What it covers: How quantities change in tandem (rates of change, concavity) and the two headline function families — polynomial and rational functions.
Exam weight: About 30–40% of the AP Precalculus exam — the largest of the three tested units.
The big question: How do the algebraic features of a function (its factors, degree, and leading term) determine the shape and behavior of its graph?
A function is increasing/decreasing as outputs rise/fall. Concave up = rate of change increasing; concave down = rate of change decreasing. A point of inflection is where concavity switches.
Rates of Change
Average rate = (f(b) − f(a))/(b − a), the secant slope. Rate at a point = the tangent slope. Linear → constant rate; quadratic → rate changes at a constant rate.
Polynomial Zeros
A degree-n polynomial has n complex zeros with multiplicity. Odd multiplicity → graph crosses the axis; even multiplicity → graph touches and turns. Non-real zeros come in conjugate pairs.
Polynomial End Behavior
Set by the leading term. Even degree → both ends same way; odd degree → opposite ways. Positive leading coefficient → right end goes to +∞.
Rational Zeros & Asymptotes
Zeros: numerator = 0 (denominator ≠ 0). Vertical asymptotes: denominator = 0 (numerator ≠ 0). Output grows without bound near a vertical asymptote.
Horizontal & Slant Asymptotes
Compare degrees. Num < den → y = 0. Num = den → ratio of leading coefficients. Num one more than den → slant asymptote (from division).
Holes
A common factor in numerator and denominator cancels, leaving a removable discontinuity (a hole) instead of a vertical asymptote at that input.
Average rate of change — the slope of the secant line, (f(b) − f(a))/(b − a), over an interval.
Concavity — concave up where the rate of change increases; concave down where it decreases.
Point of inflection — where a graph switches concavity.
Multiplicity — how many times a factor repeats; odd = cross, even = touch and turn.
End behavior — what outputs approach as x → ±∞, set by the leading term and written with limit notation.
Vertical asymptote — an input where the denominator is zero but the numerator is not.
Horizontal / slant asymptote — the end-behavior line of a rational function, found by comparing degrees.
Hole (removable discontinuity) — a single missing point caused by a canceled common factor.
Transformation — a translation (additive) or dilation/reflection (multiplicative) of a parent function.
Function model — a chosen function type that fits data, along with the assumptions it depends on.
Key themes to remember
Algebraic form reveals graphical behavior. Factors give zeros, the leading term gives end behavior, and the denominator gives asymptotes and holes.
Every feature has a "why." Don't memorize that there's an asymptote — know it's because the denominator is zero where the numerator is not.
Rates of change describe how, not just where. Increasing/decreasing tells direction; concavity tells whether that change is speeding up or slowing down.
Represent functions four ways. Graphical, numerical, analytical, and verbal — the exam moves between them constantly.
Communicate with precise language. "Increasing at a decreasing rate" and "concave down" earn points; "goes up" does not.
Common exam traps
Increasing ≠ concave up. A function can increase while concave down (rising but slowing). Direction and concavity are separate ideas.
Vertical asymptote vs. hole. A denominator zero gives a vertical asymptote only if the factor does NOT also cancel from the numerator; if it cancels, it's a hole.
Even vs. odd multiplicity. Even multiplicity makes the graph touch and bounce off the x-axis; odd multiplicity makes it cross. Don't mix them up.
Horizontal asymptote rules depend on degree. Numerator degree greater than the denominator's means there is NO horizontal asymptote (it may be a slant asymptote instead).
Inside changes act backwards. f(x − 3) shifts RIGHT by 3, not left. Horizontal transformations are the opposite of what the sign suggests.
Complex zeros are still zeros. A degree-n polynomial always has n zeros with multiplicity — but non-real ones are not x-intercepts.
Leading term, not the whole polynomial, sets end behavior. Ignore the lower-degree terms when describing behavior as x → ±∞.