AP Precalculus common mistakes
AP Precalculus · Understand the traps and how to avoid them.
When asked for the domain of a function like f(x)=sqrt(x-3)/(x-5) or g(x)=ln(2x+1), students give only one restriction (e.g., x≥3) and drop the denominator or argument restriction, or they state range values instead of domain values.
Students memorize 'domain = x-values' as a slogan but treat each restriction as an isolated rule rather than intersecting all conditions (radicand ≥0, denominator ≠0, log argument >0) into one combined inequality.
What to do instead
Before writing an answer, list every operation in the function and write its restriction on a separate line (radicand ≥0; denominator ≠0; log/ln argument >0). Then intersect the intervals on a number line and express the final answer in interval notation, checking endpoints for open vs. closed.
Students compute an average rate of change over [a,b] as f(b)−f(a) divided by b+a, or they report only the numerator f(b)−f(a) as the 'rate of change'.
The slope formula is recalled loosely, and students conflate 'change in output' with 'rate of change', especially when the interval is given as a table of values rather than a graph.
What to do instead
Write the formula explicitly as (f(b)−f(a))/(b−a) and label numerator 'change in output' and denominator 'change in input' before substituting. Practice with tables so the interval endpoints must be read from the table, not the graph.
Given a graph of f, students evaluate a composition like f(f(2)) or find f⁻¹(3) by reading the y-value at x=2 and stopping, or by reading the x-value off the wrong axis.
Composition and inverse require switching between input and output roles; students read graphs left-to-right only and never practice 'start on the y-axis' for inverse evaluation.
What to do instead
Use a two-pass routine: for f(f(2)), first read f(2) off the graph (output), then re-enter the graph treating that number as a new input. For f⁻¹(3), start on the y-axis at 3 and read across to the curve, then down to x. Do several with a marked-up graph.
Students describe end behavior of a polynomial by its degree alone (e.g., 'degree 4 so both ends up') without checking the sign of the leading coefficient, and they misreport multiplicity effects at zeros (cross vs. touch).
End-behavior rules are memorized as a single fact about even/odd degree, and multiplicity is treated as a counting exercise rather than a graphical behavior.
What to do instead
Force a two-factor check: (1) even/odd degree sets same/opposite directions, (2) sign of leading coefficient sets up/down. For zeros, state multiplicity and its parity explicitly: odd multiplicity crosses, even multiplicity touches, multiplicity ≥3 flattens. Verify with a quick sign chart.
When finding horizontal asymptotes of a rational function, students compare degrees but forget to reduce common factors first, so they report an asymptote where there is actually a hole.
The degree-comparison rule is taught as a shortcut, and students apply it to the unreduced expression without factoring, missing removable discontinuities.
What to do instead
Always factor numerator and denominator and cancel before classifying. Then state holes (at canceled factors, with the y-value from the reduced function) separately from vertical asymptotes (at remaining denominator zeros) and horizontal asymptotes (degree comparison of the reduced form).
Students solve a polynomial or rational inequality such as (x−1)(x+3)/(x−2) ≤ 0 by multiplying through or by testing only one interval, and they include x=2 in the solution set.
Students carry over equation-solving habits (multiply both sides, solve for boundary points) into inequalities, where sign changes and excluded values govern the answer.
What to do instead
Use a sign chart: find all zeros and undefined points, plot them on a number line, test one value per interval, and mark undefined points as always excluded. Write the solution in interval notation and check whether boundary zeros are included based on ≤ vs. <.
Students treat exponential growth/decay as linear, computing a constant difference per step instead of a constant ratio, and they misapply the model y=ab^t when the rate is given as a percent per different time unit (e.g., monthly rate but annual time).
Linear reasoning is over-practiced, and percent-rate-to-base conversion (1+r vs. 1−r) plus time-unit alignment are procedural steps students skip under time pressure.
What to do instead
Before computing, ask 'is the change additive or multiplicative?' and confirm by checking ratios of successive outputs. Convert rates explicitly: growth base = 1+r, decay base = 1−r, and rewrite the exponent so the time unit matches (e.g., b^(12t) for monthly compounding over t years).
Students solve log equations like log(x)+log(x−3)=1 by dropping the logs to get x+(x−3)=1, or by combining to log(x(x−3))=1 and then forgetting to check that both arguments are positive.
Log properties are recalled as 'you can just remove the log', and extraneous-solution checking is treated as optional rather than required by the domain restriction.
What to do instead
Combine logs using product/quotient rules first, rewrite in exponential form, solve, then substitute every candidate back into the original equation and reject any that makes a log argument ≤0. Make the check a mandatory written step.
Students give the general solution to a trig equation as a single value (e.g., sin θ = 1/2 gives only θ=π/6) and ignore the second solution in the period and the +2πn family.
Equation-solving habits from algebra produce one answer, and the periodic symmetry of the unit circle is not internalized as producing multiple solutions.
What to do instead
Solve on the unit circle by drawing the reference angle and marking all quadrants where the sign matches, then list all solutions in [0,2π) before adding 2πn (or πn for tangent). Check the requested interval explicitly.
When converting between rectangular and polar coordinates, students swap r and θ (writing (θ,r)) or use r=sqrt(x²+y²) without choosing the correct quadrant for θ, and they plot polar points by moving along the wrong axis first.
Polar notation is new and the ordered pair (r,θ) conflicts with the familiar (x,y) reading order; arctangent returns only two quadrants, so quadrant adjustment is easy to skip.
What to do instead
Anchor the convention verbally every time: 'r is distance from origin, θ is angle from positive x-axis.' For conversions, compute r=sqrt(x²+y²) and then adjust θ by quadrant (add π when x<0). When plotting, rotate to θ first, then move r units along that ray, including negative r (move opposite the ray).