Grade 12 Functions Graphs — How to Identify the Question Type

Grade 12 functions questions go beyond sketching — they ask you to read and compare graphs, determine equations from graphs, and reason about transformations. The four core function families (parabola, hyperbola, exponential, and logarithm) are all examinable, and questions regularly combine two or more graphs on the same axes.

Type 1: Determining the equation of a function from its graph

Trigger words: "Determine the equation", "Find a, p, q", "Write down the values of a and b"

Trigger structure: A graph is given with key features labelled (intercepts, asymptotes, turning point). You must reverse-engineer the equation.

Do not confuse with: Sketching — here the graph is given and the equation is unknown, not the other way around.

Method (no numbers — just the steps)

Identify the function family from the graph shape (parabola, hyperbola, exponential, logarithm)
Write the general form: y = a(x + p)² + q, or y = a/(x + p) + q, or y = ab^(x+p) + q
Use the asymptotes or turning point to read off p and q directly
Substitute a known point to solve for a

Grade 12Paper 1Level 3 — Complex Procedures

4.3 The graph of f(x) = a/x + q has a horizontal asymptote at y = 2 and passes through (1 ; 5). Determine the values of a and q.

Source: DBE Mathematics Paper 1, November 2022, Question 4.3

See the progression — same type, increasing difficulty

Easy

A parabola has turning point (2 ; −3) and passes through (0 ; 1). Determine the equation in the form y = a(x + p)² + q.

Practice question — not sourced from a past paper.

Medium

An exponential function passes through (0 ; 3) and (1 ; 6), and has a horizontal asymptote at y = 0. Find the equation in the form y = ab^x.

Practice question — not sourced from a past paper.

Hard

The graph of y = a/(x + p) + q has vertical asymptote x = −1, horizontal asymptote y = 3, and passes through (0 ; 1). Find a, p, and q.

Practice question — not sourced from a past paper.

Type 2: Transformations of functions

Trigger words: "Describe the transformation", "Write down the equation of g if g(x) = f(x − 2)", "Reflect"

Trigger structure: A base function f is given and a transformed version g is defined using f with operations applied to x or the output.

Do not confuse with: Finding the equation from a graph — transformations give you a rule (e.g. shift right 2) whereas finding the equation gives you coordinates.

Method (no numbers — just the steps)

f(x) + k: shift UP by k (shift DOWN if k < 0)
f(x + k): shift LEFT by k (RIGHT if k < 0)
−f(x): reflect in the x-axis
f(−x): reflect in the y-axis
a·f(x) where a > 1: vertical stretch; 0 < a < 1: vertical compression

Grade 12Paper 1Level 2 — Routine Procedures

4.2 f(x) = 2^x. Write down the equation of g if g is the reflection of f in the x-axis, shifted 3 units up.

Source: DBE Mathematics Paper 1, May/June 2023, Question 4.2

Common mistake

Confusing f(x + 2) (shift LEFT) with f(x) + 2 (shift UP) — the direction of the horizontal shift is opposite to the sign.

See the progression — same type, increasing difficulty

Easy

f(x) = x². Write down the equation of g(x) if g(x) = f(x − 1) + 4.

Practice question — not sourced from a past paper.

Medium

The graph of h(x) = (1/2)^x is reflected in the y-axis. Write down the equation of the image and state whether it is increasing or decreasing.

Practice question — not sourced from a past paper.

Hard

f(x) = log₂ x. Describe the transformation that maps f onto g(x) = −log₂(x + 3) + 1.

Practice question — not sourced from a past paper.

Type 3: Comparing and combining graphs

Trigger words: "For which values of x is f(x) > g(x)", "For which values of x is f(x) ⋅ g(x) > 0", "Find the x-coordinates of the intersection"

Trigger structure: Two graphs are drawn on the same axes. The question asks you to compare their y-values or find where one exceeds the other.

Method (no numbers — just the steps)

Find intersection points by solving f(x) = g(x)
For f(x) > g(x): identify where the graph of f is ABOVE the graph of g — read the x-interval from the diagram
For f(x) ⋅ g(x) > 0: both functions must be positive OR both negative simultaneously

Grade 12Paper 1Level 3 — Complex Procedures

4.4 The graphs of f(x) = x² − 4 and g(x) = x + 2 are given. For which values of x is f(x) ≤ g(x)?

Source: DBE Mathematics Paper 1, November 2021, Question 4.4

Common mistake

Solving the inequality algebraically without checking the graph — the graph shows which function is on top in which region.

See the progression — same type, increasing difficulty

Easy

From the graph, f(x) = x + 1 and g(x) = −x + 3. For which value of x does f(x) = g(x)?

Practice question — not sourced from a past paper.

Medium

f(x) = x² − 1 and g(x) = 3. For which values of x is f(x) < g(x)?

Practice question — not sourced from a past paper.

Hard

The graphs of f(x) = 2/x and g(x) = x − 1 are given. For which values of x is f(x) ⋅ g(x) > 0?

Practice question — not sourced from a past paper.

Words like determine and hence appear throughout this topic — see the instruction word glossary for full definitions.