ACCUPLACER AAF flashcards

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

Solving a two-step equation

For ax + b = c with a ≠ 0, undo addition or subtraction first, then division: x = (c − b)/a. Check by substituting back into the original equation.

Inequality sign reversal rule

Multiplying or dividing both sides of an inequality by a negative number flips the inequality symbol. Adding or subtracting a negative number does not flip it.

Absolute value equation cases

The equation |x| = k has solutions x = k and x = −k when k > 0, the single solution x = 0 when k = 0, and no real solution when k < 0.

No solution versus all reals

A linear equation simplifies to a false statement such as 0 = 7 when it has no solution, and to a true identity such as 0 = 0 when every real number is a solution.

Slope from two points

For points (x1, y1) and (x2, y2) with x1 ≠ x2, slope m = (y2 − y1)/(x2 − x1). A horizontal line has slope 0; a vertical line has undefined slope.

Slope-intercept form

The equation y = mx + b graphs a line with slope m and y-intercept b. The slope gives the change in y per one-unit increase in x.

Point-slope form

The equation y − y1 = m(x − x1) describes the line through point (x1, y1) with slope m, and is useful when a point and slope are known.

Parallel and perpendicular slopes

Two nonvertical lines are parallel when their slopes are equal. They are perpendicular when their slopes are negative reciprocals, so the product of the slopes is −1.

Solutions of linear systems

Two lines intersect once when slopes differ, never when slopes are equal but intercepts differ, and coincide when both slope and intercept match, giving infinitely many solutions.

Quadratic formula

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

Discriminant meaning

The value b² − 4ac from a quadratic ax² + bx + c tells the number of real roots: positive gives two, zero gives one, negative gives none.

Zero-product property

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

Difference of squares factoring

An expression in the form a² − b² factors as (a − b)(a + b). For example, x² − 9 factors as (x − 3)(x + 3).

Quadratic vertex location

The parabola y = ax² + bx + c has its vertex at x = −b/(2a). The vertex is a minimum when a > 0 and a maximum when a < 0.

Rational expression domain restriction

A rational expression is undefined wherever its denominator equals zero. Those excluded values are removed from the domain before any simplification is performed.

Solving rational equations

Multiply both sides by the least common denominator, solve the resulting polynomial equation, then substitute each candidate into the original equation to discard extraneous solutions.

Adding rational expressions

Rewrite each rational expression over a common denominator by multiplying numerator and denominator by the needed factors, then add or subtract the numerators.

Cross-multiplication for proportions

The proportion a/b = c/d, with b and d nonzero, is equivalent to ad = bc. This converts a rational equation into a linear one.

Simplifying square roots

Factor out the largest perfect-square factor: √(ab) = √a · √b for a, b ≥ 0. For example, √50 = √25 · √2 = 5√2.

Rationalizing a denominator

Multiply numerator and denominator by the radical or conjugate that removes the radical from the denominator, since multiplying by a form of 1 preserves the value.

Extraneous radical solutions

Squaring both sides of a radical equation can create solutions that fail the original equation. Always substitute each result back to check.

Principal square root of x squared

For any real number x, √(x²) equals |x|, not x, because the principal square root is defined to be nonnegative.

Fractional exponent rule

For a ≥ 0 and n a positive integer, a^(m/n) equals the nth root of a raised to the m power, or the nth root of a^m.

Multiplicity of a zero

If (x − r) appears to the m power in a factored polynomial, r has multiplicity m. Odd multiplicity crosses the axis; even multiplicity touches and turns.

End behavior of polynomials

The leading term determines end behavior: even degree makes both ends match direction, odd degree makes them opposite, and a positive leading coefficient rises on the right.

Remainder theorem

Dividing polynomial p(x) by x − r gives remainder p(r). The value r is a zero of the polynomial exactly when that remainder equals zero.

Logarithm definition

For base b > 0, b ≠ 1, the statement log_b(x) = y means b^y = x. The argument x must be positive, so the domain is x > 0.

Logarithm product and quotient rules

The log of a product equals the sum of logs, and the log of a quotient equals the difference: log_b(mn) = log_b m + log_b n and log_b(m/n) = log_b m − log_b n.

Change of base formula

To evaluate log_b(x) with a calculator, use log_b(x) = log(x)/log(b) or equivalently ln(x)/ln(b), choosing any valid base for the new logs.

Exponential growth model

The function y = a·b^x starts at a when x = 0. It models growth when b > 1 and decay when 0 < b < 1.

Unit circle coordinates

On the unit circle, the point at angle θ measured from the positive x-axis has coordinates (cos θ, sin θ), so sin²θ + cos²θ = 1.

Amplitude and period

For y = A sin(Bx), the amplitude is |A| and the period is 2π/|B|. Amplitude gives half the vertical span of the wave.