A function maps each input \(x\) in the domain to exactly one output \(y\). Key ideas: domain/range, one-to-one (horizontal line test), inverse functions.
Vertical shift: f(x) + k (up k)
Horizontal shift: f(x − h) (right h)
Reflection: −f(x) over x-axis, f(−x) over y-axis
Stretch/Compress: a·f(bx)
Example
If \(f(x)=x^2\), write the equation of the graph shifted right 3 and up 2.
Answer: \(y=(x-3)^2+2\)
Unit 2
Polynomial Functions
A polynomial of degree \(n\) has at most \(n\) real zeros. The Remainder Theorem: \(f(a)\) equals the remainder when \(f(x)\) is divided by \((x-a)\). Factor Theorem: \((x-a)\) is a factor iff \(f(a)=0\).
Quadratic Formula: x = (−b ± √(b²−4ac)) / (2a)
Discriminant: Δ = b²−4ac
Δ > 0 → two real roots
Δ = 0 → one repeated root
Δ < 0 → two complex roots
Example
Find the remainder when \(f(x)=x^3-4x+5\) is divided by \((x-2)\).
Answer: \(f(2)=8-8+5=5\)
Unit 3
Rational Functions
Vertical asymptotes occur where the denominator is zero (and numerator ≠ 0). Horizontal asymptotes compare degrees of numerator \(n\) and denominator \(m\).
HA rules:
n < m → y = 0
n = m → y = leading coeff ratio
n > m → no HA (oblique asymptote)
Hole: factor cancels in num & denom
Example
Find the horizontal asymptote of \(\dfrac{3x^2+1}{2x^2-5}\).
Answer: \(y=\dfrac{3}{2}\) (equal degrees)
Unit 4
Exponential & Logarithmic Functions
\(\log_b x = y \iff b^y = x\). Logs and exponentials are inverse functions. The natural log \(\ln x = \log_e x\).
Product: log(MN) = log M + log N
Quotient: log(M/N) = log M − log N
Power: log(Mⁿ) = n·log M
Change of base: log_b(a) = ln(a)/ln(b)
Compound: A = P(1+r/n)^(nt)
Continuous: A = Pe^(rt)
Unit circle: radius = 1. Angles in standard position. Radian measure: \(\pi\) rad = 180°.
sin²θ + cos²θ = 1
tan θ = sin θ / cos θ
Arc length: s = rθ (θ in radians)
Period of sin/cos: 2π/|b|
Period of tan: π/|b|
Special: sin30°=1/2, cos60°=1/2, tan45°=1
sin45°=√2/2, cos30°=√3/2
Fundamental identities allow simplification and solving of trigonometric equations.
Double angle:
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 2cos²θ−1 = 1−2sin²θ
tan 2θ = 2tanθ/(1−tan²θ)
Sum/Difference:
sin(A±B) = sinA cosB ± cosA sinB
cos(A±B) = cosA cosB ∓ sinA sinB