Ten lessons following Units 1 to 3 of the course: polynomial and rational functions, exponentials and logarithms, trigonometric and polar functions, plus modeling.
A reasoning guide to AP Precalculus, not a full course. Unit titles and weightings follow the College Board course page; Unit 4 is not assessed on the exam and is not covered. All numbers in examples are invented for practice.
Algebra 2 level algebra and basic geometry.
Course outline
Rates of change and average rate over an interval
Compute and interpret average rates of change for functions given in tables, graphs and formulas.
Polynomial functions: zeros, end behavior and multiplicity
Connect the degree, leading coefficient, zeros and multiplicities of a polynomial to its graph.
Rational functions: asymptotes, holes and end behavior
Find vertical and horizontal asymptotes and holes and describe rational function behavior.
Function transformations, composition and inverses
Describe shifts, stretches and reflections, compose functions and find inverses.
Exponential functions: constant ratios and modeling
Model growth and decay with exponentials and recognize them from tables and graphs.
Logarithms: inverses of exponentials and their properties
Evaluate logarithms, use their properties and solve exponential and logarithmic equations.
Trigonometric functions: the unit circle and graphs
Use the unit circle, radians and amplitude, period and midline to describe sinusoidal functions.
Trigonometric equations, identities and inverse trig functions
Solve trigonometric equations, apply core identities and use inverse trigonometric functions with correct ranges.
Polar functions and parametric extensions
Plot polar functions, convert coordinates and read rate behavior from polar graphs.
Modeling with functions and choosing the right model
Select among linear, polynomial, exponential, logarithmic and sinusoidal models and justify the choice with data.
Sources and curriculum note
Reviewed October 6, 2026. Confirm format on the College Board site for your exam year.
Read every lesson below. The interactive reader above contains the same explanations, with visual tools and quizzes.
1. Rates of change and average rate over an interval
Learning goal: Compute and interpret average rates of change for functions given in tables, graphs and formulas.
The average rate of change of f from x = a to x = b is (f(b) - f(a))/(b - a). It is the slope of the secant line through the two points. The units are output units per input unit, so a distance in miles over time in hours gives miles per hour.
A function is increasing on an interval when larger inputs give larger outputs. The rate of change tells whether it is increasing or decreasing, and whether the rates themselves are rising or falling tells concavity. A graph that bends upward is concave up: the average rates over equal-width intervals are increasing.
With a table of values, compare average rates over equal input steps. If the rates keep growing, the function is concave up on that stretch; if they keep shrinking, concave down. Unequal steps need the rate computed for each, not the raw output differences.
Rates of change summarize the function's behavior and do not require any calculus. They also set up the later ideas: linear functions have constant rate, quadratic functions have rates that change linearly, and exponential functions have rates proportional to the current value.
Worked example
f(1) = 4 and f(5) = 20 (invented). Find the average rate from x = 1 to 5.
Output change: 20 - 4 = 16.
Input change: 5 - 1 = 4.
16/4.
Average rate is 4.
Practice problem and solution
g(2) = 7 and g(6) = 19. Find the average rate of change on [2, 6].
(19 - 7)/(6 - 2) = 12/4 = 3.
Mental model: Average rate is the secant slope. Use equal steps to read concavity.
Common trap: Dividing input change by output change.
2. Polynomial functions: zeros, end behavior and multiplicity
Learning goal: Connect the degree, leading coefficient, zeros and multiplicities of a polynomial to its graph.
A polynomial's degree and leading coefficient control end behavior. An even degree has both ends in the same direction, upward for a positive leading coefficient and downward for a negative one. An odd degree has ends in opposite directions.
A real zero is an input where the function is 0, an x-intercept. The factor (x - r) gives the zero r. The multiplicity of the zero is the exponent on that factor. The graph crosses the axis at odd multiplicity and touches and turns at even multiplicity.
A polynomial of degree n has at most n real zeros and at most n - 1 turning points. Complex zeros of real polynomials come in conjugate pairs. Polynomial division and the factor theorem help find zeros: if f(r) = 0, then (x - r) is a factor.
To write a polynomial from a graph, read zeros and multiplicities, write the factors, and use one more point to find the leading constant. The sign of that constant should match the end behavior you observed.
Worked example
A polynomial has zeros at 1 (multiplicity 1) and 3 (multiplicity 2), and f(0) = 9 (invented). Find the constant.
Form k(x - 1)(x - 3)^2.
f(0) = k(-1)(9) = -9k.
-9k = 9.
k = -1.
Practice problem and solution
Zeros are 2 (multiplicity 1) and -1 (multiplicity 2). What is the degree of the simplest polynomial with these zeros?
The multiplicities add: 1 + 2 = 3.
Mental model: Degree gives ends. Even multiplicity bounces. Use a point to find the constant.
Common trap: Reading a bounce as a simple crossing.
3. Rational functions: asymptotes, holes and end behavior
Learning goal: Find vertical and horizontal asymptotes and holes and describe rational function behavior.
A rational function is a ratio of polynomials. Where the denominator is 0 and the numerator is not, there is a vertical asymptote. Where a common factor cancels, there is a hole instead, and the cancelled factor still excludes that input from the domain.
End behavior is decided by degree. If the degree of the numerator is less than the denominator, y = 0 is the horizontal asymptote. If degrees are equal, y equals the ratio of leading coefficients. If the numerator is higher by one, there is a slant asymptote found by division, and higher still gives polynomial end behavior.
To find where a rational function is positive or negative, mark every zero of numerator and denominator on a number line and test each interval. Multiplicity tells you whether the sign changes at each mark: odd changes, even does not.
For f(x) = (x^2 - 1)/(x - 1), factoring gives (x - 1)(x + 1)/(x - 1), so f(x) = x + 1 with a hole at x = 1, at the point (1, 2). For (x + 1)/(x - 1), the factor does not cancel, so x = 1 is a vertical asymptote.
Common trap: Calling a cancelled factor a vertical asymptote.
4. Function transformations, composition and inverses
Learning goal: Describe shifts, stretches and reflections, compose functions and find inverses.
For g(x) = a f(b(x - h)) + k, h shifts right, k shifts up, a stretches vertically and may reflect across the x-axis if negative, and b stretches horizontally by a factor of 1/b. Horizontal changes act on x and behave opposite to what the sign suggests.
Composition applies one function to the output of another: (f of g)(x) = f(g(x)). The domain is the set of x in the domain of g whose outputs are in the domain of f. The order matters: f of g and g of f are usually different.
An inverse undoes the function: f^-1(f(x)) = x. Only one-to-one functions have inverses. Swap x and y and solve, and note that the graphs are reflections over y = x. A restricted domain makes a non-one-to-one function invertible, such as x^2 on x >= 0.
Check composition and inverses with a point: if f(2) = 5, then f^-1(5) = 2. Use a table to read inverse values by swapping columns. These tools support the logarithm and trigonometric inverse work later in the course.
Worked example
Let f(x) = 2x + 3. Find the inverse (invented practice).
Write y = 2x + 3.
Swap: x = 2y + 3.
Solve: y = (x - 3)/2.
Inverse is (x - 3)/2.
Practice problem and solution
Let f(x) = x + 4 and g(x) = 3x. Find f(g(2)).
g(2) = 6, then f(6) = 10.
Mental model: Inside changes work opposite. Composition goes inside out. Inverses swap input and output.
Common trap: Confusing the inverse with the reciprocal.
5. Exponential functions: constant ratios and modeling
Learning goal: Model growth and decay with exponentials and recognize them from tables and graphs.
An exponential function has the form f(x) = a b^x. Equal input steps multiply the output by the same factor b. Linear functions add a constant, exponential functions multiply by a constant. A table with outputs 3, 6, 12, 24 is exponential with b = 2.
If b > 1 the function grows; if 0 < b < 1 it decays. A percent rate r gives b = 1 + r, so 5 percent growth is b = 1.05 and 20 percent decay is b = 0.8. The value a is f(0), the initial amount.
To fit a model through two points, divide the outputs to find b over the input gap, then solve for a. Interpret in context: a is the starting value, and b - 1 is the per-step percent change. Exponentials eventually exceed any polynomial.
Graphs of exponentials have a horizontal asymptote, usually y = 0 for a b^x. Transformations shift the asymptote with the vertical shift. The domain is all real numbers and the range is positive numbers for a positive a.
Worked example
A balance is 200 and grows 5 percent per year (invented). What is it after 2 years?
Factor is 1.05.
200 x 1.05 = 210.
210 x 1.05 = 220.5.
Balance is 220.50.
Practice problem and solution
A quantity is 3, 6, 12 at x = 0, 1, 2. Find f(4).
f(x) = 3 x 2^x, so f(4) = 3 x 16 = 48.
Mental model: Multiply, do not add. Percent rate to factor. Use ratios to find b.
Common trap: Using 5 percent as the factor 5 instead of 1.05.
6. Logarithms: inverses of exponentials and their properties
Learning goal: Evaluate logarithms, use their properties and solve exponential and logarithmic equations.
The logarithm base b of y is the exponent that gives y: log_b(y) = x means b^x = y. It is the inverse of the exponential, defined for positive y. Log base 10 is common and ln is log base e. Since b^0 = 1, log_b(1) = 0 always.
Properties come from exponent rules. log_b(xy) = log_b x + log_b y; log_b(x/y) = log_b x - log_b y; log_b(x^p) = p log_b x. They apply to positive arguments, and sums inside a logarithm do not split. Change of base: log_b x = ln x / ln b.
To solve an exponential equation, take a log of both sides, bring down the exponent and solve. For 3^x = 20, x = ln 20/ln 3, about 2.73. To solve a log equation, combine into one log, rewrite in exponential form and check the domain for extraneous solutions.
The graph of a logarithm is the reflection of the exponential over y = x. It has a vertical asymptote, here x = 0, grows slowly and is concave down. Logarithms turn multiplication into addition, which is why they model scales like pH and decibels.
Worked example
Solve 2^x = 12 (invented practice).
Take ln of both sides.
x ln 2 = ln 12.
x = ln 12 / ln 2.
About 3.585.
Practice problem and solution
Evaluate log_2 32.
2^5 = 32.
Mental model: A log is an exponent. Combine, rewrite, check the domain.
Common trap: Splitting log(x + y) into log x plus log y.
7. Trigonometric functions: the unit circle and graphs
Learning goal: Use the unit circle, radians and amplitude, period and midline to describe sinusoidal functions.
Radian measure is the length of the arc on the unit circle. A full turn is 2 pi, a half turn is pi and a quarter turn is pi/2. To convert degrees to radians multiply by pi/180. On the unit circle, the point at angle theta is (cos theta, sin theta).
Know the special angles: for pi/6, sin is 1/2 and cos is sqrt(3)/2; for pi/4, both are sqrt(2)/2; for pi/3, sin is sqrt(3)/2 and cos is 1/2. Tangent is sin over cos and is undefined where cos is 0.
A sinusoid f(x) = a sin(b(x - c)) + d has amplitude |a|, period 2 pi/|b|, phase shift c and midline y = d. The maximum is d + |a| and the minimum is d - |a|. Cosine is the same shape shifted by pi/2.
To write a sinusoid from data, find the midline as the average of max and min, amplitude as half the difference, period from the distance between peaks, and then choose sine or cosine with a shift. Check with a point.
Worked example
Find the amplitude and midline of y = 3 sin(2x) + 5 (invented).
Amplitude is 3.
Midline is y = 5.
Period is 2 pi/2 = pi.
Max 8, min 2.
Practice problem and solution
A sinusoid has max 9 and min 1. What is its midline y-value?
The midline is (9 + 1)/2 = 5.
Mental model: Midline is the average of max and min. Period is 2 pi over b.
Common trap: Reading the amplitude as the maximum value.
8. Trigonometric equations, identities and inverse trig functions
Learning goal: Solve trigonometric equations, apply core identities and use inverse trigonometric functions with correct ranges.
The Pythagorean identity sin^2 + cos^2 = 1 gives other forms, such as 1 + tan^2 = sec^2. Sum formulas: sin(a + b) = sin a cos b + cos a sin b and cos(a + b) = cos a cos b - sin a sin b. Double angle: sin 2a = 2 sin a cos a.
To solve a trigonometric equation, isolate the trig function, find the reference angle, use the quadrant signs and add periods. For sin x = 1/2, the solutions on [0, 2 pi) are pi/6 and 5 pi/6. For a general solution add 2 pi k.
The inverse trig functions have restricted ranges so they are functions: arcsin on [-pi/2, pi/2], arccos on [0, pi] and arctan on (-pi/2, pi/2). arcsin(1/2) is pi/6 even though 5 pi/6 also has sine 1/2.
Equations with a multiple angle, such as sin(2x) = 1/2, need the angle 2x to cover twice the interval before dividing. Always write the interval for the argument first. Check solutions in the original equation when squaring or dividing.
Worked example
Solve sin x = 1/2 on [0, 2 pi) (invented practice).
Reference angle pi/6.
Sine is positive in quadrants I and II.
x = pi/6.
x = pi - pi/6 = 5 pi/6.
Practice problem and solution
How many solutions does sin x = 1/2 have on [0, 2 pi)?
One in quadrant I and one in quadrant II.
Mental model: Write the interval. Use the ranges for inverses. Check for extraneous solutions.
Common trap: Giving only the reference angle and missing the second quadrant.
9. Polar functions and parametric extensions
Learning goal: Plot polar functions, convert coordinates and read rate behavior from polar graphs.
A polar point (r, theta) is a distance r from the origin at angle theta. Conversion: x = r cos theta, y = r sin theta, and r^2 = x^2 + y^2. A negative r plots in the opposite direction. The same point has many polar representations.
The graph of r = a is a circle of radius |a|. The graph of r = a sin theta is a circle through the origin, and r = a + b cos theta can be a limacon. Roses of the form r = a cos(n theta) have 2n petals when n is even and n petals when n is odd.
To read a polar graph, track r as theta increases: when r is increasing, the point moves away from the origin; when r is decreasing, toward the origin; where r = 0 the graph passes through the origin. This is how AP Precalculus asks you to describe polar behavior.
Rectangular and polar forms describe the same curve in different languages. Convert x^2 + y^2 = 9 to r = 3. Convert r = 2 cos theta by multiplying by r to get r^2 = 2r cos theta, so x^2 + y^2 = 2x, a circle centered (1, 0).
Worked example
Convert (3, pi/2) to rectangular form (invented practice).
x = 3 cos(pi/2) = 0.
y = 3 sin(pi/2) = 3.
Point is (0, 3).
On the y-axis.
Practice problem and solution
Convert the polar point (2, 0) to its x-coordinate.
x = 2 cos 0 = 2.
Mental model: x = r cos, y = r sin. Track r as theta grows.
Common trap: Forgetting that negative r reverses direction.
10. Modeling with functions and choosing the right model
Learning goal: Select among linear, polynomial, exponential, logarithmic and sinusoidal models and justify the choice with data.
Choose a model by the shape of the data and the context. Constant differences suggest linear, constant ratios exponential, a repeating pattern sinusoidal, and rapid growth that slows logarithmic. A context such as compound interest or cooling points to a family before any calculation.
A residual is the actual value minus the predicted value. A good model has residuals that look random, with no pattern. A curved pattern in the residuals suggests the wrong model family. Technology computes regression, but you must explain the choice.
A model has a domain where it makes sense. A polynomial that fits ten points may behave wildly outside them. Extrapolation is risky, and the AP exam expects you to comment on whether a prediction is reasonable in context. Parameters should be interpreted with units.
Piecewise or combined models can fit better: an exponential plus a constant, or a sinusoid on a rising midline. In each case, say what each part of the formula contributes. Communicating the reasoning is part of the AP score.
Worked example
A residual is actual minus predicted. If the actual is 12 and the prediction is 9 (invented), what is it?
Actual minus predicted: 3.
Positive means the model underpredicts.
Compare to other residuals.
Look for patterns.
Practice problem and solution
Values are 4, 12, 36. What is the common ratio?
12/4 = 3 and 36/12 = 3.
Mental model: Match model to pattern. Residuals show fit. Describe limits.
Common trap: Extrapolating a model far outside its data.