AP Calculus Review Notes

Overview

This lecture reviews key algebra, function, and precalculus concepts needed for AP Calculus AB, covering lines, functions, graphing, domains, parametrics, trig, exponentials, logs, and inverses.

Linear Equations and Graphing

• Slope-intercept form: y = mx + b; use point-slope when given a point and slope, then solve for y.
• Vertical lines: x = a constant; horizontal lines: y = a constant.
• Parallel lines have equal slopes; perpendicular lines have opposite reciprocals.
• Be careful with negative signs and distributing during algebraic manipulation.

Absolute Value and Piecewise Functions

• Absolute value functions can be written piecewise, split at the 'hinge point' where the inside equals zero.
• Graphs of absolute value functions are V-shaped; determine domain and range from the graph.
• For sums of absolute values, split into intervals at each hinge point and define corresponding linear pieces.

Functions: Exponential, Square Root, and Logarithmic

• Exponential functions: y = a·b^(x–h) + k; identify asymptote, domain, and range.
• For y = –2 + √(x–4): domain x ≥ 4; range y ≥ –2; no y-intercept if domain excludes x = 0.
• Logarithms: y = log_b(x–h) + k has a vertical asymptote at x = h.
• Convert between log and exponential forms for table of values and solving.

Domain and Range, Function Composition

• Domain: set of all possible x-values; check for restrictions like square roots (radicand ≥ 0) and division by zero.
• When composing functions, the domain is limited to values valid in the inner and outer function.
• Rational and radical expressions need careful domain analysis.

Parametric Equations

• Parametric equations define x(t) and y(t); plot points for various t to graph.
• Eliminate the parameter t using algebraic identities to find a Cartesian (x, y) equation.
• Typical applications: ellipses, parabolas, and other curves.

Trigonometry and Degrees/Radians

• Convert degrees to radians: multiply by π/180; radians to degrees: multiply by 180/π.
• Know common conversions (30° = π/6, –3π/4 = –135°, etc.).
• Trig function graphing: amplitude = |coefficient|; period = 2π/b; phase shift from inside function.

Inverses and Algebraic Manipulation

• Find inverse by solving for x in terms of y, then swap variables.
• Prove a function and its inverse by showing f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
• For logarithmic and exponential equations, use properties like ln(a) – ln(b) = ln(a/b) and exponent rules.

Key Terms & Definitions

• Slope-intercept form — Linear equation: y = mx + b, where m = slope, b = y-intercept.
• Piecewise function — A function defined by different expressions over different intervals.
• Asymptote — A line that a graph approaches but never touches.
• Domain — All valid x-values for a function.
• Range — All possible output (y) values of a function.
• Parametric equations — Express x and y in terms of a third variable, typically t.
• Amplitude — Height from centerline to peak in a trig function.
• Period — Length needed for a trig function to repeat.
• Inverse function — Function that reverses another function's effect.

Action Items / Next Steps

• Review and practice problems from the PDF solution key.
• Study all function types, their domains, and graphing techniques.
• Complete any assigned homework or prep for the upcoming test.