Algebra and Functions
Core concepts
You must be fluent with linear, quadratic, and simple exponential relationships, and with function notation. Know how to rearrange formulas, solve systems of two equations, factor and complete the square, and read slope and intercepts from a graph. Understand domain and range, and how transformations (shifts, reflections, stretches) change a parent graph.
Traps to watch
- Sign errors when distributing a negative across brackets.
- Dividing by a variable and losing the solution x = 0.
- Confusing slope with the reciprocal when identifying perpendicular lines.
- Forgetting that a quadratic with a negative discriminant has no real roots.
- Mixing up f(x + 2), which shifts left, with f(x) + 2, which shifts up.
Worked example
Solve 2x² − 8x + 6 = 0. Divide by 2: x² − 4x + 3 = 0. Factor: (x − 1)(x − 3) = 0, so x = 1 or x = 3. Check by substitution: 2(1) − 8 + 6 = 0 and 18 − 24 + 6 = 0. Both hold.
Rule of thumb
When a question asks for a value rather than a method, substitute early and simplify; when it asks for all solutions, factor fully and check each root in the original equation.