Grade 12 Functions and Graphs: Sketching, Transformations and Inverses

The standard forms

• Parabola: y = a(x − p)² + q, turning point (p, q).
• Hyperbola: y = a/(x − p) + q, asymptotes x = p and y = q.
• Exponential: y = a·b^(x − p) + q, asymptote y = q.
• Logarithmic: y = log_b x, the inverse of y = b^x.

Worked example: inverse of a parabola

Let f(x) = 2x². Swap x and y: x = 2y², so y = ±√(x/2). This is not a function, so restrict the domain of f, for example x ≥ 0, giving f⁻¹(x) = √(x/2) with x ≥ 0.

State the restriction and give the domain and range of the inverse — both carry marks.

Transformation rules

• y = f(x) + q shifts the graph up by q.
• y = f(x − p) shifts the graph right by p.
• y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis.
• The inverse is a reflection in the line y = x.

Reading answers off a graph

• 'For which values of x is f(x) ≥ 0' means read where the graph is on or above the x-axis.
• 'For which values of x is f(x)·g(x) < 0' means the two graphs are on opposite sides of the x-axis.
• Length of a vertical line between two graphs is f(x) − g(x), taking the upper graph first.
• Always write answers as intervals in x, not as coordinates.