Functions and graphs. Attnom Academy

M4.10.1 Chapter 1

Function notation

My book says f(x) = 2x − 1. Is f a number that multiplies x?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Yes. f is a number, and f(3) means f times 3.
You picked A. Read on. At the end, the chapter comes back to your guess.
B No. f(3) means f plus 3.
You picked B. Read on. At the end, the chapter comes back to your guess.
C No. f is the name of a rule, and f(3) is what comes out when 3 goes in.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Find f(3) when f(x) = 2x − 1
Solve f(x) = 7 for the same function

A function is a rule. You give it a number, and it gives back one number. The number you give is the input. The number you get back is the output.

The rule f: double the input, then take away 1. The output is called f(x).

We write this rule as f(x) = 2x − 1. Read it as "f of x equals 2 x minus 1". The letter f is the name of the rule. The letter x stands for any input. And f(x) is the output.

The brackets in f(x) are not a multiplication. Do not read f(x) as f times x. They show which number is the input.

f(x) = 2x − 1 as arrows. Each input has exactly one output.

In the picture the input 3 goes to the output 5. We write that as f(3) = 5. Notice that each input has one arrow only. That is what makes a rule a function.

Draw it first

From a rule to a graph

1 Work out f(x) for several inputs, and plot each pair (x, f(x)) as a point.
2 The points lie on a straight line. Join them with a ruler.
3 To read f(3), go up from x = 3 to the line, then across to the y axis. You land on 5.

The graph of a function is the set of all points (x, f(x)). That is why we often write y = f(x). The y value of the graph is the output.

Worked example

f(x) = 2x − 1. Find f(3).

Write the rule: f(x) = 2x − 1.
Put 3 in place of every x: f(3) = 2 × 3 − 1.
Multiply first: 2 × 3 = 6.
Then take away 1: 6 − 1 = 5.

Answer: f(3) = 5.

Now a rule with a square. Take care with a negative input. Put the input in a bracket, then square the whole bracket.

Worked example

f(x) = x² − 3. Find f(−2).

Write the rule: f(x) = x² − 3.
Put −2 in a bracket in place of x: f(−2) = (−2)² − 3.
Square the bracket: (−2) × (−2) = 4.
Take away 3: 4 − 3 = 1.

Answer: f(−2) = 1.

Sometimes you are told the output and must find the input. Then you write f(x) equal to that number, and solve the equation. This means undoing the rule, step by step, in the opposite order.

To find x, run the machine backwards: add 1, then divide by 2.

Worked example

f(x) = 2x − 1. Solve f(x) = 7.

Write f(x) = 7 as an equation: 2x − 1 = 7.
Add 1 to both sides: 2x = 8.
Divide both sides by 2: x = 4.
Check by putting 4 in the rule: f(4) = 2 × 4 − 1 = 7.
The output is 7, as wanted. So the input is 4.

Answer: x = 4.

See it three ways

With things

Amina doubles her number of beans, then gives one bean away. She starts with 3 beans.

As a picture

In words

Double the input, then take away 1. The input 3 gives the output 5.

In symbols

f(x) = 2x − 1, so f(3) = 2 × 3 − 1 = 5.

Try it

A table for f(x) = 2x − 1

You need: paper, a pencil.

Write the numbers 0, 1, 2, 3 and 4 in a row. These are the inputs.
Under each one, work out 2x − 1. These are the outputs f(x).
Look along the outputs. By how much does each one differ from the one before?

What you will see: The outputs go up by 2 each time, the number in front of x. When the input is 0, the output is −1, the number you take away.

If you cannot try it, this is what happens.

A wrong idea many people have

f(3) means f times 3.

Why it feels right: In algebra, letters next to each other multiply, as in 3x. So f(3) looks like a product.

The test that breaks it: Take f(x) = 2x − 1. Then f(1) = 1. If f were a number, f(1) = f × 1 would make f equal to 1. Then f(2) = f × 2 would be 2. But f(2) = 2 × 2 − 1 = 3.

What is true: f is the name of a rule, not a number. The brackets hold the input. f(3) is the output when the input is 3.

A wrong idea many people have

f(x) = 7 means find f(7).

Why it feels right: The number 7 is in the question. The expression f(7) also has a 7 in it, so they seem the same job.

The test that breaks it: For f(x) = 2x − 1, f(7) = 13. But f(4) = 7. The equation f(x) = 7 asks for the input 4.

What is true: In f(7) the 7 is the input. In f(x) = 7 the 7 is the output. Solve the equation to find the input.

Key words

function: a rule that gives exactly one output for each input.
input: the number you put into the function, shown as x in f(x).
output: the number that comes out of the function, written f(x).

Back to your guess

The answer is C: No. f is the name of a rule, and f(3) is what comes out when 3 goes in.

The letter f is a name, like the name of a machine. The brackets do not mean times. They hold the input. So f(3) means: put 3 into the rule f and see what comes out. For f(x) = 2x − 1, double 3, take away 1, and the output is 5.

Keep it

Copy this into your exercise book.

A function is a rule. Each input has one output. f(3) means put 3 in. f(x) = 7 means find the x that gives 7. Undo the rule backwards.

Draw: A machine with the arrows x, then × 2, then − 1, then f(x), and a second row going back.

M4.10.2 Chapter 2

Composite and inverse functions

If I put a number into g, and then put the answer into f, is that the same as doing f first?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Yes. Two functions give the same result in either order.
You picked A. Read on. At the end, the chapter comes back to your guess.
B No. Doing g first and then f usually gives a different result.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Only when the numbers are small.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Find fg(x) from two functions f and g
Find the inverse function f⁻¹(x)

You can join two functions, one after the other. Put a number into g. Then put the answer into f. The pair together is a new function, called a composite function.

Here g(x) = x + 3 and f(x) = 2x. The number goes through g first, which adds 3, then through f, which doubles it.

We write this composite as fg(x). It means f(g(x)). The function nearest to x goes first. So in fg(x), the function g is done first, and f second.

Worked example

f(x) = 2x and g(x) = x + 3. Find fg(4).

fg(4) means f(g(4)). Work out g(4) first.
g(4) = 4 + 3 = 7.
Now put 7 into f: f(7) = 2 × 7.
f(7) = 14.

Answer: fg(4) = 14.

Now turn the order round. The function gf(x) means g(f(x)). Now f goes first, and g second.

Worked example

f(x) = 2x and g(x) = x + 3. Find gf(4), and compare it with fg(4).

gf(4) means g(f(4)). Work out f(4) first.
f(4) = 2 × 4 = 8.
Now put 8 into g: g(8) = 8 + 3.
g(8) = 11.
But fg(4) = 14. The two are different, so the order matters.

Answer: gf(4) = 11, and it differs from fg(4) = 14.

A composite can also be written with x, so that it works for every input. Put the whole of g(x) into f, in place of each x.

Worked example

f(x) = 2x + 1 and g(x) = x − 3. Find fg(x).

fg(x) means f(g(x)). Put g(x) into f.
g(x) = x − 3, so f(g(x)) = 2(x − 3) + 1.
Multiply the bracket: 2 × x = 2x, and 2 × (−3) = −6.
So the line reads 2x − 6 + 1.
Collect the numbers: −6 + 1 = −5.
So fg(x) = 2x − 5.

Answer: fg(x) = 2x − 5

Now go the other way. Sometimes you know the output of a function and want its input. A rule that takes you back is called the inverse function. For f, we write it f⁻¹.

f(x) = 2x + 1 takes each input to its output.

The inverse f⁻¹ turns every arrow round. It takes each output back to its input.

An inverse works only when each output comes from one input only. The rule f(x) = 2x + 1 passes. But the rule x² does not: the output 4 comes from both 2 and −2. We keep to rules that can be undone.

Draw it first

Find f⁻¹ by undoing the steps

1 Write the steps of f in order: multiply by 2, then add 1.
2 Undo them in the reverse order. The last step was add 1, so undo it first with take away 1. Then undo × 2 with ÷ 2.
3 The undo row, with the letter x for the input, is the rule of f⁻¹.

You can do the same with algebra. Write y = f(x), then make x the subject. The result gives the inverse, once you name its input x again.

Worked example

f(x) = 3x − 2. Find f⁻¹(x).

Write y = 3x − 2.
Add 2 to both sides to undo the take away: 3x = y + 2.
Divide both sides by 3: x = (y + 2)/3.
Call the input x, as always. So f⁻¹(x) = (x + 2)/3.

Answer: f⁻¹(x) = (x + 2)/3

There is a way to check an inverse. Put a number into f, then put the answer into f⁻¹. You must get back the number you began with.

Worked example

f(x) = 3x − 2 and f⁻¹(x) = (x + 2)/3. Show that f⁻¹ takes f(4) back to 4.

First f(4) = 3 × 4 − 2 = 10.
Now put 10 into the inverse: f⁻¹(10) = (10 + 2)/3.
Add on the top: 10 + 2 = 12.
Divide: 12 ÷ 3 = 4.
We are back at 4, where we began.

Answer: f⁻¹(f(4)) = 4.

See it three ways

With things

Kofi thinks of a number. He doubles it and adds 1. He gets 9. What number did he think of?

As a picture

In words

Undo the steps backwards. Take away 1 from 9 to get 8, then halve 8 to get 4. Kofi thought of 4.

In symbols

f(x) = 2x + 1 and f⁻¹(x) = (x − 1)/2, so f⁻¹(9) = (9 − 1)/2 = 4.

Try it

Does the order matter?

You need: paper, a pencil.

Start with the number 5. Add 3, then multiply the result by 2. Write the answer.
Start with 5 again. This time multiply by 2 first, then add 3. Write the answer.
Compare the two answers.

What you will see: The answers are 16 and 13. They differ, so the order of the two functions changes the result.

If you cannot try it, this is what happens.

A wrong idea many people have

fg(x) means do f first, then g.

Why it feels right: We read from left to right, so the f comes first and it seems to go first.

The test that breaks it: Take f(x) = 2x and g(x) = x + 3. If f went first, fg(4) would be g(f(4)) = g(8) = 11. But fg(4) means f(g(4)) = f(7) = 14, because the brackets show g working on 4.

What is true: In fg(x) = f(g(x)), the function next to x goes first. Read it from the right: g first, then f.

A wrong idea many people have

f⁻¹(x) means 1 divided by f(x).

Why it feels right: A small raised −1 on a number gives a reciprocal. For example, 2⁻¹ is 1/2. So the same mark on f seems to mean the same.

The test that breaks it: Take f(x) = 2x + 1. Then f(3) = 7. If f⁻¹ meant 1 divided by f, then f⁻¹(7) would be 1 ÷ f(7) = 1/15. But f⁻¹(7) = 3, because f takes 3 to 7 and the inverse takes 7 back to 3.

What is true: For a function, f⁻¹ is the inverse function, which undoes f. The reciprocal 1/f(x) is a different thing.

Key words

composite function: two functions joined, with the output of one put into the next, such as fg(x) = f(g(x)).
inverse function: the function f⁻¹ that undoes f, taking each output back to its input.

Back to your guess

The answer is B: No. Doing g first and then f usually gives a different result.

The order matters. Take f(x) = 2x and g(x) = x + 3, with the input 4. Doing g first gives 7, and then f gives 14. Doing f first gives 8, and then g gives 11. Joining functions is like putting on socks and then shoes: the other way round is not the same.

Keep it

Copy this into your exercise book.

fg(x) means f(g(x)): g goes first. The order matters. The inverse f⁻¹ undoes f: reverse the steps. Check it by putting a number through f, then f⁻¹.

Draw: A machine with two rows: the steps of f going right, and the undone steps going back.

M4.10.3 Chapter 3

Cubic, reciprocal and circle graphs

What does the graph of y = 1/x look like? Does it ever touch the axes?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Two separate curves that come close to the axes but never touch them.
You picked A. Read on. At the end, the chapter comes back to your guess.
B One smooth U shape that touches the x axis.
You picked B. Read on. At the end, the chapter comes back to your guess.
C A straight line that slopes down.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Sketch y = x³ and y = 1/x
Sketch the circle x² + y² = 25
Match an equation to the shape of its graph

You know that y = 2x + 1 gives a straight line, and y = x² gives a U shaped curve. In this chapter you meet three more shapes. Each one comes from a table of values, and each has features you can learn to spot.

First, the cubic y = x³. It means x × x × x. Make a table, with negative values of x as well. Take care: three negative numbers multiplied give a negative answer.

For x from −3 to 3, the table gives y = −27, −8, −1, 0, 1, 8 and 27. The y values are negative when x is negative.

Worked example

Find y when x = −2 for y = x³.

x³ means x × x × x. So y = (−2) × (−2) × (−2).
Multiply the first two: (−2) × (−2) = 4.
Then multiply by the last: 4 × (−2) = −8.

Answer: y = −8.

The graph of y = x³. Plot the points of the table and join them with one smooth curve.

Look at A and B. They are the same distance from the origin, on opposite sides of it. The graph looks the same when it is turned half way round about the origin.

The graph of y = x³ passes through the origin. For large x it climbs steeply. For very negative x it drops steeply. It is flat for an instant at the origin, and it never turns round.

Next, the reciprocal graph y = 1/x. The reciprocal of a number is 1 divided by it. For example, the reciprocal of 4 is 0.25, and the reciprocal of −5 is −0.2.

Worked example

Find y for y = 1/x when x = 4, and when x = −5.

y = 1/x means y = 1 ÷ x. When x = 4: y = 1 ÷ 4 = 0.25.
When x = −5: y = 1 ÷ (−5). A positive divided by a negative is negative.
So y = −0.2.

Answer: When x = 4, y = 0.25. When x = −5, y = −0.2.

You cannot work out y when x = 0, because nothing can be divided by 0. So there is a gap in the graph at x = 0. The graph comes in two parts, called branches.

Draw it first

The graph of y = 1/x

1 Plot points for positive and negative x. Use small values like 1, 2 and 4.
2 Join the points with x above 0 into one smooth curve, and the points with x below 0 into a second curve. The two branches never join.

Look at how the branches behave. When x is big, y = 1/x is tiny but still positive. So the curve gets close to the x axis. When x is very small and positive, y is huge. So the curve climbs close to the y axis.

For big x, y = 1/x is close to the x axis. At x = 10 it is still 0.1.

For small x, y = 1/x is huge. The curve climbs close to the y axis but never reaches it.

The axes here are called asymptotes of the graph. An asymptote is a line that a curve gets nearer and nearer to, without ever touching it.

See it three ways

With things

Leila has a rectangle of area 1 m². Its width is x m and its length is y m. With width 2 m the length is 0.5 m. With width 4 m the length is 0.25 m.

As a picture

In words

A wider rectangle must be shorter. The length gets closer to 0 m, but it never reaches 0.

In symbols

y = 1/x, so y = 0.5 when x = 2, and y = 0.25 when x = 4.

The third shape is a circle. The points on a circle with its centre at the origin are all the same distance from the origin. Take the point P with coordinates (3, 4). Pythagoras gives its distance from the origin: 3² + 4² = 9 + 16 = 25, and √25 = 5.

Any point (x, y) that is 5 units from the origin has x² + y² = 25. That is the equation of the circle with radius 5 and centre at the origin. In general, x² + y² = r² is the circle with radius r.

Draw it first

The circle x² + y² = 25

1 Find points 5 units from the origin, such as (5, 0), (3, 4) and (4, 3). Copy them into the other quarters.
2 Join them into one smooth round curve, a circle with centre at the origin. The dashed line from the origin to P is 5 units long, because 3² + 4² = 25.

A circle is not the graph of a function. For x = 3 it has two y values, 4 and −4. A function must give one output for each input.

Worked example

Find the y values on the circle x² + y² = 25 when x = 3.

Put x = 3 into x² + y² = 25. First 3² = 9.
So 9 + y² = 25. Take away 9: y² = 16.
The square roots of 16 are 4 and −4. So y = 4 or y = −4.
The circle has two points above and below x = 3: (3, 4) and (3, −4).

Answer: y = 4 or y = −4.

Try it

Get close to zero

You need: a calculator, or paper and a pencil.

Work out 1 ÷ x for x = 1, 10, 100 and 1000.
Write the answers in a table.
Look at the answers. Do they ever reach 0?

What you will see: The answers fall from 1 to 0.001 and keep getting smaller, but each one is still above 0. The curve gets close to the x axis and never touches it.

If you cannot try it, this is what happens.

A wrong idea many people have

The two parts of y = 1/x join up across the y axis.

Why it feels right: Smooth curves usually run on without a gap, and the two parts face each other.

The test that breaks it: To cross the y axis the curve would need a value at x = 0. But 1 ÷ 0 has no answer. The curve is above the axis for x above 0 and below it for x below 0.

What is true: y = 1/x has two separate branches. The graph never touches the y axis or the x axis.

A wrong idea many people have

The circle x² + y² = 25 has radius 25.

Why it feels right: The number 25 is the number in the equation, and a radius seems to be that number.

The test that breaks it: The point (5, 0) is on the circle, because 25 + 0 = 25. It is 5 units from the origin. The point (25, 0) gives 625 + 0, which is not 25.

What is true: In x² + y² = r², the number on the right is r², the radius squared. The radius is its square root, so here it is 5.

Key words

cubic: a graph or rule with x³ as its highest power, such as y = x³.
reciprocal: the reciprocal of a number other than 0 is 1 divided by it. y = 1/x is the reciprocal graph.
asymptote: a line that a curve gets nearer and nearer to, without touching it.

Back to your guess

The answer is A: Two separate curves that come close to the axes but never touch them.

The graph of y = 1/x has two parts, one in the top right and one in the bottom left. Each part comes closer and closer to both axes. But 1/x is never 0, and x can never be 0, so the curve never touches an axis.

Keep it

Copy this into your exercise book.

Cubic: an S through the origin. Reciprocal: two branches that never touch the axes. Circle: x² + y² = r², radius r.

Draw: Three small graphs side by side: an S curve, two branches in opposite corners, and a circle round the origin.

M4.10.4 Chapter 4

Exponential graphs

What does the graph of y = 2ˣ look like? It starts small. How fast does it rise?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A A straight line, going up by 2 at each step.
You picked A. Read on. At the end, the chapter comes back to your guess.
B A curve that rises slowly, then levels off.
You picked B. Read on. At the end, the chapter comes back to your guess.
C A curve that rises slowly at first, then faster and faster.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Draw y = 2ˣ from a table of values
Tell a growth curve from a decay curve

Some things double again and again. A patch of algae on a pond covers 1 m² today. It doubles each day, so it covers 2 m² tomorrow, then 4 m², then 8 m². After x days the area is x 2s multiplied together.

We write that as y = 2ˣ. The x is the number of days. It stands in the power, not as a number to multiply by. A rule with the letter in the power is called an exponential function. The number 2 is its base.

The table for y = 2ˣ. Each y value is double the one before.

For x = 0 the table gives 1. Any number except 0, to the power 0, is 1. For negative x, use 2⁻¹ = 1/2 and 2⁻² = 1/4. These are the decimals 0.5 and 0.25.

Worked example

y = 2ˣ. Find y when x = 5, and when x = −2.

When x = 5, multiply five 2s: 2 × 2 × 2 × 2 × 2 = 32.
When x = −2, use the rule for a negative power: 2⁻² = 1/2². First 2² = 4.
So 2⁻² = 1/4 = 0.25.

Answer: When x = 5, y = 32. When x = −2, y = 0.25.

Draw it first

The graph of y = 2ˣ

1 Plot the points of the table. The first ones are very close to the x axis.
2 Join them with one smooth curve. It rises slowly, then faster and faster.

Look at what the graph does. It crosses the y axis at (0, 1), because 2⁰ = 1. It is always above the x axis, because a power of 2 is never 0 or negative. On the left it gets closer and closer to the x axis, which is an asymptote.

Worked example

The algae covers 2ˣ m² after x days. After how many days is the area 64 m²?

We want 2ˣ = 64. Try whole numbers for x.
Try x = 5: 2 × 2 × 2 × 2 × 2 = 32. This is too small.
Try x = 6: 2 × 32 = 64.
So x = 6. The area is 64 m² after 6 days.

Answer: x = 6

Now take a base between 0 and 1, such as 0.5. In y = 0.5ˣ the value is halved each time x goes up by 1. The values 1, 0.5, 0.25, 0.125 fall towards 0. A sample of material whose mass halves every hour follows this rule.

The growth curve y = 2ˣ and the decay curve y = 0.5ˣ. Both pass through A at (0, 1).

A graph y = aˣ with a above 1 is a growth curve. It rises as x increases. With a between 0 and 1 it is a decay curve. It falls as x increases. The base a must be positive. If a = 1, then y = 1 for every x, a flat line that neither grows nor decays.

Worked example

Is y = 0.9ˣ a growth curve or a decay curve?

Work out y for x = 0, 1 and 2. Each step multiplies by 0.9.
At x = 0, y = 1.
At x = 1, y = 1 × 0.9 = 0.9.
At x = 2, y = 0.9 × 0.9 = 0.81.
The values 1, 0.9, 0.81 fall. So y = 0.9ˣ is a decay curve.

Answer: It is a decay curve, because the values 1, 0.9, 0.81 fall.

See it three ways

With things

A patch of algae covers 1 m² on day 0. It doubles each day. On day 1 it covers 2 m², on day 2 it covers 4 m², and on day 3 it covers 8 m².

As a picture

In words

The area doubles each day, so it climbs faster each day. Each day adds more than the day before.

In symbols

y = 2ˣ, so y = 2 when x = 1, y = 4 when x = 2, and y = 8 when x = 3.

Try it

Fold a sheet of paper

You need: a sheet of paper.

Fold the sheet in half. Count the layers.
Fold it in half again, and count the layers. Do this up to 4 folds.
Write the number of folds and the number of layers in a table.

What you will see: The layers double with each fold: 1, 2, 4, 8, 16. After x folds there are 2ˣ layers.

If you cannot try it, this is what happens.

A wrong idea many people have

y = 2ˣ is the same as y = 2x.

Why it feels right: Both have a 2 and an x, and both go up as x gets bigger.

The test that breaks it: Take x = 5. Then 2x = 10, but 2ˣ = 32. Take x = 10. Then 2x = 20, but 2ˣ = 1 024.

What is true: In 2x the x multiplies 2. In 2ˣ the x is a power, so x 2s are multiplied together. The graph of 2x is a straight line, and the graph of 2ˣ curves up ever more steeply.

A wrong idea many people have

A decay curve reaches zero in the end.

Why it feels right: The values fall and fall, so it seems they must run out.

The test that breaks it: Take y = 0.5ˣ. At x = 10 the value is about 0.00098, which is small but above zero. Halving a positive number never gives zero.

What is true: A decay curve gets as close to the x axis as you like, but it never reaches it. The x axis is an asymptote.

Key words

exponential: a rule with the letter in the power, like y = 2ˣ.
growth: an exponential rule y = aˣ with the base a above 1. The values rise as x rises.
decay: an exponential rule y = aˣ with the base a between 0 and 1. The values fall as x rises.

Back to your guess

The answer is C: A curve that rises slowly at first, then faster and faster.

In y = 2ˣ the x is a power, so y doubles each time x goes up by 1. The values 1, 2, 4, 8, 16 climb faster and faster. The graph is a steep curve that never levels off. For negative x it comes very close to the x axis but never touches it.

Keep it

Copy this into your exercise book.

y = aˣ passes through (0, 1) and stays above the x axis. Base above 1: growth, rising faster. Base between 0 and 1: decay, falling towards the axis.

Draw: One graph with a rising curve and a falling curve crossing at (0, 1).

M4.10.5 Chapter 5

Graphs of sin, cos and tan

What does the graph of y = sin x look like, for angles from 0° to 360°?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A A straight line that rises all the way to 360.
You picked A. Read on. At the end, the chapter comes back to your guess.
B A wave that rises to 1, falls to −1, and comes back to 0.
You picked B. Read on. At the end, the chapter comes back to your guess.
C A U shaped curve like y = x².
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Sketch y = sin x and y = cos x up to 360°
Use a graph to find two angles with sin x = 0.5

In a right angled triangle, sin of an angle is a ratio of two sides. Here you use sin and cos for any angle from 0° to 360°. Check that your calculator is in degree mode. If sin 30° does not show 0.5, it is not.

Think of a wheel with radius 1. A point P sits on its rim, level with the centre, on the right. Turn the wheel anticlockwise by an angle x. Now P has a height above the centre line, and a distance across from the centre.

We define sin x as the height of P above the centre. We define cos x as its distance to the right of the centre. Below the centre the height is negative. To the left of the centre the distance is negative.

The wheel of radius 1. P is 30° round from the right, and Q is 150° round. Both are at height 0.5.

At 30° the point P is at height 0.5, so sin 30° = 0.5. At 150° the point Q is at the same height, so sin 150° = 0.5 too. As the wheel turns, the height of P rises and falls like a wave. Plot it against the angle, and you get the graph of y = sin x.

At 210° the point is 30° below the centre line, on the left. Its height is −0.5.

Worked example

Find sin 210°, using the wheel.

Work out how far 210° is past the half turn: 210° − 180° = 30°.
So P is 30° below the horizontal line, on the left. Its distance below the centre is the same as the height at 30°, which is 0.5.
Below the centre means negative. So sin 210° = −0.5.

Answer: sin 210° = −0.5.

The graph of y = sin x, with x in degrees. Each dot is the height of the wheel point after that angle.

Read the graph from left to right. At 0° it is 0. It rises to 1 at 90° and falls to 0 at 180°. Then it reaches −1 at 270° and is back to 0 at 360°. After 360° the wheel is back where it began, so the wave repeats. The length of one full wave is its period: 360° for sin and cos.

The graph of cos x is the same wave, but it starts at the top. At 0° it is 1. It falls to 0 at 90°, to −1 at 180°, rises to 0 at 270°, and is back at 1 at 360°. Cos x is the distance of P across from the centre, so it starts at 1.

The two waves. The cos wave is the sin wave moved 90° to the left.

Both waves stay between −1 and 1. So sin x = 2 has no solution at all. The graph of tan x is different. It is sin x divided by cos x. Cos x is 0 at 90° and 270°, so tan x has no value there.

The graph of y = tan x. The dashed lines at 90° and 270° are asymptotes. The curve gets close but never touches them.

The tan graph passes through 0 at 0°, 180° and 360°. It repeats every 180°, so its period is 180°. At 45° it is 1. Between 90° and 180° it is negative.

Now use the graph to solve an equation. To solve sin x = 0.5, draw the horizontal line y = 0.5 across the sin graph. Each place where it meets the curve gives an angle.

Worked example

Solve sin x = 0.5 for x between 0° and 360°.

Draw the line y = 0.5 across the graph of y = sin x.
It meets the curve twice between 0° and 360°.
The calculator gives the first angle: sin⁻¹ 0.5 = 30°.
The wave is symmetric about 90°. The second angle is 180° − 30° = 150°.

Answer: x = 30° or x = 150°.

See it three ways

With things

A valve on a bicycle wheel starts level with the centre, on the right, and the wheel turns anticlockwise. At what angles is the valve half a radius above the centre?

As a picture

In words

The valve is at that height twice in one turn. It is there once on the way up, after 30°, and once on the way down, after 150°.

In symbols

sin x = 0.5 gives x = 30° or x = 150°.

Worked example

Solve cos x = 0.5 for x between 0° and 360°.

The line y = 0.5 meets the cos curve twice between 0° and 360°.
The calculator gives the first angle: cos⁻¹ 0.5 = 60°.
The cos wave is symmetric about 180°. From 60° to 180° is 180° − 60° = 120°.
The second angle is 120° on the other side of 180°: 180° + 120° = 300°.

Answer: x = 60° or x = 300°.

Try it

Measure the sine on a circle

You need: paper, a ruler, a protractor, a pencil.

Draw a circle with radius 10 cm. Draw a horizontal line through its centre.
With the protractor, draw radius lines at 30°, 60° and 90° above the horizontal line.
Measure the height of the end of each radius above the horizontal line. Divide each height by 10.

What you will see: The heights are about 5 cm, 8.7 cm and 10 cm. Divided by 10 they give 0.5, 0.87 and 1. These are sin 30°, sin 60° and sin 90°.

If you cannot try it, this is what happens.

A wrong idea many people have

sin x = 0.5 has only one solution, 30°.

Why it feels right: The calculator gives one answer when you press sin⁻¹, so it seems that there is only one.

The test that breaks it: The wheel point is at height 0.5 at 30°. It is at the same height again at 150°. So sin 150° = 0.5 as well.

What is true: Between 0° and 360° the line y = 0.5 meets the sin wave twice. The calculator gives the first angle. Use symmetry to find the second.

A wrong idea many people have

tan x stays between −1 and 1, like sin x.

Why it feels right: Sin, cos and tan are written alike and come from the same triangle.

The test that breaks it: tan 45° = 1, but tan 80° is about 5.7. The values keep growing as the angle gets close to 90°.

What is true: Only sin x and cos x stay between −1 and 1. The tan graph climbs without limit towards the asymptotes at 90° and 270°.

Key words

period: the length of one full wave before the graph repeats. For sin and cos it is 360°.
sine wave: the wave shaped graph of y = sin x.

Back to your guess

The answer is B: A wave that rises to 1, falls to −1, and comes back to 0.

The graph of y = sin x is a smooth wave. It starts at 0 and rises to 1 at 90°. It falls through 0 at 180° to −1 at 270°, and is back at 0 at 360°. The wave never goes above 1 or below −1, and it repeats every 360°.

Keep it

Copy this into your exercise book.

Sin and cos are waves between −1 and 1, repeating every 360°. Sin starts at 0, cos starts at 1. To solve sin x = 0.5, find 30°, then use symmetry for 150°.

Draw: A sin wave and a dashed cos wave from 0° to 360°. Add a horizontal line at 0.5 that meets the sin wave twice.

M4.10.6 Chapter 6

Moving graphs

If I draw y = f(x + 3), does the graph of y = f(x) slide to the right or to the left?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A To the right, because 3 is added to x.
You picked A. Read on. At the end, the chapter comes back to your guess.
B To the left, even though 3 is added to x.
You picked B. Read on. At the end, the chapter comes back to your guess.
C It does not move. It only gets taller.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Sketch y = f(x) + a and y = f(x + a)
Sketch y = −f(x) and y = f(−x)

Once you know the shape of a graph, you can move it about. We start with one curve, y = f(x), where f(x) = x² − 2x. Its lowest point is V at (1, −1), because f(1) = 1 − 2 = −1.

Follow V as the curve moves. Each change to the rule moves V in a different way. The shape of the curve stays the same, unless the curve is turned over.

First, add a number after the function. For y = f(x) + 3, every output is 3 more than before. So every point of the curve goes up by 3. A slide like this is called a translation.

Adding 3 outside the function moves the whole curve up 3. V goes to W.

Worked example

Find the lowest point of y = f(x) + 3.

The lowest point of y = f(x) is V at (1, −1), because f(1) = −1.
For y = f(x) + 3, at x = 1 the output is f(1) + 3 = −1 + 3.
So −1 + 3 = 2, and the lowest point moves up to (1, 2).

Answer: The lowest point is (1, 2).

Now add a number inside the brackets. For y = f(x + 3), the input is changed before f works on it. The result may be a surprise: the curve slides to the left.

Here is why. V is the lowest point because f works on the input 1. For y = f(x + 3) we need x + 3 = 1, which gives x = −2. So the lowest point now stands at x = −2. The graph has moved 3 to the left.

Adding 3 inside the brackets moves the curve 3 to the left. V goes to W.

Worked example

Find the lowest point of y = f(x + 3).

The lowest point of y = f(x) is where the input of f is 1.
For y = f(x + 3), we need x + 3 = 1. So x = 1 − 3 = −2.
At x = −2 the output is f(−2 + 3) = f(1) = −1.
So the lowest point moves to (−2, −1), which is 3 to the left of V.

Answer: The lowest point is (−2, −1).

The plus sign inside the brackets moves the graph left, and a minus sign moves it right. So y = f(x − 2) slides the graph 2 to the right.

Next, the flips. The rule y = −f(x) changes the sign of every output. A point (x, y) becomes (x, −y). So the curve is turned over, like a mirror image in the x axis. We call this a reflection. Points on the x axis stay where they are.

The curve y = −f(x) is y = f(x) turned over in the x axis. V goes to W.

The rule y = f(−x) changes the sign of every input. A point (x, y) becomes (−x, y). So the curve is turned over like a reflection in the y axis. Points on the y axis stay where they are.

The curve y = f(−x) is y = f(x) turned over in the y axis. V goes to W.

Worked example

The curve y = f(x) passes through the point (3, 3). Find the matching points on y = −f(x) and on y = f(−x).

First check that (3, 3) is on the curve. f(3) = 3² − 2 × 3 = 9 − 6 = 3.
For y = −f(x) keep x = 3 and change the sign of y. So −f(3) = −3.
The matching point on y = −f(x) is (3, −3).
For y = f(−x) change the sign of x. At x = −3, f(−x) = f(3) = 3. The matching point is (−3, 3).

Answer: The points are (3, −3) and (−3, 3).

See it three ways

With things

A path is drawn on a map. The whole map is slid 3 squares up. Every point of the path stays in its column, and is 3 squares higher.

As a picture

In words

Each output has 3 added to it. Each x stays the same. So every point moves straight up by 3.

In symbols

y = f(x) + 3, so (1, −1) moves to (1, 2).

Try it

Slide and flip a curve

You need: squared paper, thin paper that you can see through, a pencil.

Draw y = x² − 2x on squared paper, and mark its lowest point V at (1, −1). Trace the curve and the axes onto the thin paper.
Slide the thin paper 3 up, keeping the axes level, and mark where V lands. Put it back, then slide it 3 to the left and mark V again.
Put it back. Fold the thin paper along the x axis and mark V. Unfold it, then fold along the y axis and mark V.

What you will see: Slides keep the curve the same way up. Folding along the x axis turns it upside down. Folding along the y axis turns it left to right.

If you cannot try it, this is what happens.

A wrong idea many people have

y = f(x + 3) moves the graph 3 to the right.

Why it feels right: Adding 3 usually makes things larger or moves them to the right along a number line.

The test that breaks it: The lowest point of y = f(x) is at x = 1. For y = f(x + 3), the lowest point needs x + 3 = 1, so it is at x = −2, which is left of 1.

What is true: A number added inside the brackets moves the graph the opposite way. The graph of f(x + 3) moves left 3. The graph of f(x − 3) moves right 3.

A wrong idea many people have

y = −f(x) and y = f(−x) are the same.

Why it feels right: Both have a minus sign next to f, and both seem to turn the graph round.

The test that breaks it: For f(x) = x² − 2x, the point V is at (1, −1). On y = −f(x) it goes to (1, 1). On y = f(−x) it goes to (−1, −1). They are different points.

What is true: A minus outside, as in −f(x), turns the graph over in the x axis. A minus inside, as in f(−x), turns it over in the y axis.

Key words

translation: a slide of a graph without turning it, such as 3 up or 3 left.
reflection: a turn over of a graph, as in a mirror, such as in the x axis or the y axis.

Back to your guess

The answer is B: To the left, even though 3 is added to x.

The graph slides 3 to the left. The curve y = f(x + 3) gives at x = −2 the same output that y = f(x) gives at x = 1. So every point of the curve arrives 3 earlier on the x axis, which is to the left.

Keep it

Copy this into your exercise book.

f(x) + a: up a. f(x + a): left a. −f(x): over in the x axis. f(−x): over in the y axis. Follow one point to check.

Draw: A curve and its four moved copies, with one marked point on each.

M4.10.7 Chapter 7

Gradients and areas under graphs

A cyclist speeds up all the time. How can I find her speed at one moment from the graph?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Divide the distance by the time, from the start to that moment.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Draw a straight line that just touches the curve at that moment, and find its gradient.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Read the height of the curve at that moment.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Estimate the gradient of a curve by drawing a tangent
Estimate the area under a curve using trapeziums
Explain what gradient and area mean on a speed time graph

A cyclist rides so that her distance after t seconds is d = t² metres. The graph of distance against time is a curve. It gets steeper, because she is speeding up.

For a straight line, the gradient is the rise divided by the run. On a distance time graph, the gradient is the speed: metres divided by seconds. A curve is steeper in some places than in others, so its gradient changes from point to point.

To find the gradient of a curve at a point, draw the tangent there. A tangent is a straight line that just touches the curve at that point, and runs the same way as the curve. The gradient of the curve at the point is the gradient of its tangent.

The tangent at P touches the curve at P only. The dashed triangle gives its rise and its run.

Worked example

Estimate the speed at 3 s, using the tangent through A (2, 3) and B (4, 15).

Read two points on the tangent: A at (2, 3) and B at (4, 15).
The rise is 15 − 3 = 12 m, and the run is 4 − 2 = 2 s.
The gradient is 12 ÷ 2 = 6. So the speed at 3 s is about 6 m/s.

Answer: The speed is about 6 m/s.

On paper you draw the tangent with a ruler. Lay the ruler so that it touches the curve at your point. Turn it until the curve seems to leave the ruler at the same angle on both sides. Then draw the line, and pick two points far apart on it.

A chord is a straight line that joins two points of the curve. It is easy to draw, but it gives only an average gradient between its two ends. It is not the gradient at one point.

The chord PQ is steeper than the tangent at P. It gives the average speed from 3 s to 5 s, which is more than the speed at 3 s.

Now to areas. A tram goes at a steady 4 m/s for 5 s. The distance it travels is speed times time, which is 4 × 5 = 20 m. On a speed time graph this is the area of a rectangle.

See it three ways

With things

A tram goes at a steady 4 m/s for 5 s. How far does it go?

As a picture

In words

Distance is speed times time. On a speed time graph that is the width times the height of the rectangle under the line, which is its area.

In symbols

distance = 4 × 5 = 20 m.

This works for any speed time graph, even a curve. The area under the graph, between the line and the time axis, is the distance travelled. For a curve we cut the area into thin strips, and treat the top of each strip as a straight line.

Each strip is then a trapezium, a shape with two parallel sides. Its area is the width times the average of the two parallel sides. If the sides are a and b and the width is w, the area is w × (a + b) ÷ 2. If one side is 0, the strip is a triangle, and the rule still works.

The speed after each second. The six strips are 1 s wide. There are seven readings but only six strips.

Worked example

The speed of a runner is read each second. It is 0, 5, 8, 9, 8, 5 and 0 m/s. Estimate the distance, using trapeziums.

Seven readings give six strips, each 1 s wide.
The area of a strip is the width times the average of its two heights.
The first strip: 1 × (0 + 5) ÷ 2 = 2.5.
The others are 6.5, 8.5, 8.5, 6.5 and 2.5.
Add the six areas: 2.5 + 6.5 + 8.5 + 8.5 + 6.5 + 2.5 = 35.

Answer: The distance is about 35 m.

This is an estimate. The curve here is a hill, and the straight tops of the strips lie just below it. So the estimate is a little too small. More, thinner strips give a better estimate.

A speed time graph has two meanings. Its gradient is the change of speed divided by the time. This is the acceleration, in metres per second per second, written m/s². The area under it is the distance travelled.

A car speeds up for 5 s, goes at a steady speed, and then slows to a stop at 20 s.

Worked example

Use the graph of the car. Find its acceleration in the first 5 s, and the total distance it travels.

The gradient of a speed time graph is the acceleration. In the first 5 s the speed rises from 0 to 10 m/s.
So the acceleration is 10 ÷ 5 = 2 m/s².
The area under the graph is a trapezium. Its parallel sides are 20 s and 10 s long, and their average is (20 + 10) ÷ 2 = 15.
The height is 10 m/s, so the area is 15 × 10 = 150.
So the car travels 150 m.

Answer: The acceleration is 2 m/s², and the distance is 150 m.

Try it

Squeeze the chord into a tangent

You need: a calculator, or paper and a pencil.

On the curve d = t², take the point at t = 3, where d = 9.
Join it to the points at t = 4, 3.5, 3.1 and 3.01. For each, work out the rise divided by the run.
Look at the gradients. What number do they come close to?

What you will see: The gradients are 7, 6.5, 6.1 and 6.01. They get closer to 6, which is the gradient of the tangent at t = 3.

If you cannot try it, this is what happens.

A wrong idea many people have

A chord gives the gradient of the curve at a point.

Why it feels right: A chord is a straight line that touches the curve at the point, and it is easy to draw.

The test that breaks it: For d = t², the chord from t = 3 to t = 5 has gradient (25 − 9) ÷ 2 = 8. But the tangent at t = 3 has gradient 6.

What is true: A chord joins two points and gives an average gradient between them. Only the tangent, which touches the curve at one point, gives the gradient at that point.

A wrong idea many people have

Seven readings make seven trapeziums.

Why it feels right: There are seven numbers in the list, so it seems that there are seven shapes.

The test that breaks it: The readings at 0, 1, 2, 3, 4, 5 and 6 s have only six gaps between them. The whole graph is 6 s wide. Seven strips would each be 6 ÷ 7 s wide, not 1 s.

What is true: Count the gaps, not the readings. With n readings there are n − 1 strips. The width of a strip is the gap between two readings.

Key words

tangent: a straight line that touches a curve at one point and runs the same way as the curve there.
chord: a straight line that joins two points of a curve.
trapezium: a shape with four sides, two of them parallel. Its area is the width times the average of the parallel sides.

Back to your guess

The answer is B: Draw a straight line that just touches the curve at that moment, and find its gradient.

On a distance time graph the gradient is the speed. A curve has a different gradient at every point. To find it at one moment, draw the tangent there. This is the straight line that just touches the curve. Then work out its gradient.

Keep it

Copy this into your exercise book.

Gradient of a curve at a point: the gradient of the tangent. Area under a speed time graph: distance. Trapezium: width times the average of the two sides.

Draw: A curve with a tangent at one point, and a hill shaped curve cut into strips.

M4.10.8 Chapter 8

Solving with graphs and trials

Two graphs cross at a point. What does that point have to do with solving an equation?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Nothing. The crossing point only shows where the graphs are close.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The y value of the point is the solution of the equation.
You picked B. Read on. At the end, the chapter comes back to your guess.
C The x value of the point is a solution of the equation.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Solve an equation by finding where two graphs meet
Find a root between two numbers by trying values

Some equations are hard to solve with algebra, but a graph can help. Take x² = x + 2. Draw two graphs, y = x² and y = x + 2. At a point where they cross, the two y values are equal. So x² = x + 2 there.

In general, to solve f(x) = g(x), draw y = f(x) and y = g(x) on one grid. The x values of the crossing points are the solutions. Each crossing point is called an intersection.

The graphs of y = x² and y = x + 2 cross at A and B.

Worked example

Use the graphs of y = x² and y = x + 2 to solve x² = x + 2.

Draw y = x² and y = x + 2 on one grid.
The graphs cross at A (−1, 1) and B (2, 4).
The x values of the crossing points are the solutions: x = −1 and x = 2.
Check x = 2: x² = 4, and x + 2 = 2 + 2 = 4. The two sides agree.

Answer: x = −1 or x = 2.

The same idea works for any pair of graphs. To solve 2ˣ = 10, draw y = 2ˣ and the flat line y = 10, and read the x value where they meet. A graph gives an estimate, because you can read it only to the accuracy of the drawing.

To get closer, use trials. A root of f is a value of x with f(x) = 0. It is where the graph of y = f(x) crosses the x axis. Take f(x) = x³ + x − 3. Its graph crosses the x axis once, near x = 1.2.

The graph of y = x³ + x − 3 crosses the x axis at R, so R is a root of f.

Work out f(x) for some values of x. Suppose f is negative at one value and positive at another. If the graph has no break between them, it must cross the x axis between those values. There is a root there.

Worked example

f(x) = x³ + x − 3. Show that f has a root between 1 and 2.

Work out f at x = 1 and x = 2.
f(1) = 1 + 1 − 3 = −1.
f(2) = 8 + 2 − 3 = 7.
f(1) is negative and f(2) is positive. The graph has no break, so it crosses the x axis between 1 and 2.

Answer: There is a root between 1 and 2.

Now close in on the root. Try values between 1 and 2, in steps of 0.1. Look for the place where f changes sign.

Worked example

f(x) = x³ + x − 3 has a root between 1 and 2. Find it to one decimal place.

Try values in steps of 0.1. We look for a change of sign.
f(1.2) = 1.728 + 1.2 − 3 = −0.072, which is negative.
f(1.3) = 2.197 + 1.3 − 3 = 0.497, which is positive.
The root is between 1.2 and 1.3. To decide how to round, try the middle value: f(1.25) = 1.953125 + 1.25 − 3 = 0.203125.
f(1.25) is positive, so the root is between 1.2 and 1.25. It is below 1.25, so to one decimal place the root is 1.2.

Answer: x = 1.2 (to one decimal place).

The middle value matters. Between 1.2 and 1.3, the answer to one decimal place could be 1.2 or 1.3. Trying the middle value, 1.25, shows on which side of it the root lies.

See it three ways

With things

Amina has a plant 10 cm high that grows 2 cm each day. Ravi has a plant 4 cm high that grows 4 cm each day. When are the two plants the same height?

As a picture

In words

The lines cross where the two heights are equal. That is after 3 days, when both plants are 16 cm high.

In symbols

10 + 2d = 4 + 4d, so 6 = 2d and d = 3. The height is 10 + 2 × 3 = 16 cm.

Try it

Close in on a square root

You need: a calculator, or paper and a pencil.

We want x with x² = 7. Try x = 2 and x = 3 first.
Then try 2.5, 2.6, 2.7 and 2.65. For each, write whether x² is lower or higher than 7.
Find the two values closest together that have 7 between their squares.

What you will see: 2.6² = 6.76 is low and 2.65² = 7.0225 is high, so the solution is between 2.6 and 2.65. To one decimal place, it is 2.6.

If you cannot try it, this is what happens.

A wrong idea many people have

The solutions of x² = x + 2 are the y values of the crossing points.

Why it feels right: The crossing points are where the graphs meet, and the heights seem to be the answers.

The test that breaks it: The crossing points are (−1, 1) and (2, 4). Put x = 4 into the equation: 4² = 16, but 4 + 2 = 6. They are not equal. Put x = 2: 2² = 4 and 2 + 2 = 4.

What is true: The equation asks for the x values where the two rules agree. Read the x value of each crossing point, going down to the x axis.

A wrong idea many people have

If f changes sign between two numbers, there must be a root between them.

Why it feels right: The graph goes from below the axis to above it, so it seems it must cross the axis.

The test that breaks it: Take f(x) = 1/x. Then f(−1) = −1 and f(1) = 1, a change of sign. But 1/x is never zero. The graph has a break at x = 0 and jumps across the axis without touching it.

What is true: A sign change shows a root only when the graph has no break between the two numbers. Graphs made only of numbers and positive whole number powers of x, such as x³ + x − 3, have no breaks.

Key words

intersection: a point where two graphs cross.
root: a value of x for which f(x) = 0. It is where the graph of y = f(x) meets the x axis.

Back to your guess

The answer is C: The x value of the point is a solution of the equation.

At a crossing point, the two graphs have the same x and the same y. So at that x, the two rules give the same output. That means the x value solves the equation that says the two rules are equal.

Keep it

Copy this into your exercise book.

Solve f(x) = g(x): read the x values where the graphs cross. Roots: where f changes sign, if there is no break. Narrow the gap by trying values.

Draw: A line and a curve crossing at two points, with their x values marked on the x axis.

From a rule to a graph

A table for f(x) = 2x − 1

Find f⁻¹ by undoing the steps

Does the order matter?

The graph of y = 1/x

The circle x² + y² = 25

Get close to zero

The graph of y = 2ˣ

Fold a sheet of paper

Measure the sine on a circle

Slide and flip a curve

Squeeze the chord into a tangent

Close in on a square root

Key words, A to Z