High school · Aligned to AP Precalculus
Algebra I to AP Precalculus — study guide & practice questions
Linear equations and functions through AP Precalculus: polynomial, rational, exponential, log and trig functions.
Study guide
Foundations: Expressions & Properties
- Order of operations (PEMDAS) and the distributive/commutative/associative properties govern how expressions simplify.
- Combine only like terms, and translate word phrases carefully — "less than" reverses word order.
Solving Linear Equations
- Use inverse operations to isolate the variable; distribute and combine like terms before moving terms across the equals sign.
- Absolute value equations split into two cases; literal equations solve for one variable in terms of the others.
Solving Linear Inequalities
- Solve like an equation, except multiplying or dividing by a negative flips the inequality sign.
- Compound inequalities combine two conditions with "and" (overlap) or "or" (union).
Linear Functions & Graphing
- Slope m = (y2 − y1) / (x2 − x1); a function assigns exactly one output to each input.
- A line can be graphed or written from slope-intercept, standard, or point-slope form.
Systems of Linear Equations
- Solve a system by graphing, substitution, or elimination; the solution is where both equations are satisfied at once.
- Parallel lines give no solution; identical lines give infinitely many.
Exponents & Exponential Functions
- The product, quotient and power rules combine exponents; scientific notation writes numbers as a × 10ⁿ.
- Growth/decay is modeled by y = a(1 ± r)ᵗ, graphed as a curve approaching a horizontal asymptote.
Polynomials & Factoring
- Multiply polynomials with FOIL or distribution; always factor out the GCF first.
- Recognize difference-of-squares and perfect-square-trinomial patterns before trial and error.
Quadratic Functions & Equations
- y = ax² + bx + c graphs as a parabola with axis of symmetry x = −b/(2a).
- Solve by factoring, the quadratic formula, or completing the square — the discriminant predicts how many real solutions exist.
Radicals & Absolute Value
- Simplify √a by pulling out perfect-square factors; solve radical equations by isolating the radical and squaring both sides, then check for extraneous solutions.
- |expr| = k splits into expr = k or expr = −k; |expr| < k becomes −k < expr < k, while |expr| > k becomes expr < −k or expr > k.
Sequences & Data Analysis
- Arithmetic sequences add a constant difference d (aₙ = a₁ + (n−1)d); geometric sequences multiply by a constant ratio r (aₙ = a₁·r^(n−1)).
- A scatter plot's line of best fit shows a trend, but correlation never proves causation.
Course modules
10 foundation modules free to read here; 4 AP-level and advanced modules with Pro. Every module ends with a quiz in the app.
1. Foundations: Expressions & Properties Free
Order of operations, the properties that justify every algebraic move, and translating words into expressions.
Read the 4 lessons →2. Solving Linear Equations Free
Isolating a variable through one-step, multi-step, literal, and absolute value equations.
Read the 4 lessons →3. Solving Linear Inequalities Free
One-variable, multi-step, compound, and absolute value inequalities — and when to flip the sign.
Read the 4 lessons →4. Linear Functions & Graphing Free
Function notation, slope, the three forms of a line, and writing an equation from given information.
Read the 4 lessons →5. Systems of Linear Equations Free
Solving systems by graphing, substitution, and elimination, plus systems of inequalities.
Read the 4 lessons →6. Exponents & Exponential Functions Free
The laws of exponents, scientific notation, and modeling growth and decay.
Read the 4 lessons →7. Polynomials & Factoring Free
Operating on polynomials and the factoring patterns that undo multiplication.
Read the 4 lessons →8. Quadratic Functions & Equations Free
Graphing parabolas and three methods for solving quadratic equations.
Read the 4 lessons →9. Radicals & Absolute Value Free
Simplifying square roots, solving radical and absolute value equations, and absolute value inequalities.
Read the 4 lessons →10. Sequences & Data Analysis Free
Arithmetic and geometric sequences, and reading trends in scatter plot data.
Read the 4 lessons →11. AP Precalc: Function Analysis, Composition & Inverses Pro · AP Precalc · Unit 1–2
Rates of change, composing and inverting functions, and the transformation rules that shift, stretch, and reflect any graph.
12. AP Precalc: Polynomial & Rational Functions Pro · AP Precalc · Unit 1
End behavior, zeros and multiplicity, the factor theorem, and asymptotes and holes of rational functions.
13. AP Precalc: Exponential & Logarithmic Functions Pro · AP Precalc · Unit 2
Exponential models, logarithm rules, solving exponential and log equations, and recognizing exponential data.
14. AP Precalc: Trigonometric & Polar Functions Pro · AP Precalc · Unit 3
Radians and the unit circle, sinusoidal models, solving trig equations, and polar coordinates.
Sample practice questions
From module 1, Foundations: Expressions & Properties. Pick your answers, then check them in the app, with an explanation for every question.
1. Simplify: 6x − 2(x − 4).
2. Translate: "the quotient of a number and 6, increased by 3."
Sample flashcards
4 of the cards in this course's deck. Tap a card to flip it.
Solving EquationsSolve for x: 3x + 5 = 20.
x = 5 (subtract 5 from both sides to get 3x = 15, then divide by 3).
Slope & GraphingWhat does the slope of a line measure, and how is it calculated from two points?
The rate of change (steepness/direction) of a line; slope m = (y2 − y1) / (x2 − x1).
Slope & GraphingWhat is slope-intercept form, and what do m and b represent?
y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis).
Slope & GraphingWhat does it mean for two lines to be parallel vs. perpendicular, in terms of slope?
Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals of each other (product = −1).
The full course is in LectureMate's Course Library: every module quiz with explanations, the complete flashcard deck with spaced repetition, and a 6-step strategy for AP Precalculus questions. Foundation modules are free; AP-level and advanced modules need Pro.