Algebra and Functions
Algebra and Functions
This domain assesses symbolic fluency: simplifying expressions, solving equations and inequalities, interpreting functions, and connecting graphs, tables, and formulas. Items often require choosing a representation or explaining a transformation.
Core rules
- Distribute carefully: a(b + c) = ab + ac. Combine like terms only when variable parts match.
- A linear function f(x) = mx + b has slope m = (y2 − y1)/(x2 − x1) and y-intercept b. Parallel lines have equal slopes; perpendicular slopes are negative reciprocals.
- Solve systems by substitution or elimination; the solution is the intersection point.
- For a quadratic ax^2 + bx + c = 0, factor when possible, complete the square, or use x = [−b ± √(b^2 − 4ac)]/(2a). The discriminant b^2 − 4ac tells the number of real roots: positive, zero, or negative gives two, one, or no real roots.
- Function domain restrictions come from denominators (not zero) and even roots (radicand nonnegative).
- Exponential functions have constant ratios, not constant differences; their graphs have a horizontal asymptote.
Worked example
Solve 2x^2 − 5x − 3 = 0. Find factors of (2)(−3) = −6 that add to −5: −6 and 1. Rewrite: 2x^2 − 6x + x − 3 = 0. Factor by grouping: 2x(x − 3) + 1(x − 3) = (2x + 1)(x − 3) = 0. So x = −1/2 or x = 3. Substituting either value into the original equation gives 0.
Traps
When multiplying an inequality by a negative number, reverse the inequality sign. Squaring both sides can introduce extraneous solutions, so check in the original equation. Do not confuse f(x + 2) with f(x) + 2: the first shifts the graph left, the second shifts it up. In a system, “no solution” means parallel lines, not that you made an arithmetic error.