Functions & Graphs for Matric Maths

Key concepts

• Parabola: y = a(x − p)² + q, axis of symmetry, turning point
• Hyperbola: y = a/(x − p) + q, asymptotes
• Exponential and log graphs and their inverses
• Transformations: shifts, reflections, stretches
• Average gradient between two points
• Inverse functions as a reflection in the line y = x
• Domain and range from an equation or a sketch
• Restricting the domain of y = ax² so that its inverse is a function
• y = bˣ and its inverse y = log_b x
• Reading inequalities such as f(x) ≥ g(x) from graphs

Worked examples

Given f(x) = x² − 4x − 5, find the turning point.

Complete the square: (x − 2)² − 9. Turning point is (2, −9).

Find the inverse of f(x) = 3ˣ and state its domain.

Swap x and y: x = 3ʸ, so y = log₃ x. f⁻¹(x) = log₃ x, with domain x > 0 (the range of f becomes the domain of the inverse).

g(x) = 2/(x − 1) + 3. Write down the asymptotes and find the intercepts.

Asymptotes: x = 1 and y = 3. y-intercept: g(0) = 2/(−1) + 3 = 1, so (0, 1). x-intercept: 2/(x − 1) = −3, so x − 1 = −2/3 and x = 1/3, giving (1/3, 0).

Find the average gradient of f(x) = x² − 4x − 5 between x = 1 and x = 4.

f(1) = −8 and f(4) = −5. Average gradient = (f(4) − f(1)) / (4 − 1) = (−5 + 8) / 3 = 1.

f(x) = 2x² for x ≥ 0. Find f⁻¹(x).

Swap x and y: x = 2y², so y² = x/2 and y = √(x/2). Take the positive root because the restricted domain x ≥ 0 of f becomes the range y ≥ 0 of the inverse.

Common mistakes

• Confusing the axes of symmetry of f and f⁻¹.
• Forgetting domain restrictions on logs and inverses.
• Reading the y-intercept off the wrong axis on a hyperbola.
• Leaving out the base when writing a log inverse, e.g. log x instead of log₃ x.
• Giving the range of a hyperbola without excluding the horizontal asymptote.
• Reading f(x) > g(x) the wrong way round: it means the graph of f lies above the graph of g.

Exam tips

• Always label intercepts, turning points and asymptotes on every sketch — these earn marks.
• When asked for an inverse, swap x and y and solve.
• Substitute any given point into the equation to find unknowns before you do anything else.
• For 'for which values of x' questions, mark the relevant part of the sketch and decide whether the endpoints are included.

In past papers

• NSC Nov 2024 Paper 1 Q4–Q5 — parabola and hyperbola interpretation

Frequently asked questions

How are inverses tested in Paper 1?

Usually one full question on the inverse of an exponential or quadratic — domain restrictions, sketches and 'is the inverse a function?'.

What is the difference between a function and its inverse?

The inverse undoes the function: the x- and y-values swap, so the graph is reflected in the line y = x. The domain of the function becomes the range of the inverse.

Why must a parabola's domain be restricted for its inverse to be a function?

A parabola is many-to-one, so its reflection fails the vertical line test. Restricting y = ax² to x ≥ 0 or x ≤ 0 makes it one-to-one, so the inverse is a function.

Which graphs are examined in Paper 1?

Straight lines, parabolas, hyperbolas, exponential graphs and their inverses (including logarithmic graphs), and cubic graphs in the calculus section. Trigonometric graphs are examined in Paper 2.