Algebra and Functions
Algebra and Functions - what the exam tests
This domain covers linear equations and inequalities, systems, quadratic and exponential relationships, function notation, and slope. The exam rewards translating a situation into an equation and reading key features-slope, intercept, vertex, domain-from an expression or graph.
Core rules
- Slope: m = (y₂ − y₁)/(x₂ − x₁); slope-intercept form y = mx + b. Parallel lines share m; perpendicular slopes are negative reciprocals.
- Linear equations: isolate the variable with inverse operations, then substitute back to check.
- Inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign.
- Systems: use substitution or elimination. Parallel lines (same slope, different intercept) give no solution; identical lines give infinitely many.
- Quadratics: factor, complete the square, or use x = [−b ± √(b² − 4ac)]/(2a). The discriminant b² − 4ac tells you 2, 1, or 0 real roots; the vertex is at x = −b/(2a).
- Functions: f(x) is an output, not a multiplication. Exclude values that make a denominator zero or an even root negative.
- Exponential models: y = a·b^x grows when b > 1 and decays when 0 < b < 1.
Worked example
Solve 2x + 3y = 12 and x − y = 1. From the second equation, x = y + 1. Substituting: 2(y + 1) + 3y = 12, so 5y + 2 = 12 and y = 2. Then x = 3. Check: 2(3) + 3(2) = 12.
Traps that cost marks
- Dropping a negative sign when distributing, especially −(x − 4).
- Treating f(x + 1) as f(x) + 1.
- Assuming a quadratic always has two real solutions when the discriminant is negative.
- Solving for x but answering the question asked about y.