Grade 11 Functions Graphs — How to Identify the Question Type

Grade 11 Functions and Graphs covers the parabola, the hyperbola, and the exponential graph. The same diagram can be asked about in several different ways — an intercept, a turning point, an asymptote, or the equation itself — and each of those needs a different method. Work out which one is being asked for before you calculate anything.

Type 1: x- and y-intercepts

Trigger words: "Determine the x-intercept(s)", "Calculate the coordinates of A and B", "Where does the graph cut the y-axis?"

Trigger structure: Any time the question asks where the graph crosses an axis — not where it turns, and not where it levels off.

Do not confuse with: The turning point (Type 2) or an asymptote (Type 3) — intercepts are points ON the axes, the turning point and asymptotes usually aren't.

Method (no numbers — just the steps)

For the x-intercept(s): set y = 0 (or f(x) = 0) and solve for x
For the y-intercept: set x = 0 and solve for y
Write the answer as coordinates, e.g. (2 ; 0)

Grade 11Paper 1Level 2 — Routine Procedures

Given: f(x) = −2x² + x + 6 Determine the x-intercepts of f.

Source: DBE Mathematics Paper 1, Grade 11, November 2016, Question 2.3

Common mistake

Substituting x = 0 to find the x-intercept (the two intercepts are found by setting the OTHER variable to zero, not the one you're solving for).

See the progression — same type, increasing difficulty

Easy

Given: g(x) = x² + x − 6 Determine the x-intercepts of g.

Practice question — not sourced from a past paper.

Medium

Given: h(x) = (8)/(x + 2) − 1 Determine the x-intercept of h.

Practice question — not sourced from a past paper.

Hard

Given: f(x) = x² − 2x − 8 and g(x) = x + 1 Determine the coordinates of the point(s) where the graphs of f and g intersect.

Practice question — not sourced from a past paper.

Type 2: Turning point of a parabola

Trigger words: "Determine the coordinates of the turning point", "the minimum/maximum value of f"

Trigger structure: Only applies to parabolas — hyperbolas and exponential graphs don't have a turning point.

Do not confuse with: An intercept — the turning point is rarely on an axis unless the question is specifically constructed that way.

Method (no numbers — just the steps)

Write f(x) in the form a(x + p)² + q, by completing the square, OR
Use x = −b / (2a) to find the x-coordinate of the turning point
Substitute that x-value back into f(x) to find the y-coordinate
Write the answer as coordinates

Grade 11Paper 1Level 2 — Routine Procedures

Given: f(x) = −2x² + x + 6 Calculate the coordinates of the turning point of f.

Source: DBE Mathematics Paper 1, Grade 11, November 2016, Question 2.1

Common mistake

Reporting only the y-coordinate (the maximum or minimum value) when the question asks for the full coordinates of the turning point.

See the progression — same type, increasing difficulty

Easy

Given: f(x) = x² − 6x + 5 Determine the coordinates of the turning point of f.

Practice question — not sourced from a past paper.

Medium

Given: g(x) = −2x² + 8x − 3 Determine the maximum value of g.

Practice question — not sourced from a past paper.

Hard

Given: f(x) = x² + bx + c has a turning point at (3 ; −4). Determine the values of b and c.

Practice question — not sourced from a past paper.

Type 3: Asymptotes (hyperbola and exponential graphs)

Trigger words: "Write down the equations of the asymptotes", "the equation of the horizontal/vertical asymptote"

Trigger structure: A hyperbola has two asymptotes (one vertical, one horizontal); an exponential graph has one (horizontal only).

Do not confuse with: Intercepts — asymptotes are lines the graph approaches but never touches, so they're never 'on' the graph itself.

Method (no numbers — just the steps)

For a hyperbola in the form y = a/(x + p) + q: the vertical asymptote is x = −p, the horizontal asymptote is y = q
For an exponential graph in the form y = a·b^x + q: the horizontal asymptote is y = q (there is no vertical asymptote)
Write each asymptote as an equation, not just a number

Grade 11Paper 1Level 1 — Knowledge

Given: f(x) = −3/(x + 2) + 1 Write down the asymptotes of f.

Source: DBE Mathematics Paper 1, Grade 11, November 2017, Question 1.3

Common mistake

Writing the asymptote as a bare number (e.g. '2') instead of a full equation (e.g. 'y = 2').

See the progression — same type, increasing difficulty

Easy

Given: f(x) = 4/x − 3 Write down the equations of the asymptotes of f.

Practice question — not sourced from a past paper.

Medium

Given: g(x) = 2(3)^x − 5 Write down the equation of the asymptote of g.

Practice question — not sourced from a past paper.

Hard

The graph of k(x) = a/(x + p) + q has asymptotes x = 2 and y = −1, and passes through the point (3 ; 1). Determine the values of a, p and q.

Practice question — not sourced from a past paper.

Type 4: Determining the equation of a function from its graph

Trigger words: "Determine the equation of f", "Determine the values of a, p and q"

Trigger structure: The graph (or a description of it — intercepts, asymptotes, turning point) is given, and you must reconstruct the algebraic equation.

Do not confuse with: Reading a single feature off an already-given equation (Types 1–3) — this type runs in the opposite direction, from picture to equation.

Method (no numbers — just the steps)

Identify the general form for the type of graph shown (parabola, hyperbola, or exponential)
Use any asymptotes or the turning point to fix as many constants as possible directly
Substitute any given point(s) on the graph into the remaining equation
Solve for the unknown constant(s)

Grade 11Paper 1Level 3 — Complex Procedures

Sketched below is the graph of g(x) = a/(x − p) + q. C(2 ; 6) is the point of intersection of the asymptotes of g. B(5/2 ; 0) is the x-intercept of g. Determine the equation of g in the form g(x) = a/(x − p) + q.

Source: DBE Mathematics Paper 1, Grade 11, November 2015, Question 4.1

Common mistake

Substituting a point into the general form before first using the asymptotes/turning point to reduce the number of unknowns — this makes the algebra far longer than it needs to be.

See the progression — same type, increasing difficulty

Easy

The graph of f(x) = x² + q has a y-intercept at (0 ; −4). Determine the equation of f.

Practice question — not sourced from a past paper.

Medium

The graph of g(x) = a·2^x has a y-intercept at (0 ; 3). Determine the equation of g.

Practice question — not sourced from a past paper.

Hard

The graph of h(x) = a/(x + p) + q has asymptotes x = −1 and y = 2, and a y-intercept at (0 ; 5). Determine the equation of h.

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.