Grade 10 Functions Graphs — How to Identify the Question Type

Grade 10 functions cover four graph types: linear (straight line), quadratic (parabola), hyperbola, and exponential. Identifying which type you are dealing with is the first step before answering anything.

Type 1: Identifying and sketching a linear function

Trigger words: "Sketch", "Draw the graph of", "y = mx + c"

Trigger structure: The equation has x to the first power only — no x², no x in the denominator.

Do not confuse with: A quadratic — if there is an x², the graph is a parabola, not a straight line.

Method (no numbers — just the steps)

Identify m (gradient) and c (y-intercept) from y = mx + c
Plot the y-intercept (0 ; c)
Use the gradient (rise over run) to find a second point
Draw a straight line through both points
Label intercepts if asked

Grade 10Paper 1Level 2 — Routine Procedures

Sketch the graph of f(x) = 2x − 3, clearly showing all intercepts.

Practice question — not sourced from a past paper.

See the progression — same type, increasing difficulty

Easy

Draw the graph of y = 3x + 1.

Practice question — not sourced from a past paper.

Medium

Determine the equation of the straight line passing through (0 ; 4) and (2 ; 0).

Practice question — not sourced from a past paper.

Hard

The graph of f(x) = mx + c passes through (−1 ; 5) and (3 ; −3). Determine the values of m and c.

Practice question — not sourced from a past paper.

Type 2: Identifying and sketching a quadratic function (parabola)

Trigger words: "Sketch", "Draw", "y = ax² + bx + c", "y = a(x + p)² + q"

Trigger structure: The equation contains x².

Do not confuse with: Linear — if there is no x², it is a straight line. At Grade 10, parabolas are mainly y = ax² or y = ax² + q.

Method (no numbers — just the steps)

Identify a (determines opening direction and stretch), p (horizontal shift), q (vertical shift)
Find the turning point (−p ; q) from the completed-square form, or (0 ; q) for y = ax² + q
Find the y-intercept (substitute x = 0)
Find x-intercepts if possible (set y = 0)
Sketch the curve through these key points

Grade 10Paper 1Level 2 — Routine Procedures

5.1 Sketch the graph of f(x) = −x² + 4, showing all intercepts and the turning point.

Source: DBE Mathematics Paper 2, November 2020, Question 5.1

Common mistake

Confusing the sign of p — in y = a(x + 2)² + q the turning point is at x = −2, not x = 2.

See the progression — same type, increasing difficulty

Easy

Sketch g(x) = x² − 9. Show the intercepts and turning point.

Practice question — not sourced from a past paper.

Medium

Sketch h(x) = 2(x − 1)² − 8. Show the intercepts and turning point.

Practice question — not sourced from a past paper.

Hard

The parabola f(x) = ax² + q has x-intercepts at x = −2 and x = 2, and passes through (1 ; −3). Determine the values of a and q.

Practice question — not sourced from a past paper.

Type 3: Reading information from a graph

Trigger words: "Read off", "Write down the domain", "Write down the range", "For which values of x is f(x) > 0"

Trigger structure: A graph is provided and the question asks you to extract information from it rather than draw it.

Do not confuse with: Sketching — these questions require interpretation, not drawing.

Method (no numbers — just the steps)

Domain: read the x-values over which the graph exists (left to right)
Range: read the y-values the graph takes (bottom to top)
f(x) > 0: identify where the graph is above the x-axis
f(x) < 0: identify where the graph is below the x-axis
Intercepts: read exact coordinates from the graph

Grade 10Paper 1Level 1 — Knowledge

The graph of f is a straight line with x-intercept at 3 and y-intercept at −6. Write down the domain and range of f.

Practice question — not sourced from a past paper.

See the progression — same type, increasing difficulty

Easy

A parabola has a turning point at (0 ; −4) and opens upward. Write down the range of this function.

Practice question — not sourced from a past paper.

Medium

Given the graph of f(x) = x² − 4, for which values of x is f(x) < 0?

Practice question — not sourced from a past paper.

Hard

The graphs of f(x) = x + 2 and g(x) = x² − 4 are drawn on the same set of axes. For which values of x is f(x) > g(x)?

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.