Precalculus | Index

Precalculus pulls together the functions you met in Algebra 2 and Trigonometry. It gets you ready for calculus.

The course starts with function analysis and with polynomial and rational functions. It goes on to exponential and log models, matrices, conic sections, vectors and parametric equations. It ends with the binomial theorem and a first look at limits.

It’s for high school precalculus and AP Precalculus students, college precalculus students and anyone getting ready for calculus.

What to know first: Algebra 2 and Trigonometry, especially functions, logarithms and the unit circle.

Lessons for this course are coming soon. The list below shows every planned lesson in teaching order.

Units in this course

Unit 1: Function Analysis

  1. Parent Functions — the shape, domain and key points of each basic function family
  2. Even and Odd Functions (coming soon) — test a function for symmetry with its graph or by finding f(−x)
  3. Difference Quotient (coming soon) — simplify the difference quotient, the first step toward the derivative
  4. Rates of Change of Functions (coming soon) — compare rates of change over intervals and see how a changing rate bends a graph up or down

Also see: transformations, composition and inverse functions in Algebra 2, Unit 1.

Unit 2: Polynomial and Rational Functions

  1. Descartes’ Rule of Signs (coming soon) — predict how many positive and negative real zeros a polynomial can have
  2. Polynomial Inequalities (coming soon) — solve polynomial inequalities with the zeros and a sign chart
  3. Slant Asymptotes (coming soon) — find the slant (oblique) asymptote of a rational function with long division
  4. Rational Inequalities (coming soon) — solve inequalities with the variable in a denominator, using a sign chart
  5. Partial Fraction Decomposition (coming soon) — split a rational expression into a sum of simpler fractions

Also see: polynomial division, the rational root theorem and graphing polynomials in Algebra 2, Unit 4, and rational functions in Algebra 2, Unit 6.

Unit 3: Exponential and Logarithmic Models

  1. Logarithmic Scales (coming soon) — read and compare values on log scales like pH, decibels and earthquake magnitude
  2. Semi-Log Plots (coming soon) — see why exponential data makes a straight line when the y-axis uses a log scale
  3. Exponential Regression (coming soon) — fit an exponential model to data and use it to make predictions
  4. Logistic Functions (coming soon) — model growth that levels off at a carrying capacity

Also see: logarithms and the number e in Algebra 2, Unit 7, and Exponential Growth and Decay and Compound Interest (Algebra 1, Unit 6).

Unit 4: Matrices and Systems

  1. Matrix Operations (coming soon) — matrix size, adding and subtracting matrices, and multiplying by a scalar
  2. Matrix Multiplication (coming soon) — when two matrices can be multiplied and how to find each entry
  3. Solving Systems with Augmented Matrices (coming soon) — use row operations to solve a system of linear equations
  4. Determinant of a Matrix (coming soon) — find the determinant of a 2 × 2 or 3 × 3 matrix
  5. Inverse Matrices (coming soon) — find the inverse of a matrix and use it to solve a system
  6. Cramer’s Rule (coming soon) — solve a system of linear equations with determinants
  7. Matrix Transformations (coming soon) — use a 2 × 2 matrix to rotate, reflect or stretch points in the plane

Also see: systems of equations in three variables in Algebra 2, Unit 2.

Unit 5: Conic Sections

  1. Focus and Directrix of a Parabola (coming soon) — write and graph a parabola from its focus and directrix
  2. Ellipses (coming soon) — graph an ellipse and find its center, vertices and foci
  3. Hyperbolas (coming soon) — graph a hyperbola and find its vertices, foci and asymptotes
  4. Identifying Conic Sections (coming soon) — tell which conic an equation makes and complete the square to write it in standard form

Also see: the equation of a circle in Geometry, Unit 10.

Unit 6: Vectors and Parametric Equations

  1. Magnitude and Direction of a Vector (coming soon) — write a vector in component form and find its length and direction
  2. Adding and Subtracting Vectors (coming soon) — add, subtract and scale vectors with drawings and with components
  3. Dot Product (coming soon) — find the dot product and the angle between two vectors
  4. Vector Word Problems (coming soon) — solve force, velocity and navigation problems with vectors
  5. Parametric Equations (coming soon) — graph a curve from x(t) and y(t), and eliminate the parameter
  6. Projectile Motion with Parametric Equations (coming soon) — model the path of a thrown or launched object

Also see: polar coordinates and graphing polar equations in Trigonometry, Unit 9.

Unit 7: Sequences, Series and the Binomial Theorem

  1. Permutations and Combinations (coming soon) — count arrangements and selections with factorials
  2. Binomial Theorem (coming soon) — expand a power of a binomial with binomial coefficients and Pascal’s triangle
  3. Mathematical Induction (coming soon) — prove that a statement is true for every positive whole number

Also see: Arithmetic Sequences and Geometric Sequences (Algebra 1), and sigma notation and series in Algebra 2, Unit 8.

Unit 8: Introduction to Limits

  1. Introduction to Limits (coming soon) — what a limit means and how to estimate one from a graph or a table
  2. One-Sided Limits (coming soon) — limits from the left and the right, and when a limit does not exist

Also see: limit laws, evaluating limits and continuity in Calculus 1, Unit 1.

For Teachers: Standards Alignment

Lessons follow the AP Precalculus course framework and a typical college precalculus course. AP Precalculus Unit 3, on trig and polar functions, is in Trigonometry. Many topics from AP Precalculus Units 1 and 2, such as end behavior, rational functions and logarithms, are in Algebra 2. This course links to them. It also covers the Common Core plus (+) standards for vectors and matrices (N-VM), the binomial theorem (A-APR.5) and ellipses and hyperbolas (G-GPE.3), plus parabolas from a focus and directrix (G-GPE.2).

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