Algebra Formulas

Combining Like Terms

ax + bx = (a + b)x

Combining like terms merges coefficients of identical variables to simplify expressions and solve equations more efficiently.

FOIL Method

(a + b)(c + d) = ac + ad + bc + bd

The FOIL Method expands products of two binomials by multiplying each term in the first binomial by each term in the second.

Quadratic Formula

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

The Quadratic Formula solves any equation of the form ax^2 + bx + c = 0 in one step, giving both solutions at once without factoring.

Discriminant

\Delta = b^2 - 4ac

The discriminant, given by b^2 - 4ac, tells you whether a quadratic equation has real solutions and how many without actually solving it.

Completing the Square

x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4}

Completing the square rewrites a quadratic as a perfect square plus a constant, making it easier to solve and find the parabola's vertex.

Vertex Form

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

Vertex Form reveals a parabola's vertex and axis of symmetry at a glance, making it ideal for transformations and optimization problems.

Vertex of a Parabola

h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a}

Find the lowest or highest point of a parabola using the vertex formula, which works for any quadratic equation in standard form.

Sum and Product of Roots

r_1 + r_2 = -\frac{b}{a}, \quad r_1 r_2 = \frac{c}{a}

Quickly find how a quadratic's roots add up and multiply together without solving for them — use when you need those relationships.

Function Notation

y = f(x)

Function notation y = f(x) is a way to write and evaluate functions, showing how input values transform into output values using a concise symbolic format.

Domain and Range

f : D \to R

Domain lists all valid inputs for a function; range lists all possible outputs—use this to understand what a function can do.

Inverse Function

f\big(f^{-1}(x)\big) = x

The inverse function undoes another function: applying it after the original function returns your input value unchanged.

Composite Function

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

Composite functions apply one function to the output of another, letting you combine multiple transformations into a single operation.

Direct Variation

y = kx

Direct variation describes how one quantity changes proportionally with another, letting you find unknown values when two variables stay linked by a constant ratio.

Inverse Variation

y = \frac{k}{x}

Inverse variation describes when two quantities always multiply to give a constant value, with one variable increasing as the other decreases.

Absolute Value Equations

|ax + b| = c \iff ax + b = \pm c

Solve equations with absolute value by splitting into two cases where the expression inside equals the positive value or its negative.

Piecewise Functions

f(x) = \begin{cases} f_1(x) & x \in D_1 \\ f_2(x) & x \in D_2 \end{cases}

A piecewise function uses different formulas for different input ranges, letting you define relationships that change behavior at specific boundaries.

Function Transformations

y = a\,f\big(b(x - h)\big) + k

Function transformations describe how shifting, stretching, and flipping graphs of functions create new related functions from a parent function.

Even and Odd Functions

f(-x) = f(x) \quad \text{or} \quad f(-x) = -f(x)

Determine whether a function is even (f(-x) = f(x)) or odd (f(-x) = -f(x)) to identify graph symmetry and simplify calculations.

Average Rate of Change

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

Average Rate of Change measures how quickly a function changes between two points, showing the slope of the line connecting them.

Difference Quotient

\frac{f(x + h) - f(x)}{h}

The Difference Quotient measures how much a function changes over a small interval, forming the foundation for calculating derivatives in Calculus.

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