Math 30 Midterm Cheat Sheet

Functions

• Operations/Composition of a function

  • (f + g)(x) = f(x) + g(x)

  • (f - g)(x) = f(x) - g(x)

  • (f o g)(x) = f(g(x))

• Inverse functions

  • Swap x and y → Range becomes domain et vice versa

• Domain/Range

  • Think logically and then apply math

Transformations

• Parent function transformations/combining transformations

  • 1. Stretches

    • Graph and equation = opposites

    • Mapping = rep of graph

    • Variable replacement = opposite of graph

  • 2. Reflections

  • 3. Translations

• Radical transformations

  • Domain restrictions

• Restrictions to make a function

  • Vertical line test, restrictions on domains

Exponents/Logarithms

• Asymptotes

  • If exponential: Horizontal asymptote

    • y = b

  • If logarithmic: Vertical asymptote

    • x = -b to find asymptote

• Log rules

• Change of base

  • Sometimes just plug and chug based on available answers

• Applications

  • y = Ab^(t/p)

    • t and p same unit or doesn’t work

      • t = time spent

      • p = length of each cycle

    • b = 1 + (r/cycle) unless decay

    • if calculating interest: b = 1 +i/2

Trigonometry

• General

  • Amp = (Max - Min)/2

  • HT = Phase shift

  • d = (Max + Min)/2

  • Domain never affected, everything can affect the zeros

  • b = time to complete a cycle

• Sine Functions → y = a(sin[b(x - c)] + d

  • Period = 2π, = 2π/b

  • Amp = 1

  • D: (-∞, ∞), R: [-1, 1]

  • x-int = πn, n ∈ I

  • y-int = 0

• Cosine Functions → y = a(cos[b(x - c)] + d

  • Period = 2π, = 2π/b

  • Amp = 1

  • D: (-∞, ∞), R: [-1, 1]

  • x-int = π/2 + πn, n ∈ I

  • y-int = 1

• Tangent Functions → y = a(tan[b(x - c)] + d

  • Period = π, = π/b

  • Amp = N/A

  • D: (x ≠ π/2 + πn, n ∈ I), R: (-∞, ∞)

  • x-int = πn, n ∈ I

  • y-int = 0

• Squaring functions

  • Take outcome of function and square after