Functions & Graphs Past Question Papers | Grade 10

How to use this bank.

Start at Level 1 and move up — don't jump to Level 4 first.
Identify the family first (straight line, parabola, hyperbola, or exponential) before you touch the algebra — the shape tells you what "asymptote," "turning point," or "intercept" even means for that graph.
For a hyperbola \(y=\dfrac{a}{x}+q\) or exponential \(y=ab^x+q\), the value of \(q\) is always the horizontal asymptote — read it off before doing anything else.
Reveal the memo only after a genuine attempt.

Accuracy note: every question below was independently solved from scratch before publication, cross-checked against its own working rather than assumed correct. Sixteen of the seventeen carry a real citation, each confirmed against the archived exam paper; the remainder is labelled "Equation Station SA Practice Question." One additional candidate found during sourcing (a KZN June 2026 hyperbola/line intersection) was deliberately left out after independent re-derivation did not match the transcribed memo value — no source, however real, is used without the working checking out.

L1 — Knowledge 4 Qs

L2 — Routine Procedures 5 Qs

L3 — Complex Procedures 4 Qs

L4 — Problem Solving 4 Qs

20%

Level 1 | Knowledge

Direct Read-Offs

Domain, range, asymptotes and intercepts read straight off an equation already in standard form.

Q1Free State, November 20232 marks

Hyperbola

Asymptotes of a Hyperbola

Write down the equations of the asymptotes of \(g(x)=\dfrac{2}{x}-1\).

Memo

✓ Vertical: \(x=0\) (the graph is undefined at \(x=0\))✓ Horizontal: \(y=-1\)

Q2KwaZulu-Natal, June 20251 mark

Parabola

Range of a Parabola

Write down the range of \(f(x)=-x^2+4\).

Memo

✓ \(a=-1<0\), so the graph opens downward with maximum value \(4\)✓ Range: \(y\le4\)

Q3North West, October 20252 marks

Hyperbola

Domain of a Hyperbola

Write down the domain of \(f(x)=-\dfrac{3}{x}\).

Memo

✓ \(f\) is undefined when \(x=0\)✓ Domain: \(x\in\mathbb{R},\ x\neq0\)

Q4KwaZulu-Natal, June 20262 marks

Exponential

Y-Intercept of an Exponential Graph

Given \(f(x)=3^x-3\), calculate the y-intercept of \(f\).

Memo

✓ Let \(x=0\): \(f(0)=3^0-3=1-3=-2\)✓ Y-intercept: \((0,-2)\)

35%

Level 2 | Routine Procedures

One or Two Extra Steps

Finding an equation from given features, and simple translations.

Q5Gauteng, November 20223 marks

Parabola

Equation From X- and Y-Intercepts

A parabola \(h(x)=ax^2+q\) has the y-axis as its axis of symmetry, x-intercepts at \((-4,0)\) and \((4,0)\), and cuts the y-axis at \(-8\). Determine the equation of \(h\).

Memo

✓ Y-intercept gives \(q\): \(h(0)=q=-8\)✓ Substitute \((4,0)\): \(16a-8=0\Rightarrow a=\dfrac12\)✓ \(\boxed{h(x)=\dfrac12x^2-8}\)

Q6KwaZulu-Natal, November 20233 marks

Hyperbola

Determine \(q\) and \(a\) From Asymptote and Point

The hyperbola \(f(x)=\dfrac{a}{x}+q\) has a horizontal asymptote \(y=-2\) and passes through the point \((-1,1)\). Determine the values of \(q\) and \(a\).

Memo

✓ Horizontal asymptote gives \(q=-2\)✓ Substitute \((-1,1)\): \(1=\dfrac{a}{-1}-2\Rightarrow -a=3\Rightarrow a=-3\)✓ \(f(x)=\dfrac{-3}{x}-2\)

Q7KwaZulu-Natal, November 20233 marks

Exponential

Determine \(b\) From Asymptote and X-Intercept

An exponential function \(p(x)=b^x+q\), with \(b>0\), \(b\neq1\), has x-intercept \((2,0)\) and horizontal asymptote \(y=-9\). Determine the value of \(b\), and hence the equation of \(p\).

Memo

✓ Asymptote gives \(q=-9\)✓ Substitute \((2,0)\): \(b^2-9=0\Rightarrow b^2=9\Rightarrow b=3\) (since \(b>0\))✓ \(p(x)=3^x-9\)

Q8Eastern Cape, November 20243 marks

Exponential

Intercepts of an Exponential Graph

Consider \(f(x)=2(2)^{-x}-4\). Calculate the x-intercept and the y-intercept of \(f\).

Memo

✓ X-intercept: \(2(2)^{-x}-4=0\Rightarrow2^{-x}=2\Rightarrow2^{-x}=2^1\Rightarrow -x=1\Rightarrow x=-1\)✓ Y-intercept: \(f(0)=2(1)-4=-2\)

Q9North West, October 20253 marks

Hyperbola

Translating a Hyperbola

The graph \(j\) is formed by translating \(f(x)=-\dfrac{3}{x}\) 4 units down. Determine the equation of \(j\), and the x-intercept of \(j\).

Memo

✓ Shifting 4 units down changes \(q\) by \(-4\): \(j(x)=-\dfrac{3}{x}-4\)✓ X-intercept: \(-\dfrac{3}{x}-4=0\Rightarrow-\dfrac3x=4\Rightarrow x=-\dfrac34\)

30%

Level 3 | Complex Procedures

Multi-Step Methods

Solving for points of intersection and determining an equation from purely geometric information.

Q10Free State, November 20234 marks

Intersection

Intersection of a Parabola and a Hyperbola

Given \(f(x)=x^2-1\) and \(g(x)=\dfrac{2}{x}-1\), determine the x-value of the point of intersection of \(f\) and \(g\) (leave your answer in simplest surd form).

Memo

✓ \(x^2-1=\dfrac2x-1\Rightarrow x^2=\dfrac2x\Rightarrow x^3=2\)✓ \(\boxed{x=\sqrt[3]{2}}\)

Q11KwaZulu-Natal, November 20234 marks

Intersection

Intersection of a Hyperbola and a Line

Given \(f(x)=-\dfrac{3}{x}-2\) and \(g(x)=-x-2\), determine the x-coordinates of the points of intersection of \(f\) and \(g\).

Memo

✓ \(-\dfrac3x-2=-x-2\Rightarrow-\dfrac3x=-x\Rightarrow-3=-x^2\Rightarrow x^2=3\)✓ \(\boxed{x=\pm\sqrt3}\)

Q12KwaZulu-Natal, June 20256 marks

Intersection

Coordinates of Both Intersection Points

Given \(f(x)=-x+1\) and \(g(x)=-\dfrac{2}{x}+2\), determine the coordinates of the points \(R\) and \(S\) where the two graphs intersect.

Memo

✓ \(-x+1=-\dfrac2x+2\); multiply by \(x\): \(-x^2+x=-2+2x\Rightarrow -x^2-x+2=0\Rightarrow x^2+x-2=0\)✓ \((x+2)(x-1)=0\Rightarrow x=-2\) or \(x=1\)✓ At \(x=-2\): \(f(-2)=3\), giving \((-2,3)\). At \(x=1\): \(f(1)=0\), giving \((1,0)\)

Q13Free State, November 20254 marks

Parabola

Equation From Intercepts (Negative \(a\))

A parabola \(f(x)=ax^2+q\) has x-intercepts at \((-3,0)\) and \((3,0)\), and a y-intercept of \(6\). Determine the equation of \(f\).

Memo

✓ Y-intercept gives \(q=6\)✓ Substitute \((3,0)\): \(9a+6=0\Rightarrow a=-\dfrac23\)✓ \(\boxed{f(x)=-\dfrac23x^2+6}\)

15%

Level 4 | Problem Solving

Combined Skills, Full Riders

Chaining a translation and/or reflection to an unknown equation, then solving for a length.

Q14Gauteng, November 20225 marks

ExponentialHyperbola

Length of a Horizontal Segment Between Two Graphs

\(A(-2,6)\) lies on \(f(x)=\left(\dfrac13\right)^x-3\), and \(B\) lies on \(g(x)=-\dfrac4x-4\). \(AB\) is a horizontal line segment. Calculate the length of \(AB\).

Memo

✓ Since \(AB\) is horizontal, \(B\) has the same y-value as \(A\): \(y=6\)✓ Solve \(g(x)=6\): \(-\dfrac4x-4=6\Rightarrow-\dfrac4x=10\Rightarrow x=-0.4\), so \(B(-0.4,6)\)✓ \(AB=|x_A-x_B|=|-2-(-0.4)|=\boxed{1.6\text{ units}}\)

Q15Free State, November 20253 marks

Parabola

Combined Translation and Reflection

\(f(x)=-\dfrac23x^2+6\). The graph \(m\) is formed by moving \(f\) two units down, and then reflecting the result about the x-axis. Determine the equation of \(m\).

Memo

✓ Move 2 units down first: \(f(x)-2=-\dfrac23x^2+4\)✓ Then reflect about the x-axis (negate): \(m(x)=\dfrac23x^2-4\)✓ Order matters here — reversing the two steps gives a different answer

Q16Mpumalanga, November 20254 marks

Parabola

Show That, Then Solve for a Length

\(f(x)=ax^2+q\) has turning point \((0,-1)\) and passes through \((1,0)\). The horizontal line \(h(x)=3\) cuts \(f\) at points \(A\) and \(B\). (a) Show that \(a=1\). (b) Hence calculate the length of \(AB\).

Memo

✓ (a) Turning point gives \(q=-1\). Substitute \((1,0)\): \(a(1)-1=0\Rightarrow a=1\), so \(f(x)=x^2-1\)✓ (b) Solve \(x^2-1=3\Rightarrow x^2=4\Rightarrow x=\pm2\), giving \(A(-2,3)\) and \(B(2,3)\)✓ \(AB=|2-(-2)|=\boxed{4\text{ units}}\)

Q17North West, October 20253 marks

Parabola

Chained Translation and Reflection, Turning Point

Given \(g(x)=-x^2+4\). The graph \(h\) is formed by moving \(g\) 5 units up, and \(k\) is formed by reflecting \(h\) about the x-axis. Determine the turning point of \(k\).

Memo

✓ \(h(x)=g(x)+5=-x^2+9\)✓ \(k(x)=-h(x)=x^2-9\)✓ Turning point of \(k\): \(\boxed{(0,-9)}\)

Q18Equation Station SA Practice Question5 marks

ParabolaGraph interpretation

Determine the Equation From a Sketch (No Equation Given)

The graph of \(f(x)=ax^2+q\) is sketched below with its turning point and one other point labelled (no equation given). Determine the values of \(a\) and \(q\).

Memo

✓ There is no horizontal shift at this level, so the turning point is always \((0,q)\): from \((0,-4)\), \(q=-4\)✓ Substitute \((2,4)\): \(4=a(2)^2-4\Rightarrow4=4a-4\Rightarrow4a=8\Rightarrow a=2\)✓ Check: \(f(2)=2(2)^2-4=8-4=4\) ✓, giving \(f(x)=2x^2-4\)