Summary
Precalculus prepares high school students for calculus by studying functions and their graphs in depth. It moves from basic function ideas through polynomial, exponential, logarithmic and trigonometric functions, and ends with applications such as vectors, polar coordinates and conic sections. The focus is on reading formulas, tables and graphs, and on modeling real situations.
Chapter 1: Functions and Chapter 2: Linear Functions lay the foundation. Students learn function notation, evaluating versus solving, and domain and range in interval notation. They also use the vertical and horizontal line tests and meet the toolkit functions, such as , and . Piecewise functions and average rate of change, , follow. The linear chapter covers slope, writing equations from two points, and parallel and perpendicular lines, where perpendicular slopes multiply to . It also covers intersections and break-even comparisons, such as two truck-rental plans that cost the same at 100 miles. It ends with data fitting: interpolation, least-squares regression and the correlation coefficient .
Chapter 3: Polynomial and Rational Functions studies how a polynomial behaves far to the left and right, which matches its leading term. It also covers degree and turning points, vertex form of a quadratic, and the quadratic formula. Students see how the multiplicity of a zero shapes the graph at that intercept. Tools for finding zeros include long division, the factor theorem, the rational roots theorem and conjugate pairs of complex zeros, and the chapter treats slant asymptotes. Chapter 4 turns to exponential and logarithmic functions. Topics include growth and decay, , compound interest, the number , log properties, change of base, and solving exponential equations.
Chapter 5: Trigonometric Functions of Angles starts from the distance formula and the circle equation. It then builds radian measure, arclength , sine and cosine on the unit circle, the Pythagorean Identity, and exact values at , and . Chapter 6 treats periodic functions. Students find amplitude, midline and period, using , and model situations such as a Ferris wheel ride on the London Eye. The chapter also covers the tangent graph and inverse trigonometric functions on restricted domains.
Chapter 7: Trigonometric Equations and Identities covers solving trig equations by factoring and by substituting identities. It presents sum, difference and double angle identities and shows how to rewrite as a single sinusoid. It also builds models with a changing midline and damped harmonic motion. Chapter 8: Further Applications of Trigonometry presents the Law of Sines and the Law of Cosines and introduces polar coordinates. It covers the polar form of complex numbers with Euler's Formula, DeMoivre's Theorem, and vectors, including components, magnitude, direction and the dot product.
Chapter 9: Conics defines the ellipse and hyperbola by their distances to two foci. Students write standard-form equations, complete the square to find the center, and locate vertices, foci and asymptotes. These curves are tied to slicing a double cone with a plane.
Key ideas:
- A function gives exactly one output for each input, and graphs can be tested for this property.
- Rate of change links linear functions, slope and data models.
- A polynomial's end behavior and the shape of its graph at each intercept follow from its leading term and the multiplicity of each zero.
- Logarithms are inverses of exponentials and are used to solve for unknown exponents.
- Sine and cosine are coordinates on the unit circle, and sinusoids model periodic motion.
- Trig identities turn hard equations into ones that can be factored.
- The Laws of Sines and Cosines solve any triangle, and vectors and polar form describe direction and magnitude.
- Conics are defined by distances to foci and are graphed from their standard-form equations.