Functions: Algebra, Graphs and Transformations
Core ideas
A function pairs each input with exactly one output; the domain is the set of permitted inputs and the range the set of outputs. For HSC Advanced you must move fluently between algebraic and graphical forms: linear and quadratic functions, polynomials, absolute value, hyperbolas, circles and semicircles, exponentials and logarithms, and inverse functions.
Key skills: completing the square to find the vertex and axis of symmetry; using the discriminant to count roots; solving inequalities by sign analysis rather than by cancelling; finding domains from denominators, even roots and logarithms; and applying transformations - vertical and horizontal shifts, dilations, and reflections. Remember the asymmetry: adding inside the bracket moves the graph left, and horizontal dilations scale by the reciprocal.
Traps
- Dividing an inequality by a negative reverses the sign, and multiplying an inequality by an unknown quantity requires cases.
- Cancelling a common factor in a rational function removes a point from the domain - the graph has a hole, not a vertical asymptote.
- Domain of f(g(x)) needs x in the domain of g and g(x) in the domain of f.
- For inverses: reflect in y = x, swap x and y, and restrict the domain when the original is not one-to-one.
Worked rule
For y = a(x − h)² + k, the vertex is (h, k) and the axis is x = h. Example: y = 2x² − 12x + 7 = 2(x − 3)² − 11, so the minimum value is −11 at x = 3, and the range is y ≥ −11. Sketching the vertex first, then intercepts, prevents sign errors.
Rule of thumb
Before sketching, state the domain, then test one point on each side of every vertical asymptote or intercept.