Precalculus Review Notes

Unit 1: Functions and Graphs

Definition of a Function: A function is a relation that assigns exactly one output for each input.

Types of Functions:

  • Linear Functions: Graphs as straight lines, defined by the equation y=mx+by = mx + b where $m$ is the slope and $b$ is the y-intercept.

  • Quadratic Functions: Defined by y=ax2+bx+cy = ax^2 + bx + c, produces parabolic graphs.

  • Polynomial Functions: Functions involving terms up to xnx^n (e.g., y=3x3+2x2+x+1y = 3x^3 + 2x^2 + x + 1).

  • Composition of Functions: If f(x)f(x) and g(x)g(x) are functions, the composition is defined as f(g(x))f(g(x)).

  • Inverse Functions: Functions that reverse the effect of the original function, denoted as f−1(x)f^{-1}(x). The input-output relation is switched.

Unit 2: Trigonometry

Basic Trigonometric Ratios:

  • Sine: extsin(heta)=oppositehypotenuseext{sin}( heta) = \frac{\text{opposite}}{\text{hypotenuse}}

  • Cosine: extcos(heta)=adjacenthypotenuseext{cos}( heta) = \frac{\text{adjacent}}{\text{hypotenuse}}

  • Tangent: exttan(heta)=oppositeadjacentext{tan}( heta) = \frac{\text{opposite}}{\text{adjacent}}

Unit Circle: Circle of radius 1 centered at the origin.

Key angles: 0,π6,π4,π3,π20, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}

Corresponding coordinates give values of sin and cos.

Unit 3: Complex Numbers

Definition of Complex Numbers: Form a+bia + bi where $a$ is the real part and $bi$ is the imaginary part.

Operations:

  • Addition: (a+bi)+(c+di)=(a+c)+(b+d)i(a + bi) + (c + di) = (a+c) + (b+d)i

  • Multiplication: $$(a + bi)(c