Algebra and Functions
What this domain tests
Algebra and Functions tests translating verbal relationships into expressions and equations, solving linear equations and inequalities, systems, quadratic relationships, function notation, slope and intercepts, and interpreting graphs.
Core rules
- Maintain equivalence: whatever operation is applied to one side of an equation must be applied to the other; multiplying or dividing an inequality by a negative reverses the inequality sign.
- Slope m=(y2−y1)/(x2−x1); parallel lines have equal slopes, and perpendicular slopes are negative reciprocals.
- A linear function has a constant rate of change and form y=mx+b. A quadratic can be factored, completed to square, or solved with the quadratic formula; its graph is a parabola with axis x=−b/(2a).
- Function notation: f(a) means substitute a for the input, not multiply f by a.
- For a system, substitution or elimination yields the intersection; no solution means parallel lines, and infinitely many solutions mean the same line.
Worked example
Solve 3(x−2)+4=2x+7. Distribute: 3x−6+4=2x+7, so 3x−2=2x+7. Subtract 2x: x−2=7. Add 2: x=9. Check: 3(7)+4=25 and 2(9)+7=25.
Traps
Watch sign errors when distributing a negative. Do not assume a graph’s scale is one unit per tick. In inequalities, reversing the sign is required only for multiplication or division by a negative. When solving a quadratic, a common error is losing the negative root or dividing by a variable and dropping x=0. In function questions, f(x+h) is not f(x)+h.