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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)

  1. Identify m (gradient) and c (y-intercept) from y = mx + c
  2. Plot the y-intercept (0 ; c)
  3. Use the gradient (rise over run) to find a second point
  4. Draw a straight line through both points
  5. Label intercepts if asked
Grade 10Paper 1Level 2, Routine Procedures
In the diagram, g(x) = −2x² + 8 and f(x) = mx + c. T(0 ; 8) is the y-intercept of g. f passes through T and through R, the other x-intercept of g. 5.4 Determine the equation of f.

Source: DBE Mathematics Paper 1, Grade 10, November 2017, Question 5.4

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)

  1. Identify a (determines opening direction and stretch), p (horizontal shift), q (vertical shift)
  2. Find the turning point (−p ; q) from the completed-square form, or (0 ; q) for y = ax² + q
  3. Find the y-intercept (substitute x = 0)
  4. Find x-intercepts if possible (set y = 0)
  5. Sketch the curve through these key points
Grade 10Paper 1Level 2, Routine Procedures
In the diagram, g(x) = ax² + q. T(0 ; 8) is the y-intercept of g and S(2 ; 0) is an x-intercept of g. 5.3 Calculate the values of a and q.

Source: DBE Mathematics Paper 1, Grade 10, November 2017, Question 5.3

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)

  1. Domain: read the x-values over which the graph exists (left to right)
  2. Range: read the y-values the graph takes (bottom to top)
  3. f(x) > 0: identify where the graph is above the x-axis
  4. f(x) < 0: identify where the graph is below the x-axis
  5. Intercepts: read exact coordinates from the graph
Grade 10Paper 1Level 2, Routine Procedures
Sketched below are the graphs of f(x) = ax² + q and g(x) = (1/2)^x − 4. A and B are the x-intercepts of f. The graphs intersect at A and point E(1 ; 3) lies on f. C is the turning point of f and D is the y-intercept of g. 5.4 For which values of x is: 5.4.1 f(x) > 0? 5.4.2 f decreasing?

Source: DBE Mathematics Paper 1, Grade 10, November 2018, Question 5.4

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.