric equations require symmetry and periodicity. If \( \sin(\theta)=\frac{1}{2} \), one solution is \( \theta=\frac{\pi}{6} \), but another solution in one full cycle is \( \theta=\frac{5\pi}{6} \). Then the pattern repeats every \(2\pi\). If \( \tan(\theta)=1 \), solutions repeat every \(\pi\), not \(2\pi\). Always pay attention to the requested interval.

Identities help rewrite expressions into equivalent forms. The Pythagorean identity \( \sin^2(\theta)+\cos^2(\theta)=1 \) is the most important. It allows you to replace one expression with another depending on the problem. Sum formulas and double-angle formulas are useful when angles are combined or doubled. A good strategy is to rewrite everything in terms of sine and cosine first, simplify, then convert back if the final form requires tangent, secant, cosecant, or cotangent.

Inverse trigonometric functions have restricted ranges. This is necessary so that each inverse function is actually a function. \( \arcsin(x) \) returns values in \( \left[-\frac{\pi}{2},\frac{\pi}{2}\right] \). \( \arccos(x) \) returns values in \( [0,\pi] \). \( \arctan(x) \) returns values in \( \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \). Because of these restrictions, \( \arcsin(\sin x) \) does not always equal \(x\).

Polar coordinates represent a point using \( (r,\theta) \), where \(r\) is distance from the pole and \(\theta\) is direction. The same point can have more than one polar representation. A negative \(r\) value moves the point in the opposite direction of the angle. Converting between polar and rectangular forms uses \( x=r\cos(\theta) \), \( y=r\sin(\theta) \), and \( r^2=x^2+y^2 \). These formulas come directly from right-triangle trigonometry.

How to Use This AP® Precalculus Cheat Sheet

A cheat sheet is most useful when it becomes an active recall tool. Do not only read the formulas. Cover the explanation and try to rebuild the idea from memory. For example, when you see the sinusoidal model \( f(x)=A\sin(B(x-C))+D \), ask yourself what each parameter does, how to find it from a graph, and what it means in context. If you cannot explain a formula in words, you probably do not understand it well enough for a free-response question.

1. Scan the cheat-sheet cards first. Mark every formula or rule that feels unfamiliar. Do not slow down yet. The first pass should reveal your weak areas.
2. Use the flashcards for recall. Say the answer out loud before revealing it. If you miss a card, write the formula and one sentence explaining it.
3. Complete the quiz without notes. The quiz is short, but it checks several high-frequency concepts. Treat wrong answers as a study map.
4. Read the detailed unit explanation. Focus only on your weak unit first. Deep review is more useful than rereading everything equally.
5. Create one worked example per formula. A formula becomes easier to remember when you have used it in a real problem.
6. Practice AP-style justification. Write complete sentences explaining model choice, domain restrictions, units, and reasonableness.
7. Check your score readiness. After timed practice, use the AP Precalculus score calculator to estimate how your raw performance might translate.

For a seven-day review plan, use one major theme per day. Day 1: rates of change and polynomial behavior. Day 2: rational functions, asymptotes, and holes. Day 3: transformations, model choice, and function composition. Day 4: exponentials and logarithms. Day 5: sinusoidal models and the unit circle. Day 6: identities, inverse trig, and polar coordinates. Day 7: mixed practice and FRQ writing. The goal is not to relearn the entire course in a week. The goal is to reduce avoidable mistakes and strengthen the topics most likely to appear.

Active recall beats passive rereading. Every formula should be practiced in three ways: from an equation, from a graph, and from a context.

Free-Response Strategy for AP® Precalculus

Free-response questions reward communication. Your work should show what you know, even if your final answer is not perfect. A strong solution names the function type, identifies the relevant formula, substitutes values clearly, includes units, and explains the conclusion. If a question asks for a justification, a number alone is not enough. You need a sentence that connects your calculation to the mathematical meaning.

FRQ 1 usually tests function concepts. You may see composition, inverse functions, zeros, transformations, rates of change, or interpretation of graphs and tables. For this question, be careful with notation. If the question asks for \( f(g(2)) \), evaluate inside first. If it asks for an inverse value, remember that inverse notation swaps input and output.

FRQ 2 usually focuses on modeling a non-periodic context. This may involve polynomial, rational, exponential, or logarithmic models. The important skill is not only computing the model but also justifying why it is appropriate. If data shows a constant ratio, say that this supports an exponential model. If residuals have a pattern, say that the model may not be appropriate. If the context has a limited domain, state it.

FRQ 3 usually focuses on periodic modeling. Expect amplitude, period, midline, phase shift, maximum, minimum, and interpretation of rates of change. Sketching one cycle can help. Label the midline and key points. If the problem asks about concavity or rate of change, connect your answer to the graph: sinusoidal functions change fastest at midline crossings and have zero instantaneous rate at maxima and minima.

FRQ 4 usually focuses on symbolic manipulation. This can include logarithmic equations, exponential forms, trig identities, inverse trig restrictions, and algebraic rewriting. Write each step clearly. Do not jump from the first line to the final answer if the transformation is not obvious. Points are often awarded for valid intermediate steps.

When a calculator is allowed, use it strategically. A graphing calculator can help solve equations, find intersections, evaluate functions, and check models. However, calculator output still needs interpretation. If you use a calculator to find a zero, say what equation you solved or what graphs you intersected. If you round, keep enough decimal places during calculations and round only at the end.

FRQ writing checklist: formula, substitution, domain, units, conclusion, reasonableness. If your solution includes those six elements where appropriate, it is much easier for a grader to award credit.