Comprehensive Guide to ACT Algebra

Slope

The rate of change of a line, calculated as rise over run.

Slope-Intercept Form

A linear equation in the form y = mx + b, where m is the slope and b is the y-intercept.

Point-Slope Form

A linear equation in the form y - y₁ = m(x - x₁), useful when a point and slope are known.

Standard Form

A linear equation expressed as Ax + By = C, often used in systems of equations.

Direction Flip Rule

When multiplying or dividing both sides of an inequality by a negative number, flip the inequality sign.

Solid Circle

Indicates a value is included, used for ≤ or ≥ in graphing inequalities.

Open Circle

Indicates a value is excluded, used for < or > in graphing inequalities.

System of Equations

A set of two or more equations with the same variables whose solution is the intersection point.

Substitution Method

A method of solving systems of equations by isolating a variable and substituting it into another equation.

Elimination Method

A method of solving systems of equations by adding or subtracting equations to eliminate a variable.

Coincident Lines

Lines that lie on top of each other, having the same slope and y-intercept, resulting in infinitely many solutions.

Parallel Lines

Lines that never intersect, having the same slope but different y-intercepts, resulting in no solution.

Discriminant

The expression b² - 4ac in the quadratic formula that determines the nature of the roots of the quadratic equation.

GCF (Greatest Common Factor)

The largest factor that divides two or more numbers, important for factoring polynomials.

Difference of Squares

A polynomial factoring pattern of the form a² - b² = (a + b)(a - b).

Perfect Square Trinomials

A pattern used for factoring, where a² + 2ab + b² = (a + b)².

Quadratic Formula

A formula used to find the roots of a quadratic equation: x = (-b ± √(b² - 4ac)) / 2a.

Vertex

The peak or valley of a parabola, found at x = -b/(2a).

Axis of Symmetry

The vertical line that divides a parabola into two mirror images; it passes through the vertex.

Radical Expressions

Expressions that involve roots, particularly square roots.

Rational Exponents

Exponents expressed as fractions to indicate roots, e.g., x^(m/n) = root(n)(x^m).

Exponential Functions

Functions where the variable is in the exponent, indicating constant multiplicative growth or decay.

Growth and Decay Formula

A = P(1 ± r)ⁱ, where A is the final amount, P is the principal, r is the rate, and t is time.

Function Transformations

Changes made to the function f(x) that affect the position of its graph.

Distributing the Negative

Multiplying each term in parentheses by -1; incorrect distribution can lead to inaccuracies.

FOIL

A mnemonic for multiplying two binomials (First, Outer, Inner, Last).

Variable in Denominator

Important consideration; do not multiply by an unknown variable in inequalities without knowing its sign.