AP Precalculus

What this covers

• Analysing polynomial and rational functions: multiplicity and sign behaviour, end behaviour from the ratio of leading terms, distinguishing holes from vertical asymptotes, and slant asymptotes via division
• Describing rates of change: average rate of change over an interval, when a rate of change is itself increasing or decreasing, and connecting that to concavity and points of inflection
• Exponential and logarithmic modelling: constructing models from two points or a ratio pattern, solving equations with log properties, and reading linearised (semi-log) plots where log y against x is linear
• Sinusoidal functions: finding amplitude, midline, period from 2π/b, and phase shift with b properly factored out; building a sine or cosine model from a real context and interpreting it back
• Polar functions: sketching r = f(θ), interpreting negative r values, and describing where the distance from the origin is increasing or decreasing as θ increases
• Function composition, inverses and their domain restrictions, plus transformations of parent functions in the order they are applied
• Unit 4 material as needed for class grades: parametric curves, vector components, and matrix multiplication as a transformation

Where learners get stuck

Confusing "the function is increasing" with "the rate of change is increasing"

Both phrases contain "increasing", and in Algebra 2 students only ever described the function itself. AP Precalculus scores free-response points specifically for the second idea — concavity described in rate-of-change language — so students write a true statement that earns nothing.

Reading the phase shift straight off y = sin(2x − π) as π

Students transfer the vertical-shift habit of just taking the constant. Because the horizontal dilation acts first, the b must be factored out: sin(2(x − π/2)), giving a shift of π/2. This error also reverses when the shift is written inside a fraction.

Assuming every zero of a rational function's denominator gives a vertical asymptote

Denominator zeros are taught as "where it's undefined" long before factoring the numerator becomes routine. When a factor cancels, the graph has a removable hole with a finite limiting value, and students lose points describing end behaviour that isn't there.

Applying log properties to sums, e.g. writing log(x + y) as log x + log y

The correct rule log(xy) = log x + log y is memorised as "logs turn things into addition", and the distinction between the argument being a product versus a sum gets lost once equations get long.

What a session looks like

Sessions run as spoken conversation with Evelyn. You bring a problem set, a graph you can't interpret, or a topic from class, and work through it out loud — stating what the leading terms tell you about end behaviour, or talking through how you'd justify an answer in words the way a free-response question demands. Evelyn asks for your reasoning before confirming anything, so gaps in the middle steps surface rather than being papered over by a correct final answer. Graphs and equations can be described verbally or worked alongside on paper.

Helpful to know first

• Fluent factoring of quadratics and simple cubics, including grouping and difference of squares
• Confident exponent rules, including negative and rational exponents
• Function notation, evaluating f(g(x)), and reading domain and range from a graph
• Basic right-triangle trigonometry and the unit circle values for multiples of 30° and 45°
• Solving linear systems and manipulating equations with fractions

Questions

Do I need AP Precalculus before AP Calculus AB?

It is not a formal requirement, but the AB course assumes fluency with the function families this course covers — especially rational function behaviour, log and exponential manipulation, and trigonometric identities. Students who skip straight from Algebra 2 usually lose time relearning these while also learning limits and derivatives.

Is Unit 4 on the exam?

No. The exam assesses Units 1–3. Unit 4 (parametric equations, vectors, and matrices) is part of the course and may appear in your class grade, and it is useful preparation for later courses, but exam questions come from the first three units.

What is the exam format and can I use a calculator?

There are multiple-choice and free-response sections, each split into a part where a graphing calculator is permitted and a part where it is not. The calculator-active questions typically involve modelling and regression-style work; the no-calculator parts test algebraic manipulation and reasoning, so both need practising separately.

How is this different from my school's regular precalculus class?

The content overlaps heavily, but AP Precalculus puts far more weight on describing function behaviour in writing — rates of change, concavity, end behaviour — and on building and interpreting models from data rather than only solving for x.

My child is strong at algebra but struggling with the trigonometry unit. Is that normal?

It's common. Unit 3 shifts from algebraic manipulation to reasoning about periodic behaviour and angle measure, and students who succeeded by pattern-matching procedures often stall there. Targeted work on the unit circle and on translating between a context and a sinusoidal model usually addresses it.