ACCUPLACER AAF cheat sheet

Solving ax + b = c: subtract b from both sides, then divide by a, giving x = (c − b)/a when a ≠ 0.

An equation has no solution when simplification yields a false statement such as 0 = 5; it has infinitely many solutions when it yields a true identity such as 0 = 0.

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol.

For a compound inequality a < x < b, perform the same operation on all three parts to isolate x in the middle.

Absolute value equation |x| = k with k > 0 splits into two cases: x = k or x = −k; if k < 0 there is no solution.

Inequality |x| < k (k > 0) is equivalent to −k < x < k; |x| > k is equivalent to x < −k or x > k.

The slope between two points is m = (y2 − y1)/(x2 − x1); a horizontal line has slope 0 and a vertical line has undefined slope.

Two lines are parallel when their slopes are equal and perpendicular when the product of their slopes is −1.

Slope-intercept form y = mx + b: m is the slope (rate of change) and b is the y-intercept (value when x = 0).

Point-slope form y − y1 = m(x − x1) builds a line's equation from one point and the slope.

Standard form Ax + By = C has slope −A/B and y-intercept C/B (when B ≠ 0).

The x-intercept is found by setting y = 0; the y-intercept is found by setting x = 0.

A linear model for a real situation takes the form y = (starting value) + (rate)(x), where the rate carries units per input unit.

A system of two linear equations has one solution if slopes differ, none if slopes are equal and intercepts differ, and infinitely many if the equations describe the same line.

The graph of y = |x − h| + k is a V-shape with vertex at (h, k), opening upward; y = −|x − h| + k opens downward.

For a linear inequality in two variables, the boundary line is solid for ≤ or ≥ and dashed for < or >; shade the side that satisfies a test point.

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) / (2a).

The discriminant b² − 4ac determines the number of real roots: positive gives two, zero gives one, negative gives none.

Zero-product property: if (x − r)(x − s) = 0 then x = r or x = s.

Sum and product of roots: r + s = −b/a and rs = c/a for ax² + bx + c = 0.

Difference of squares a² − b² factors as (a − b)(a + b); perfect square trinomial a² ± 2ab + b² factors as (a ± b)².

To factor x² + bx + c, find two numbers whose product is c and whose sum is b.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); the vertex is a maximum when a < 0 and a minimum when a > 0.

Completing the square converts ax² + bx + c into a(x − h)² + k, revealing the vertex (h, k).

A rational expression is undefined where its denominator equals zero; those values are excluded from the domain.

Simplify by factoring numerator and denominator completely, then cancel common factors - never cancel terms that are added or subtracted.

Multiply rational expressions by multiplying numerators and denominators; divide by multiplying by the reciprocal of the divisor.

To add or subtract rational expressions, first rewrite each over the least common denominator.

Solve rational equations by multiplying both sides by the LCD, then check every candidate solution in the original equation to reject extraneous roots.

A proportion a/b = c/d solves by cross-multiplication: ad = bc (with b, d ≠ 0).

Complex fraction (a/b)/(c/d) simplifies to ad/(bc) when b, c, d ≠ 0.

Direct variation y = kx and inverse variation y = k/x; find k from a given pair, then use it to find unknown values.

√(ab) = √a · √b and √(a/b) = √a / √b for a ≥ 0, b > 0; use these to simplify radicals.

Combine like radicals: 3√2 + 5√2 = 8√2, but √2 + √3 cannot be combined.

Rationalize a denominator by multiplying numerator and denominator by the radical (or conjugate) that clears it.

√(x²) = |x|, not x, because the principal square root is never negative.

To solve √(x + a) = b, isolate the radical, square both sides, then check for extraneous solutions.

Fractional exponents: a^(m/n) = (n√a)^m = n√(a^m), valid for a ≥ 0 when n is even.

The domain of √(f(x)) requires f(x) ≥ 0; the domain of 1/√(f(x)) requires f(x) > 0.

Pythagorean relation for distance: d = √((x2 − x1)² + (y2 − y1)²).

End behavior follows the leading term: even degree matches ends, odd degree has opposite ends; a positive leading coefficient rises to the right.

A zero at x = r with multiplicity m: odd multiplicity crosses the x-axis, even multiplicity touches and turns.

The degree of a polynomial bounds its number of real zeros and its number of turning points (degree − 1).

Remainder Theorem: dividing p(x) by (x − r) leaves remainder p(r); r is a root exactly when p(r) = 0.

Factor Theorem: (x − r) is a factor of p(x) if and only if p(r) = 0.

For a polynomial with real coefficients, complex roots occur in conjugate pairs.

The y-intercept of a polynomial is its constant term; the x-intercepts are its real zeros.

Adding polynomials combines like terms; multiplying uses the distributive property, with degree of the product equal to the sum of degrees.

Exponential growth/decay: y = a·b^x, where a is the initial value; growth when b > 1, decay when 0 < b < 1.

Log definition: log_b(x) = y means b^y = x, with b > 0, b ≠ 1, and x > 0.

Product, quotient, and power rules: log_b(mn) = log_b m + log_b n; log_b(m/n) = log_b m − log_b n; log_b(m^p) = p·log_b m.

Change of base: log_b(x) = log(x)/log(b) or ln(x)/ln(b).

log_b(1) = 0 and log_b(b) = 1 for every valid base b.

The functions b^x and log_b(x) are inverses, so log_b(b^x) = x and b^(log_b x) = x for x > 0.

Solve exponential equations by writing both sides with the same base or by taking a logarithm of both sides.

Continuous growth model: A = Pe^(rt), where r is the continuous growth rate and t is time.

Right-triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Pythagorean identity: sin²θ + cos²θ = 1; also tan θ = sin θ / cos θ.

Radian-degree conversion: 180° = π radians, so multiply degrees by π/180 to get radians.

On the unit circle, a point at angle θ has coordinates (cos θ, sin θ), and the radius is 1.

Exact values: sin 30° = 1/2, cos 30° = √3/2, sin 45° = cos 45° = √2/2, sin 60° = √3/2, cos 60° = 1/2.

Sine and cosine have period 2π; tangent has period π. Sine is odd and cosine is even.

Amplitude of y = A sin(Bx) is |A| and the period is 2π/|B|.

Law of Sines: a/sin A = b/sin B = c/sin C; Law of Cosines: c² = a² + b² − 2ab·cos C.