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Functions & Graphs — Grade 10

Meet the concept of a function and the four basic graph families, and learn how the parameters a and q reshape and shift them. Notes, past papers and a quiz — everything for this topic is one click away.

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Grade 10 CAPS Mathematics

Functions & Graphs

What a function is, the four basic graph families, and the parameters \(a\) and \(q\).

What Is a Function?

Every graph you sketch this year is a picture of a function.

The definition

A function is a rule connecting an input value to an output value, where each input gives exactly one output. Feed the same input in twice and you always get the same output back.

Function notation

We write \(y=f(x)\): \(x\) is the input, \(f(x)\) is the output. \(f(3)\) means “substitute \(x=3\) into the rule \(f\) and evaluate” — it is not \(f\) multiplied by \(3\).

Why "one output" matters

\(x=y^2\) is not a function: if \(x=4\), \(y\) could be \(2\) or \(-2\) — one input, two possible outputs. A function specifically rules this out.

Four representations

The same function can be shown as a table, a graph, words, or a formula. CAPS expects you to move between all four, e.g. building a table of values from a formula, then plotting it.

Worked example Level 1

Given \(g(x)=4x+1\), evaluate \(g(2)\).

Show solution
  1. 1Substitute \(x=2\) into the rule: \(g(2)=4(2)+1\).
  2. 2\(g(2)=8+1=\boxed{9}\).
Worked example Level 2

Given \(f(x)=2x^2-3x+1\), evaluate \(f(-2)\) and \(f(3)\).

Show solution
  1. 1Substitute \(x=-2\): \(f(-2)=2(-2)^2-3(-2)+1=2(4)+6+1\).
  2. 2\(f(-2)=8+6+1=\boxed{15}\).
  3. 3Substitute \(x=3\): \(f(3)=2(3)^2-3(3)+1=2(9)-9+1\).
  4. 4\(f(3)=18-9+1=\boxed{10}\).

The Vertical-Line Test

A quick, visual check for the "one output" rule on a graph.

Vertical line test comparing a parabola and a circle A vertical dashed line crossing a parabola once, showing it is a function, and a circle where a vertical line crosses twice, showing it is not a function. crosses once: function crosses twice: not a function
Why it works: a vertical line marks one x-value. If it touches the graph more than once, that single input has more than one output.
The test

Slide a vertical line across the graph. If it ever crosses more than once, the graph is not a function.

Domain

All allowed input (\(x\)) values. Read left-to-right along the x-axis.

Range

All actual output (\(y\)) values the function produces. Read bottom-to-top along the y-axis.

Circles and sideways parabolas fail

A circle and \(x=y^2\) both fail the test — a big reason they never appear as CAPS "functions," only as relations.

Quick Check

Which of these is not a function?

If \(f(x)=2x^2-3x+1\), what is \(f(2)\)?

The Four Basic Graph Families

Every graph this year is a transformed version of one of these four.

FamilyBasic equationBasic shape
Straight line\(y=x\)A line through the origin, gradient 1
Parabola\(y=x^2\)A U-shape, turning point at the origin
Hyperbola\(y=\dfrac{1}{x}\)Two curved branches, never touching the axes
Exponential\(y=b^x\ (b>0)\)A curve that grows (or decays) away from a horizontal asymptote

The Straight Line: \(y=x\)

The basic straight line graph y equals x A straight line through the origin rising from bottom left to top right, with gradient 1. (0,0)
Key facts: passes through the origin, gradient \(=1\). Domain: \(x\in\mathbb{R}\). Range: \(y\in\mathbb{R}\).
Why it's straight

Every step of \(1\) in \(x\) changes \(y\) by the same fixed amount (the gradient) — a constant rate of change draws a straight line, never a curve.

Domain and range, both unrestricted

You can substitute any real \(x\) with no division or square root to break, and the line keeps rising or falling forever — so both domain and range are all real numbers.

Worked example Level 1

Given \(f(x)=-2x+4\), determine the x- and y-intercepts.

Show solution
  1. 1y-intercept: let \(x=0\). \(f(0)=-2(0)+4=4\), so \(\boxed{(0,4)}\).
  2. 2x-intercept: let \(f(x)=0\). \(0=-2x+4 \Rightarrow 2x=4 \Rightarrow x=2\), so \(\boxed{(2,0)}\).
Worked example Level 2

Given \(g(x)=\dfrac{3}{2}x-6\), determine the x- and y-intercepts.

Show solution
  1. 1y-intercept: let \(x=0\). \(g(0)=\dfrac{3}{2}(0)-6=-6\), so \(\boxed{(0,-6)}\).
  2. 2x-intercept: let \(g(x)=0\). \(0=\dfrac{3}{2}x-6 \Rightarrow \dfrac{3}{2}x=6 \Rightarrow x=6\times\dfrac{2}{3}=4\), so \(\boxed{(4,0)}\).
Quick Check

What is the y-intercept of \(f(x)=-3x+6\)?

For \(y=mx+c\), if \(m<0\), the line:

The Parabola: \(y=x^2\)

The basic parabola y equals x squared An upward-opening parabola with its turning point at the origin. TP (0,0)
Key facts: turning point at the origin, axis of symmetry \(x=0\). Domain: \(x\in\mathbb{R}\). Range: \(y\ge0\).
Why range is \(y\geq0\)

Squaring a real number can never give a negative result, so \(x^2\) is always \(\geq0\) — the graph can never dip below the x-axis.

Why it's symmetric

\((-x)^2=x^2\) for every \(x\), so the graph gives identical outputs for opposite inputs — a mirror image on either side of \(x=0\).

Worked example Level 1

Given \(f(x)=x^2-9\), determine the turning point and the x-intercepts.

Show solution
  1. 1There is no horizontal shift at this level, so the turning point is always on the y-axis: \(\boxed{(0,-9)}\).
  2. 2x-intercepts: let \(f(x)=0\). \(x^2-9=0 \Rightarrow x^2=9 \Rightarrow x=\pm3\).
  3. 3So the x-intercepts are \(\boxed{(-3,0)}\) and \(\boxed{(3,0)}\).
Worked example Level 2

Given \(f(x)=3x^2-12\), determine the turning point and the x-intercepts.

Show solution
  1. 1There is no horizontal shift at this level, so the turning point is always on the y-axis: \(\boxed{(0,-12)}\).
  2. 2x-intercepts: let \(f(x)=0\). \(3x^2-12=0 \Rightarrow x^2=4 \Rightarrow x=\pm2\).
  3. 3So the x-intercepts are \(\boxed{(-2,0)}\) and \(\boxed{(2,0)}\).
Worked example Level 3

Given \(f(x)=2x^2+8\), determine the turning point and the x-intercepts.

Show solution
  1. 1Turning point: \(\boxed{(0,8)}\).
  2. 2x-intercepts: let \(f(x)=0\). \(2x^2+8=0 \Rightarrow x^2=-4\).
  3. 3No real number squares to give a negative result, so this parabola has \(\boxed{\text{no x-intercepts}}\) — it opens upward with its turning point already above the x-axis, so it never crosses it.
Quick Check

For \(y=ax^2+q\) with \(a<0\), the graph:

What is the turning point of \(f(x)=-4x^2+9\)?

The Hyperbola: \(y=\dfrac{1}{x}\)

The basic hyperbola y equals one over x Two curved branches of the hyperbola y equals one over x, one in the top-right region and one in the bottom-left region, approaching but never touching the x and y axes. y=1/x
Key facts: asymptotes \(x=0\) and \(y=0\) (dashed). Domain: \(x\in\mathbb{R},\,x\neq0\). Range: \(y\in\mathbb{R},\,y\neq0\).
Why \(x\neq0\)

Division by zero is undefined, so \(x=0\) can never be substituted — the graph is broken into two branches either side of it.

Why \(y\neq0\)

\(\dfrac{1}{x}=0\) has no solution: no real \(x\) makes a fraction with a nonzero numerator equal to zero. The curve gets arbitrarily close to the x-axis but never touches it.

Asymptote

A line the graph approaches but never reaches or crosses. Always draw asymptotes as dashed lines — they are guides, not part of the graph.

Worked example Level 1

Given \(f(x)=\dfrac{4}{x}-1\), determine the equations of the asymptotes and the x-intercept.

Show solution
  1. 1Vertical asymptote: always \(x=0\) at this level — no parameter shifts it sideways yet.
  2. 2Horizontal asymptote: \(y=q=-1\).
  3. 3x-intercept: let \(f(x)=0\). \(\dfrac{4}{x}-1=0 \Rightarrow \dfrac{4}{x}=1 \Rightarrow x=4\), so \(\boxed{(4,0)}\).
Quick Check

For \(y=\dfrac{a}{x}+q\), the vertical asymptote at this level is always:

For \(g(x)=\dfrac{5}{x}-3\), the horizontal asymptote is:

The Exponential: \(y=b^x\)

The basic exponential graph y equals b to the power x, with b greater than 1 A curve that starts close to the x-axis on the left, passes through (0,1), and rises steeply to the right. (0,1)
Key facts: horizontal asymptote \(y=0\) (dashed), y-intercept \((0,1)\), always above the x-axis. Domain: \(x\in\mathbb{R}\). Range: \(y>0\).
Why it's always positive

A positive base \(b\) raised to any real power \(x\) stays positive — \(b^x\) can shrink toward \(0\) but can never become \(0\) or negative, which is exactly why \(y=0\) is an asymptote, not a crossable line.

Growth vs. decay

\(b>1\): the graph grows, rising left to right. \(0decays, falling left to right while staying above \(y=0\).

Worked example Level 1

Given \(f(x)=2\cdot3^x-6\), determine the y-intercept and the equation of the asymptote, and state whether the graph is increasing or decreasing.

Show solution
  1. 1y-intercept: let \(x=0\). \(f(0)=2(3)^0-6=2(1)-6=-4\), so \(\boxed{(0,-4)}\).
  2. 2Asymptote: \(y=q=\boxed{-6}\).
  3. 3Since \(b=3>1\), the graph is \(\boxed{\text{increasing}}\) (growth).
Quick Check

For \(y=b^x\) with \(0<b<1\), the graph:

The y-intercept of \(h(x)=b^x\) (any \(b>0\), \(b\neq1\)) is always:

The Parameters \(a\) and \(q\)

One equation, \(y=a\cdot f(x)+q\), controls how every basic graph is reshaped and moved.

Comparing y equals x squared with y equals negative x squared plus 3 The basic parabola y equals x squared as a dashed curve, and the reflected, shifted parabola y equals negative x squared plus 3 as a solid blue curve with turning point at (0,3). y=x² y=-x²+3
Reading the diagram: \(a=-1\) flips the parabola upside down, and \(q=3\) shifts it up 3 units — the turning point moves from \((0,0)\) to \((0,3)\).
The parameter \(q\)
  • \(q>0\): shift up
  • \(q<0\): shift down
The parameter \(a\)
  • \(a<0\): reflects the graph
  • \(|a|>1\): stretches it
  • \(0<|a|<1\): compresses it
Why it works
q is added after the rule runs, so every output shifts up or down by the same amount — the whole graph translates vertically. a multiplies the output, so it scales every y-value by the same factor: negative flips the graph, and a factor bigger than 1 in size pushes points further from the x-axis while a fraction pulls them closer.
Worked example Level 1

Given \(f(x)=-2x^2+3\), describe the effect of \(a\) and \(q\) compared to \(y=x^2\), and write down the turning point.

Show solution
  1. 1\(a=-2\): the negative sign reflects the graph in the x-axis, and \(|a|=2\) stretches it vertically.
  2. 2\(q=3\): the graph shifts 3 units up.
  3. 3Since there is no horizontal shift yet, the turning point is \(\boxed{(0,3)}\).
Worked example Level 1

Given \(g(x)=\dfrac{3}{x}-2\), determine the equations of the asymptotes.

Show solution
  1. 1The vertical asymptote of \(y=\dfrac{a}{x}+q\) is always \(x=0\) — there is no parameter shifting it sideways yet.
  2. 2The horizontal asymptote is \(y=q=-2\).
  3. 3So the asymptotes are \(\boxed{x=0}\) and \(\boxed{y=-2}\).
Quick Check

In \(y=a\cdot f(x)+q\), if \(q=-5\), the graph shifts:

If \(a=-1\) is applied to \(y=x^2\), the new graph is:

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What CAPS Expects You to Know

The Grade 10 Functions knowledge statements this page is built from.

  1. 1

    Understand the concept of a function, where a certain quantity (output value) uniquely depends on another quantity (input value), and work with tables, graphs, words and formulae, converting flexibly between them.

  2. 2

    Point-by-point plot the basic graphs \(y=x^2\), \(y=\dfrac{1}{x}\) and \(y=b^x\) (\(b>0\)), discovering their shape, domain, range, asymptotes, axes of symmetry, turning points and intercepts.

  3. 3

    Investigate the effect of the parameters \(a\) and \(q\) on \(y=a\cdot f(x)+q\) for \(f(x)=x\), \(f(x)=x^2\), \(f(x)=\dfrac{1}{x}\) and \(f(x)=b^x\) (\(b>0\)).

  4. 4

    Sketch graphs, find the equations of given graphs, and interpret graphs, based on the effects observed above.

How to Use This Lesson

A few practical notes before you start.

  • Learn the four basic shapes cold before worrying about \(a\) and \(q\) — you need to recognise them instantly.
  • There is no horizontal shift yet at this level — every hyperbola's vertical asymptote is \(x=0\), and every parabola's turning point has x-coordinate 0. That changes in Grade 11.
  • Practise reading domain and range straight off a sketch, not just from the equation.
  • Try each worked example yourself before pressing “Show solution.”

Functions & Graphs, Family by Family

Meet each of the four basic graph families below. Free, independent short videos — not made by Equation Station SA.

Straight Line

\(y=ax+q\) — a is the gradient, q is the y-intercept.

Khan Academy · Slope-intercept form

Parabola

\(y=ax^2+q\) — turning point \((0,q)\), axis of symmetry the y-axis.

Miss Martins Maths and Science · Parabola (Introduction)

Hyperbola

\(y=\dfrac{a}{x}+q\) — asymptotes \(x=0\) and \(y=q\).

Lisa Oswald · Grade 10 Functions - Hyperbola

Exponential

\(y=a\cdot b^x+q\) — horizontal asymptote \(y=q\); grows if \(b>1\), decays if \(0

Khan Academy · Graphing exponential functions

Common Exam Mistakes

Avoid these errors. They cost marks every year.

Confusing which family an equation belongs to

Check the form of the equation carefully — \(x^2\) means parabola, \(1/x\) means hyperbola, \(b^x\) means exponential.

Forgetting q sets the asymptote

Learners often forget the y-value of a hyperbola or exponential's horizontal asymptote is exactly \(q\) — no extra work needed.

Stopping at the x-value for an intersection

An intersection question wants full coordinates — always substitute the x-value back in to find y.

Doing combined transformations in the wrong order

"Shift then reflect" and "reflect then shift" give different equations — always follow the exact wording of the question.

This Topic in Other Grades

Functions & Graphs builds across all three FET grades — preview where this is heading.

Grade 11

The horizontal shift p, and average gradient

Study this

Grade 12

Logarithmic graphs, inverse functions, full Paper 1 depth

Study this

Practise This Topic

You've done the notes above — now practise and test yourself.

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Past Papers
Functions & Graphs Grade 10 Past Papers

17 exam-style Grade 10 Functions questions arranged by cognitive level, with real citations from the DBE/provincial archive.

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Functions & Graphs Grade 10 Test Your Knowledge

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Student Questions

Frequently Asked Questions

Straight answers to common Grade 10 CAPS questions about functions, the four basic graph families, and a and q.

What does Grade 10 Functions and Graphs cover?

Grade 10 covers the concept of a function, the vertical-line test, and the four basic graph families: the straight line y=x, the parabola y=x^2, the hyperbola y=1/x, and the exponential y=b^x, together with the parameters a and q.

Is there a horizontal shift in Grade 10?

No. The horizontal shift parameter p is only introduced in Grade 11. In Grade 10 every hyperbola's vertical asymptote is x=0 and every parabola's axis of symmetry is the y-axis.

What do a and q do to a graph?

a controls the shape and direction of the graph (steepness, and whether it opens up/down, grows/decays, or is reflected). q shifts the whole graph vertically and sets the horizontal asymptote or the constant term.

Which resource should I open first for this topic?

Everything you need to learn the topic is on this page already. Once you've been through the notes above, work through the Past Question Papers, then finish with the Test Your Knowledge quiz as a self-check.