What is covered in Grade 10 Mathematics Functions?

Function concept and notation. Effect of a and q on parabola (y = ax² + q), hyperbola (y = a/x + q), and exponential (y = ab^x + q). Sketch, find equations and interpret graphs.

Is Grade 10 Functions tested in Paper 1 or Paper 2?

Grade 10 Mathematics Functions is assessed in Paper 1 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Functions taught in Grade 10 Mathematics?

Functions is taught in Term 2 of Grade 10 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 10 Mathematics Functions on the CAPS curriculum?

Yes. Functions is part of the official CAPS Mathematics curriculum for Grade 10, Term 2 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 10 Mathematics study material free?

Yes. All chapter notes, definitions, formulas, and worked examples on MathSciBuddy are completely free to read. No account or subscription is needed to access CAPS-aligned study notes.

Grade 10 Mathematics

Grade 10 · Term 2Mathematics

Functions

We study the four core function families: linear ($y=mx+c$), quadratic ($y=ax^2+q$), hyperbolic ($y=\frac{a}{x}+q$), and exponential ($y=ab^x+q$). For each we learn key features, transformations, and how to sketch accurate graphs.

5.1 Function Concepts & Linear/Quadratic Functions 5.2 Hyperbolic & Exponential Functions

Week 5

5.1 Function Concepts & Linear/Quadratic Functions

Define a function: each input produces exactly one output; use function notation $f(x)$
Investigate the effect of $a$ and $q$ on $y=ax^2+q$: sketch, intercepts, domain, range, axis of symmetry, turning point
Sketch $y=x$ and $y=x^2$ point-by-point; then generalise to $y=ax+q$ and $y=ax^2+q$

🌍

Real-World Connection

A function is like a vending machine: every input produces exactly one output. No button gives two different snacks. If pressing the same button could give cola OR orange juice, that machine would be a relation, not a function.

Definition

Function

A function is a relationship where each input (from the domain) produces exactly ONE output (in the range). We write $y=f(x)$ .

f: x \mapsto f(x)

Definition

Domain and Range

Domain = the set of all allowable inputs. Range = the set of all resulting outputs.

\text{Domain} \subseteq \mathbb{R},\quad \text{Range} \subseteq \mathbb{R}

Property / Rule

Linear Function $y=ax+q$

The graph is a straight line. The parameter $a$ is the gradient: $a>0$ the line rises left to right; $a<0$ it falls; $a=0$ it is horizontal. The parameter $q$ is the $y$ -intercept. Find the $x$ -intercept by setting $y=0$ .

y\text{-intercept: }(0;\;q)\quad x\text{-intercept: }\left(-\tfrac{q}{a};\;0\right)\;(a\neq0)

Property / Rule

Parabola $y=ax^2+q$

Shape is a parabola. If $a>0$ : opens upward, minimum TP at $(0;\,q)$ . If $a<0$ : opens downward, maximum TP at $(0;\,q)$ . Axis of symmetry: $x=0$ .

\text{TP: }(0;\,q);\quad\text{Range: }y\geq q\text{ (if }a>0\text{)}

💡 Tip

For $y=ax^2+q$ : the $y$ -intercept is always $(0;\,q)$ . Find $x$ -intercepts by setting $y=0$ : $ax^2=-q\Rightarrow x^2=-q/a$ . Real $x$ -intercepts only exist if $-q/a\geq0$ .

Worked Examples

Worked Example

Plot $y=2x-4$ point-by-point

Problem

y=2x-4

Worked Example

Find the equation of a linear function from a graph

Problem

y

Worked Example

Sketch $f(x)=2x^2-8$ — key features and table

Problem

f(x)=2x^2-8

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge2 marks

Evaluate

f(3)

f(x)=x^2-2x+1

L1 · Knowledge2 marks

State the turning point and axis of symmetry of

y=3x^2-5

L2 · Routine Procedures3 marks

Find the

x

- and

y

-intercepts of

y=-x^2+4

L2 · Routine Procedures2 marks

(2;5)

y=3x^2-7

L3 · Complex Procedures5 marks

Sketch

h(x)=-2x^2+8

, showing all intercepts and turning point.

L3 · Complex Procedures4 marks

f(x)=ax^2-3

passes through

(2;9)

, find

a

and state the range.

L4 · Problem Solving4 marks

Find

x

where

f(x)=g(x)

f(x)=x^2-1

and

g(x)=3

L4 · Problem Solving5 marks

The graph of

y=ax^2+q

has range

y\leq5

and passes through

(3;-13)

. Find

a

and

q

Week 6

5.2 Hyperbolic & Exponential Functions

Plot $y=\dfrac{1}{x}$ point-by-point; then study the effect of $a$ and $q$ on $y=\dfrac{a}{x}+q$
Plot $y=b^x$ ($b>0$, $b\neq1$) point-by-point; then study $y=ab^x+q$
Identify asymptotes, domain, range, intercepts; sketch and find equations from given graphs

🌍

Real-World Connection

The exponential function models compound interest, population growth, and radioactive decay. The hyperbola models any inverse relationship — drive twice as fast and you take half the time. Both have asymptotes: lines the curve approaches but never touches.

Property / Rule

Hyperbola $y=\dfrac{a}{x}+q$

Two asymptotes: $x=0$ (y-axis) and $y=q$ (horizontal). Domain: $x\neq0$ . Range: $y\neq q$ . If $a>0$ : curves in Q1 and Q3. If $a<0$ : curves in Q2 and Q4.

\text{Asymptotes: }x=0\text{ and }y=q

Property / Rule

Exponential $y=ab^x+q$

If $b>1$ : exponential growth. If $0<b<1$ : exponential decay. Horizontal asymptote: $y=q$ . Domain: $\mathbb{R}$ . Range: $y>q$ if $a>0$ ; $y<q$ if $a<0$ .

\text{Asymptote: }y=q;\quad b>0,\;b\neq1

💡 Tip

y-intercept of $y=ab^x+q$ : substitute $x=0$ to get $y=a\cdot b^0+q=a+q$ . The asymptote is $y=q$ , NOT $y=0$ .

Worked Examples

Worked Example

Plot $y=\dfrac{1}{x}$ and $y=2^x$ point-by-point

Problem

y=\dfrac{1}{x}

Worked Example

Sketch and identify key features

Problem

f(x)=\dfrac{2}{x}

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge2 marks

State the asymptotes of

y=\dfrac{3}{x}+2

L1 · Knowledge2 marks

Find the

y

-intercept of

y=2\cdot3^x-1

L2 · Routine Procedures3 marks

Find the

x

-intercept of

y=\dfrac{4}{x}-1

L2 · Routine Procedures2 marks

Describe the transformation from

y=2^x

y=2^x-3

L3 · Complex Procedures4 marks

Find the equation of a hyperbola with asymptotes

x=0

and

y=-2

passing through

(1;3)

L3 · Complex Procedures3 marks

Given

y=\left(\tfrac{1}{2}\right)^x

, describe the function and find

y

when

x=-3

L4 · Problem Solving5 marks

For

f(x)=\dfrac{a}{x}+q

: the graph passes through

(2;7)

and

(-1;-2)

. Find

a

and

q

L4 · Problem Solving4 marks

Solve

2^{x+1}=\dfrac{1}{4}

without a calculator.

Euclidean Geometry