📋 Exam 1 Review — Formula Sheet

All key formulas from 6 sections at a glance.

§1.fn

Domain of a Rational Function

f(x) = \frac{p(x)}{q(x)} \implies \text{exclude all } x \text{ where } q(x) = 0

Set the denominator equal to zero and solve — those x-values are NOT in the domain

Domain of a Square Root Function

f(x) = \sqrt{g(x)} \implies g(x) \geq 0

The expression under the radical must be zero or positive

Interval Notation Quick Reference

(a, b) \;\text{open} \quad [a, b] \;\text{closed} \quad [a, b) \;\text{half-open} \quad (-\infty, \infty) \;\text{all reals}

Infinity always gets a parenthesis, never a bracket

§1.comp

Composition & Transformations

Average Rate of Change

\frac{f(b) - f(a)}{b - a}

Slope of the secant line between (a, f(a)) and (b, f(b))

Composition of Functions

(f \circ g)(x) = f(g(x))

Apply g first, then f — read right to left

Vertical & Horizontal Shifts

f(x) + k \;\text{(up/down)} \qquad f(x - h) \;\text{(right } h \text{)}

Inside changes move opposite: x - h shifts RIGHT h units

Stretches & Reflections

a \cdot f(x) \;\text{(vert. stretch)} \qquad f(bx) \;\text{(horiz. compress by } \tfrac{1}{b}\text{)}

-f(x) reflects over x-axis; f(-x) reflects over y-axis

Absolute Value (Vertex Form)

f(x) = a|x - h| + k, \quad \text{vertex } (h, k)

a > 0 opens up (V), a < 0 opens down (∧)

§1.inv

Inverse Functions

Inverse Verification

f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x

Both compositions must equal x — checking only one isn't enough

Finding an Inverse Algebraically

y = f(x) \;\longrightarrow\; \text{swap } x \text{ and } y \;\longrightarrow\; \text{solve for } y = f^{-1}(x)

Replace f(x) with y, swap every x and y, then isolate y

Domain–Range Relationship

\text{Domain of } f = \text{Range of } f^{-1}, \quad \text{Range of } f = \text{Domain of } f^{-1}

Inputs and outputs swap roles when you invert

§2.lin

Linear Functions

Slope

m = \frac{y_2 - y_1}{x_2 - x_1}

Rise over run — the rate of change between any two points

Slope-Intercept Form

y = mx + b

m = slope, b = y-intercept

Point-Slope Form

y - y_1 = m(x - x_1)

Use when you know the slope and one point $(x_1, y_1)$

Parallel & Perpendicular Slopes

\text{Parallel: } m_1 = m_2 \qquad \text{Perpendicular: } m_1 \cdot m_2 = -1

Perpendicular slopes are negative reciprocals (e.g. $\frac{2}{3}$ and $-\frac{3}{2}$ )

§3.cq

Complex Numbers & Quadratics

Standard Form of a Complex Number

z = a + bi, \quad i^2 = -1

Complex Conjugate Product

(a + bi)(a - bi) = a^2 + b^2

Multiply by the conjugate to clear i from a denominator

Vertex Form of a Quadratic

f(x) = a(x - h)^2 + k

Vertex at (h, k); axis of symmetry x = h

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant

\Delta = b^2 - 4ac

Δ > 0 → two real roots; Δ = 0 → one repeated root; Δ < 0 → two complex roots

§3.poly

Polynomials

End Behavior (Leading Term Test)

f(x) = a_n x^n + \cdots \implies \text{end behavior matches } a_n x^n

Even degree → both ends same direction; odd degree → opposite ends

Division Algorithm

f(x) = d(x) \cdot q(x) + r(x)

dividend = divisor × quotient + remainder (degree of r < degree of d)

Remainder Theorem

f(c) = \text{remainder when } f(x) \div (x - c)

Evaluate f(c) without plugging in — just read the remainder from synthetic division

Factor Theorem

(x - c) \text{ is a factor of } f(x) \iff f(c) = 0

Rational Zero Theorem

\text{Possible rational zeros} = \pm\frac{p}{q}

p = factors of the constant term, q = factors of the leading coefficient