Algebra and Functions
What this domain covers
Algebra and Functions is also about 30% of the exam. Questions ask you to simplify expressions, solve linear equations and inequalities, solve systems of two equations, evaluate and interpret functions, and connect a linear equation to its graph.
Rules worth memorizing
- Distribute every term: −3(2x − 5) = −6x + 15, not −6x − 15.
- Inequalities: solving works like equations, except multiplying or dividing both sides by a negative number reverses the inequality symbol.
- Slope: m = (y₂ − y₁)/(x₂ − x₁). Horizontal lines have slope 0; vertical lines have undefined slope.
- Forms: slope-intercept y = mx + b; point-slope y − y₁ = m(x − x₁). Parallel lines share m; perpendicular slopes are negative reciprocals.
- Systems: substitute when one variable is isolated, eliminate when coefficients cancel. Same slope with different intercepts means no solution.
- Functions: f(3) means substitute 3 for x. The domain is the set of legal inputs, so a denominator cannot be zero.
- Quadratics: factor and set each factor equal to zero, or use x = [−b ± √(b² − 4ac)] / (2a).
Worked example
Adult tickets cost $8 and child tickets cost $5. A total of 200 tickets sold for $1,330. Let a = adult and c = child.
a + c = 200 and 8a + 5c = 1330. Substitute c = 200 − a: 8a + 5(200 − a) = 1330, so 8a + 1000 − 5a = 1330, then 3a = 330, giving a = 110 and c = 90. Check: 8(110) + 5(90) = 880 + 450 = 1330.
Traps that cost marks
- Dropping the sign when distributing a negative across parentheses.
- Forgetting to flip the inequality sign when dividing by a negative.
- Computing slope as run over rise instead of rise over run.
- Reading f(x) as f multiplied by x.
- Finding one x-intercept of a quadratic and missing the second.