Praxis Algebra I flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Distributive property

For real numbers a, b, and c, a(b + c) = ab + ac. It lets you remove parentheses by multiplying each term inside by the outside factor.

Rational vs irrational numbers

A rational number can be written as p/q with integers p and q, q ≠ 0; its decimal terminates or repeats. Irrational numbers such as √2 or π do neither.

Absolute value as distance

|a| is the distance from a to 0 on the number line, so |a| ≥ 0. Similarly, |a − b| is the distance between the points a and b.

Order of operations

Evaluate grouping symbols first, then exponents and roots, then multiplication and division left to right, then addition and subtraction left to right.

Negative exponent rule

For any nonzero base x and positive integer n, x⁻ⁿ = 1/xⁿ. A negative exponent indicates a reciprocal, not a negative value.

Difference of squares

a² − b² factors as (a + b)(a − b). This pattern applies whenever a binomial is a difference of two perfect squares, such as x² − 49 = (x + 7)(x − 7).

Perfect square trinomial

(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². The middle term is twice the product of the two terms being squared.

Combining like terms

Terms may be added or subtracted only when their variable parts are identical. For example, 7x² and 3x² combine to 10x², but 7x² and 3x do not.

Solving a literal equation

Treat every other variable as a constant and use inverse operations to isolate the requested variable, as when solving A = P + Prt for r to get r = (A − P)/(Pt).

Inequality reversal rule

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol, so −2x > 6 becomes x < −3.

Absolute value equation

The equation |x| = c with c > 0 has two solutions, x = c and x = −c. If c < 0 it has no solution, since absolute value is never negative.

No solution vs all reals

Solving an equation may yield a false statement such as 3 = 7, meaning no solution, or a true statement such as 0 = 0, meaning every real number is a solution.

Slope formula

For two points (x₁, y₁) and (x₂, y₂) with x₁ ≠ x₂, slope m = (y₂ − y₁)/(x₂ − x₁). It measures vertical change per unit of horizontal change.

Perpendicular line slopes

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so their product is −1. A horizontal line and a vertical line are also perpendicular.

Point-slope form

y − y₁ = m(x − x₁) is the equation of the line through point (x₁, y₁) with slope m. It rearranges directly into slope-intercept form.

System solution types

A linear system has one solution when slopes differ, no solution when slopes are equal but intercepts differ, and infinitely many solutions when the equations describe the same line.

Vertical line test

A graph represents a function only if no vertical line crosses it more than once. A single vertical line touching two points means one input has two outputs.

Rational exponent rule

For positive x and integers m and n with n > 1, x^(m/n) = ⁿ√(xᵐ) = (ⁿ√x)ᵐ. The denominator is the root and the numerator is the power.

Simplifying square radicals

Factor out the largest perfect square from the radicand and take its root: √72 = √(36 · 2) = 6√2. Only radicals with matching radicands can be added or subtracted.

Exponential growth function

f(x) = a · bˣ has initial value a and growth factor b. When b > 1 the function grows; when 0 < b < 1 it decays toward zero.

Compound interest formula

A = P(1 + r/n)^(nt) gives the balance after t years when principal P earns annual rate r compounded n times per year.

Rationalizing a denominator

Rewrite a fraction so no radical remains in the denominator, usually by multiplying numerator and denominator by that radical: 1/√3 = √3/3.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac))/(2a). It solves every quadratic equation, factorable or not.

Discriminant

The value b² − 4ac for ax² + bx + c = 0. Positive gives two real solutions, zero gives one repeated real solution, negative gives no real solutions.

Axis of symmetry

For y = ax² + bx + c, the vertical line x = −b/(2a) passes through the vertex and divides the parabola into mirror-image halves.

Zero product property

If a product equals zero, at least one factor equals zero. Solving (x − r)(x − s) = 0 therefore gives x = r or x = s.

Vertex form of a parabola

y = a(x − h)² + k has vertex (h, k) and axis x = h. The sign of a sets direction, and |a| controls how narrow the parabola is.

Arithmetic sequence formula

The nth term is aₙ = a₁ + (n − 1)d, where d is the constant common difference. Consecutive terms differ by d, so the sequence graphs as a line.

Geometric sequence formula

The nth term is aₙ = a₁ · rⁿ⁻¹, where r is the constant common ratio found by dividing any term by the previous one.

Average rate of change

Over the interval [a, b], the average rate of change of f is (f(b) − f(a))/(b − a), the slope of the secant line through those two graph points.

Interquartile range

IQR = Q3 − Q1, the spread of the middle half of an ordered data set. Values beyond 1.5 · IQR from the quartiles are often treated as outliers.

Conditional relative frequency

In a two-way table, it is a cell count divided by its row or column total. It estimates a probability given that the row or column condition holds.