Maths · 3 Functions

3.1 Graphs of functions

Before you start

Plotting a straight line from $y = m x + c$ , and what the gradient and intercept mean.
Substituting a value into a function, e.g. finding $f (2)$ when $f (x) = x^{2} - 3$ .
Reading coordinates off a pair of axes.

Syllabus objectives covered (3)

Recognise the following function types from the shape of their graphs: (a) linear f(x) = ax + b (b) quadratic f(x) = ax² + bx + c (c) cubic f(x) = ax³ + bx² + cx + d (d) reciprocal f(x) = a/x (e) exponential f(x) = aˣ with 0 < a < 1 or a > 1 (f) trigonometric f(x) = a sin(bx); a cos(bx); tan x.
Determine one or two of a, b, c or d for the graphs above.
Determine values in a function from its graph.

Part 1 of 3

Six shapes to know on sight

This subtopic is about recognition. Given a curve, you should be able to name the type of function within a couple of seconds, from its overall shape alone.
Linear, $f (x) = a x + b$ . A straight line. $a$ is the gradient and $b$ is the $y$ -intercept.
Quadratic, $f (x) = a x^{2} + b x + c$ . A parabola — one U-shaped turn, symmetrical about a vertical line through its vertex. It opens upwards if $a > 0$ and downwards if $a < 0$ .
Cubic, $f (x) = a x^{3} + b x^{2} + c x + d$ . Typically an S-shape with two turning points, though it can flatten into a single bend. It has rotational rather than reflective symmetry.
The six shapes. Learn them as pictures, not as equations.
Reciprocal, $f (x) = a / x$ . Two separate curved branches that never touch the axes, in opposite quadrants. The axes are asymptotes.
Exponential, $f (x) = a^{x}$ . A curve that always passes through $(0, 1)$ , because $a^{0} = 1$ for any $a$ . If $a > 1$ it rises steeply to the right; if $0 < a < 1$ it falls to the right.
Either way it flattens towards the $x$ -axis on one side without ever reaching it, and it never goes below the $x$ -axis.
Trigonometric, $f (x) = a \sin (b x)$ , $a \cos (b x)$ , $\tan x$ . Sine and cosine are repeating waves; tangent repeats too but shoots off to infinity at regular intervals.
The quickest ways to tell them apart: count the turning points (0 for linear, 1 for quadratic, up to 2 for cubic), check for repetition (trigonometric), and check whether the curve avoids the axes entirely (reciprocal).

Part 2 of 3

Finding the constants from a graph

Once you have identified the type, you can often work out one or two of the constants by reading specific features off the graph.
Linear. Read $b$ straight off as the $y$ -intercept. Get $a$ from the gradient — pick two clear points and use rise over run.
Quadratic. The $y$ -intercept gives $c$ directly, since $f (0) = c$ . The sign of $a$ comes from which way the parabola opens.
Exponential. The curve always passes through $(0, 1)$ , so use another point instead. If it passes through $(2, 9)$ then $a^{2} = 9$ , giving $a = 3$ .
Trigonometric. Two features carry the information, and the syllabus notes mention both by name.
The amplitude is $a$ — half the vertical distance between the maximum and the minimum. For $y = 3 \sin x$ the curve runs from $- 3$ to $3$ , so the amplitude is 3.
The period is how far along the $x$ -axis the pattern takes to repeat. For $y = \sin (b x)$ the period is $360 ° / b$ , so a curve repeating every $120 °$ has $b = 3$ .
The general method whenever a reading does not give a constant directly: substitute a clear point from the graph into the function and solve for the unknown.

Part 3 of 3

Reading values off a graph

Two directions of reading, and it is worth being clear which a question wants.
**Given $x$ , find $f (x)$ .** Go up from the value on the $x$ -axis to the curve, then across to the $y$ -axis. This is evaluating the function.
**Given $f (x)$ , find $x$ .** Go across from the value on the $y$ -axis to the curve, then down to the $x$ -axis. This is solving an equation.
The second reading can have more than one answer. A horizontal line often cuts a quadratic twice, a cubic three times, and a sine wave many times — so check the whole graph before answering.
Draw the construction lines on the graph. They show the examiner your method, and a mark is often available for a correct line even if the reading is slightly off.
Read to the accuracy the grid allows, and use the syllabus rule of 3 significant figures unless the question says otherwise.
Two readings are worth naming. A zero (or root) is where the curve crosses the $x$ -axis, so $f (x) = 0$ . The ** $y$ -intercept** is where it crosses the $y$ -axis, so $x = 0$ and the value is $f (0)$ .

Worked examples

Example 1. A curve passes through $(0, 1)$ and $(3, 8)$ , rises steeply to the right, and approaches the $x$ -axis without touching it as $x$ becomes large and negative. Identify the type of function and find its equation.

Show the working

Use the shape clues. Passing through $(0,1)$ , rising to the right, and flattening towards the $x$ -axis on the left is the exponential pattern.
So the function has the form $f (x) = a^{x}$ , with $a > 1$ because it rises to the right.
The point $(0,1)$ tells you nothing extra — every exponential passes through it — so use the other point.
$a^{3} = 8$
Solve by taking the cube root.
$a = \sqrt[3]{8} = 2$
Check with the original point: $2^{3} = 8$ . Correct.

Answer: It is an exponential function, $f (x) = 2^{x}$ .

Example 2. The graph of $y = a \sin (b x)$ has a maximum value of $4$ , a minimum value of $- 4$ , and completes one full cycle between $x = 0 °$ and $x = 180 °$ . Find $a$ and $b$ .

Show the working

Take the amplitude first. It is half the distance from the minimum to the maximum.
That distance is $4 - (- 4) = 8$ , so the amplitude is $4$ and therefore $a = 4$ .
Now the period, which is the $x$ -distance for one complete cycle — here $180 °$ .
For $y = a \sin (b x)$ the period is $360 ° / b$ , so set that equal to $180 °$ .
$\frac{360}{b} = 180 \Rightarrow b = 2$
Sense-check: $b = 2$ means the curve completes two full cycles in $360 °$ , which matches one cycle in $180 °$ .

Answer: $a = 4$ and $b = 2$ , so $y = 4 \sin (2 x)$ .

Example 3. The graph of $y = f (x)$ is a parabola with vertex $(2, - 4)$ , crossing the $x$ -axis at $x = 0$ and $x = 4$ . Write down the zeros of $f$ , the value of $f (0)$ , and the number of solutions of $f (x) = - 4$ and of $f (x) = 2$ .

Show the working

The zeros are where the curve crosses the $x$ -axis, so $x = 0$ and $x = 4$ .
$f (0)$ is the $y$ -value when $x = 0$ . The curve crosses the $x$ -axis there, so $f (0) = 0$ .
For $f (x) = - 4$ , draw the horizontal line $y = - 4$ . It touches the parabola exactly at the vertex $(2, - 4)$ and nowhere else, so there is one solution.
For $f (x) = 2$ , the line $y = 2$ lies above the vertex and cuts the upward parabola on both sides, so there are two solutions.
The lesson to carry forward: how many times a horizontal line meets the curve is how many solutions the equation has.

Answer: Zeros at $x = 0$ and $x = 4$ ; $f (0) = 0$ ; $f (x) = - 4$ has one solution; $f (x) = 2$ has two solutions.

Where people go wrong

Mistake: Calling any curve with a bend a quadratic.

Instead: Count the turning points — a quadratic has exactly one, a cubic up to two.

A cubic with a gentle S can look parabolic at a glance. Check whether the two ends go the same way (quadratic) or opposite ways (cubic).

Mistake: Using $(0,1)$ to find $a$ for an exponential function.

Instead: Every exponential passes through $(0,1)$ , so use a different point.

$a^{0} = 1$ whatever $a$ is, so that point carries no information about $a$ .

Mistake: Confusing amplitude with the distance from maximum to minimum.

Instead: The amplitude is half that distance.

Amplitude is measured from the centre line to the peak, not from peak to trough.

Mistake: Giving only one solution when reading $f (x) = k$ from a graph.

Instead: Draw the horizontal line right across and count every intersection.

A quadratic usually gives two and a sine curve several within a given range. Marks are lost for the ones you did not look for.

Mistake: Drawing a reciprocal graph touching the axes.

Instead: The branches approach the axes but never meet them.

$a / x$ is undefined at $x = 0$ and can never equal zero, so both axes are asymptotes.

Check yourself

How many turning points does a cubic graph typically have, and how does that distinguish it from a quadratic?
Answer
Up to two, whereas a quadratic has exactly one. A cubic's two ends also go in opposite vertical directions.
Which two function types on the list have asymptotes?
Answer
The reciprocal function, whose branches approach both axes, and the exponential function, which approaches the $x$ -axis on one side. The tangent function also has vertical asymptotes.
Through which point does every graph of $f (x) = a^{x}$ pass, and why?
Answer
$(0, 1)$ , because $a^{0} = 1$ for any positive $a$ .
A curve $y = a \cos (b x)$ ranges from $- 2$ to $2$ and repeats every $90 °$ . Find $a$ and $b$ .
Answer
Amplitude $= 2$ , so $a = 2$ . Period $360 / b = 90$ , so $b = 4$ .
For a quadratic $f (x) = a x^{2} + b x + c$ , which constant can you read directly off the graph, and how?
Answer
$c$ , from the $y$ -intercept, since $f (0) = c$ .
Explain the difference between finding $f (3)$ from a graph and solving $f (x) = 3$ .
Answer
For $f (3)$ , go up from $x = 3$ to the curve and across to the $y$ -axis. For $f (x) = 3$ , go across from $y = 3$ to the curve and down to the $x$ -axis, and there may be more than one answer.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. A curve passes through the origin, has a maximum at $(90 °, 5)$ and a minimum at $(270 °, - 5)$ , and returns to zero at $x = 360 °$ .

(a) State the type of function shown. 1 mark No calculator

Mark scheme

B1 trigonometric / sine

(b) Write down the amplitude and the period of the curve. 2 marks No calculator

Mark scheme

B1 amplitude 5
B1 period 360°

(c) Give the equation of the curve in the form $y = a \sin (b x)$ . 2 marks No calculator

Mark scheme

B1 a = 5; FT their amplitude
B1 b = 1, giving y = 5 sin x oe

Question 2. The diagram shows the graph of $y = f (x)$ , a quadratic curve with a minimum at $(- 1, - 9)$ and crossing the $x$ -axis at $x = - 4$ and $x = 2$ .

(a) Write down the zeros of $f$ . 1 mark No calculator

Mark scheme

B1 -4 and 2, both required

(b) Write down the number of solutions of $f (x) = - 9$ . 1 mark No calculator

Mark scheme

B1 1

(c) Explain why $f (x) = - 12$ has no solutions. 1 mark No calculator

Mark scheme

B1 -12 is below the minimum value of -9, so the line y = -12 does not meet the curve oe

Question 3. Four curves are shown. Curve A is a straight line, curve B has two separate branches that approach but never touch either axis, curve C passes through $(0,1)$ and decreases towards the $x$ -axis as $x$ increases, and curve D has two turning points.

(a) State the type of function shown by each of curves B, C and D. 3 marks No calculator

Mark scheme

B1 B reciprocal
B1 C exponential
B1 D cubic

(b) For curve C, state what can be said about the value of $a$ in $f (x) = a^{x}$ . 1 mark No calculator

Mark scheme

B1 0 < a < 1 oe

Words to know

Amplitude: Half the vertical distance between the maximum and minimum of a trigonometric graph.
Period: The horizontal distance after which a repeating graph starts to repeat itself.
Zero (root): A value of $x$ where the curve crosses the $x$ -axis, so $f (x) = 0$ .
Asymptote: A line that a curve approaches ever more closely but never touches.
Turning point: A point where a curve changes from increasing to decreasing, or the reverse.

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