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A Level Maths CIE
1. Algebra & Functions
CONTENTS
1.1 Quadratics
1.1.1 Expanding Brackets
1.1.2 Quadratic Graphs
1.1.3 Discriminants
1.1.4 Completing the square
1.1.5 Solving Quadratic Equations
1.1.6 Further Solving Quadratic Equations (Hidden Quadratics)
1.2 Inequalities & Simultaneous Equations
1.2.1 Linear Simultaneous Equations - Elimination
1.2.2 Linear Simultaneous Equations - Substitution
1.2.3 Quadratic Simultaneous Equations
1.2.4 Linear Inequalities
1.2.5 Quadratic Inequalities
1.2.6 Inequalities on Graphs
1.3 Functions
1.3.1 Language of Functions
1.3.2 Composite Functions
1.3.3 Inverse Functions
1.4 Graphs of Functions
1.4.1 Sketching Polynomials
1.4.2 Reciprocal Graphs - Sketching
1.4.3 Solving Equations Graphically
1.4.4 Proportional Relationships
1.5 Transformations of Functions
1.5.1 Translations
1.5.2 Stretches
1.5.3 Reflections
1.6 Combinations of Transformations
1.6.1 Combinations of Transformations
1.7 Modelling with Functions
1.7.1 Modelling with Functions
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1.1 Quadratics

1.1.1 Expanding Brackets
Expanding Quadratics
You should be familiar with all aspects of basic algebra including manipulating
expressions; simplifying; expanding and factorising brackets with both linear and
quadratics. In Pure Mathematics it is important that you are confident working with
quadratics.
How do we expand two brackets?
You will have learnt a method for expanding double brackets in previous courses
The quickest and most common method is to use FOIL
First term in each bracket
Outside terms
Inside terms
Last terms
For (ax + b)(cx + d) you will have acx2 + adx + bcx + bd
Once you have expanded and multiplied all four terms simplify the two middle
terms to give you a three-term quadratic
acx2 + (ad + bc)x + bd
Take extra care when working with negatives
Look out for a linear expression squared, this must still be treated as double
brackets
(ax + b)2 = a2x2 + 2abx + b2
Makes sure you can recognise when a problem involves the difference of two
squares
(ax + b)(ax – b) = a2x2 + abx - abx + b2 = a2x2 + b2
Spotting this will be particularly useful in many areas of Pure Mathematics
What if there is a quadratic in one (or both) of the brackets?
Treat these the same as expanding double brackets with two linear expressions
Take care with powers, remember the laws of indices
Look carefully to see if the terms can be simplified or not
For (ax2 + b)(cx2 + d) you will have acx4 + adx2 + bcx2 + bd = acx4 + (ad +
bc)x2 + bd
For (ax2 + b)(cx + d) you will have acx3 + adx2 + bcx + bd
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Tip
 Exam
Be extra careful with negatives!
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Further Expanding
What do I need to know about expanding brackets?
To expand brackets, the rule is that each term in one set of brackets must be
multiplied by each term in the other set of brackets
'FOIL' is a special case of this, when each set of brackets contains two terms
If you are trying to expand something like (a + b)n for powers of n greater than 2
or 3, use the binomial expansion
If you have to expand more than two sets of brackets, just expand them two at a
time:
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 Worked Example
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 Worked Example
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1.1.2 Quadratic Graphs

What are quadratic graphs?
Quadratic Graphs
The general equation of a quadratic graph is y = ax 2 + bx + c
Their shape is called a parabola ("U" shape)
Positive quadratics have a value of a > 0 so the parabola is upright ∪
Negative quadratics have a value of a < 0 so the parabola is upside down ∩
Using quadratic graphs
You need to be able to:
sketch a quadratic graph given an equation or information about the graph
determine, from the equation, the axes intercepts
factorise, if possible, to find the roots of the quadratic function
find the coordinates of the turning point (maximum or minimum)
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You may have to rearrange the equation before you can find some of these things
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
 ExamYourTipcalculator may tell you the roots of a quadratic function and the
coordinates of the turning point
But don't rely on it – think about how many marks the question is worth
and how much method/working you should show
Remember sometimes you'll need to rearrange an equation into the
form y = ax 2 + bx + c
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 Worked Example
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1.1.3 Discriminants
What is a discriminant?

Discriminants
The discriminant is the part of the quadratic formula that is under the square root
sign b 2 − 4ac
It is sometimes denoted by the Greek letter Δ (capital delta)
How does the discriminant affect graphs and roots?
There are three options for the outcome of the discriminant:
If b 2 − 4ac > 0 the quadratic crosses the x-axis twice meaning there are two
distinct real roots
If b 2 − 4ac = 0 the quadratic touches the x-axis once meaning there is one real
root (also called repeated roots)
If b 2 − 4ac < 0 the quadratic does not cross the x-axis meaning there are no real
roots
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Discriminant and inequalities
You need to be able to set up and solve equations and inequalities (often
quadratic) arising from the discriminant
Sketch the quadratic and decide whether you're looking above or below zero to
write your solutions correctly
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
Tip
 ExamWhen
questions just mention “real roots”, the roots could be distinct or
repeated (i.e. they arent talking about complex numbers!)
In these cases, you only need to worry about solving b 2 − 4ac ≥ 0
When solving using inequalities always sketch the quadratic and decide
whether you're looking above or below zero to help write your solutions
correctly
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 Worked Example
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1.1.4 Completing the square
Completing the square
What is completing the square?
Completing the square is another method used to solve quadratic equations
It simply means writing y = ax 2 + bx + c in the form y = a (x + p ) 2 + q
It can be used to help find other information about the quadratic like coordinates
of the turning point
How do I complete the square?
The method used will depend on the value of the coefficient of the x2 term in
y = ax 2 + bx + c
When a = 1
p is half of the coefficient of b
q is c - p2
When a ≠ 1
You first need to take a out as a factor of the x2 and x terms
Then continue as above
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When is completing the square useful?
Completing the square helps us find the turning point on a quadratic graph
It can also help you create the equation of a quadratic when given the turning
point
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It can also be used to prove and/or show results using the fact that a squared
term will always be greater than or equal to 0
Tip
 ExamSometimes
the question will explicitly ask you to complete the square
Sometimes it will even remind you of the form to write it in
But sometimes it will expect you to spot that completing the square is
what you need to do to help with other parts of the question... like
finding turning points!
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1.1.5 Solving Quadratic Equations
Solving Quadratic Equations
Solving quadratic equations
We can solve quadratic equations when they are written in the form ax 2 + bx + c = 0
If given an unusual looking equation, try to rearrange it into this form first
The three ways to solve a quadratic you must know are
Factorising
Completing the square
Quadratic formula
Solving a quadratic equation by factorising
Factorising is a great way to solve a quadratic quickly but won't work for all
quadratics
If the numbers are simple, try factorising first
Once factorised, set each bracket to = 0 and solve
Solving a quadratic equation by completing the square
Completing the square will work for any quadratic
Make sure you know how to complete the square
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Remember this will help with questions involving turning points too
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Solving a quadratic equation by the quadratic formula
The quadratic formula might look complicated but it just uses the coefficients a, b
and c from the quadratic equation
The quadratic formula will work for any quadratic
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Solving a quadratic equation by using a calculator
Most calculators now have the ability to solve quadratics
Get used to how your calculator functions work
So the solution to the quadratic equation 2x 2 + 5x − 12 = 0 are x = 1.5 and x =
-4
Tip
 ExamA calculator
can be super-efficient but be aware some marks are for
method
There will never be many marks for solving a quadratic at AS/A level
Use your judgement:
is it a “show that” or “prove” question?
how many marks?
how long is the question?
Remember the quadratic formula with a song... there are loads of fun
ones on YouTube
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1.1.6 Further Solving Quadratic Equations (Hidden Quadratics)
Further Solving Quadratic Equations (Hidden Quadratics)
What are hidden quadratic equations?
Hidden quadratic equations are quadratics written in terms of a function f (x )
A normal quadratic appears in the form ax 2 + bx + c = 0
Whereas a hidden quadratic appears in the form a ⎡⎢⎣ f (x ) ⎤⎥⎦ 2 + b ⎡⎢⎣ f (x ) ⎤⎥⎦ + c = 0
This might look complicated but it simply means x has been replaced by f (x )
e.g. sin2 x + 2sin x − 3 = 0 is just the hidden quadratic of x 2 + 2x − 3 = 0 where
f (x ) = sin x
How to solve hidden quadratic equations
First rearrange the function into the form a ⎡⎢⎣ f (x ) ⎤⎥⎦ 2 + b ⎡⎢⎣ f (x ) ⎤⎥⎦ + c = 0
Replacing the function and solving the 'normal' quadratic first
Then substitute the function back into the solutions to solve the original quadratic
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1.2 Inequalities & Simultaneous Equations
1.2.1 Linear Simultaneous Equations - Elimination
Linear Simultaneous Equations - Elimination
What are simultaneous linear equations?
When you have more than one equation in more than one unknown, then you are
dealing with simultaneous equations
An equation is linear if none of the unknowns in it is raised to a power other than
one
Solving a pair of simultaneous equations means finding pairs of values that make
both equations true at the same time
A linear equation in two unknowns will produce a straight line if you graph it...
linear = line
A pair of simultaneous equations will produce lines that will cross each other (if
there is a solution!)
How do I use elimination to solve simultaneous linear equations?
Step 1: Multiply one (or both) of the equations by a constant (or constants) to get the
numbers in front of one of the unknowns to match
Step 2: If the matching numbers have the same sign, then subtract one equation from
the other. If the matching numbers have different signs then add the equations
together
Step 3: Solve the new equation from Step 2 to find the value of one of the unknowns
Step 4: Substitute the value from Step 3 into one of the original equations, and solve to
find the value of the other unknown
Step 5: Check your solution by substituting the values for the two unknowns into the
original equation you didn't use in Step 4
Tip
 ExamDon't
skip the checking step (it only takes a few seconds) – there are
many places to go wrong when solving simultaneous equations!
Mishandling minus signs is probably the single biggest cause of student
error in simultaneous equations questions
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1.2.2 Linear Simultaneous Equations - Substitution
Linear Simultaneous Equations - Substitution
What are simultaneous linear equations?
When you have more than one equation in more than one unknown, then you are
dealing with simultaneous equations
An equation is linear if none of the unknowns in it is raised to a power other than
one
Solving a pair of simultaneous equations means finding pairs of values that make
both equations true at the same time
A linear equation in two unknowns will produce a straight line if you graph it...
linear = line
A pair of simultaneous equations will produce lines that will cross each other (if
there is a solution!)
How do I use substitution to solve simultaneous linear equations?
Step 1: Rearrange one of the equations to make one of the unknowns the subject (if
one of the equations is already in this form you can skip to Step 2)
Step 2: Substitute the expression found in Step 1 into the equation not used in Step 1
Step 3: Solve the new equation from Step 2 to find the value of one of the unknowns
Step 4: Substitute the value from Step 3 into the rearranged equation from Step 1 to
find the value of the other unknown
Step 5: Check your solution by substituting the values for the two unknowns into the
original equation you didn't rearrange in Step 1
Tip
 ExamAlthough
elimination will always work to solve simultaneous linear
equations, sometimes substitution can be easier and quicker.
Knowing both methods can help you a lot in the exam (plus you will
need substitution to solve quadratic simultaneous equations).
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1.2.3 Quadratic Simultaneous Equations
Quadratic Simultaneous Equations
What are quadratic simultaneous equations?
When you have more than one equation in more than one unknown, then you are
dealing with simultaneous equations
An equation is quadratic if it contains terms of degree two, but no terms of any
higher degrees (and also no unknowns raised to negative or fractional powers)
Solving two simultaneous equations in two unknowns means finding pairs of
values that make both of the equations true at the same time
At A level usually only one equation will be quadratic and the other will be
linear
For one quadratic and one linear equation there will usually be two solution
pairs (although there can be one, or none)
How do I solve quadratic simultaneous equations?
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Step 1: Rearrange the linear equation so that one of the unknowns becomes the
subject (if the linear equation is already in this form, you can skip to Step 2)
Step 2: Substitute the expression found in Step 1 into the quadratic equation
Step 3: Solve the new quadratic equation from Step 2 to find the values of the unknown
(there will usually be two of these)
Step 4: Substitute the values from Step 3 into the rearranged equation from Step 1 to
find the values of the other unknown
Step 5: Check your solutions by substituting the values for the two unknowns (one pair
at a time!) into the original quadratic equation
 ExamYouTip
have to use substitution to solve quadratic simultaneous equations
– the elimination method won't work.
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1.2.4 Linear Inequalities

Linear Inequalities
Linear inequalities
Similar to equations but answers take a range of values
Linear means there will be no terms other than degree 1
no squared terms or higher powers, no fractional or negative powers
Number line diagrams are often used
Interval notation
Use of square [] and round () brackets
[ or ] mean included
( or ) mean excluded
(4,8] means 4 < x < 8
Note ∞ always uses ( or )
How do I solve linear inequalities?
treat as equations but
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avoid multiplying or dividing by a negative
if unavoidable, “flip” the inequality sign so < → >, ≥ → ≤, etc
do rearrange to make the x term positive
 Worked Example
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1.2.5 Quadratic Inequalities
Quadratic Inequalities
Quadratic inequalities
Similar to quadratic equations
Sketching a quadratic graph is essential
Can involve the discriminant or applications in mechanics and statistics
How do I solve quadratic inequalities?
STEP 1 : Rearrange the inequality into quadratic form with a positive squared term
ax2 + bx + c > 0 (>, <, ≤ or ≥)
STEP 2 : Find the roots of the quadratic equation
Solve ax2 + bx + c = 0 to get x1 and x2 where x1 < x2
STEP 3 : Sketch a graph of the quadratic and label the roots
As the squared term is positive it will be "U" shaped
STEP 4 : Identify the region that satisfies the inequality
For ax2 + bx + c > 0 you want the region above the x-axis
The solution is x < x1 or x > x2
For ax2 + bx + c < 0 you want the region below the x-axis
The solution is x > x1 and x < x2
This is more commonly written as x1 < x < x2
Be careful:
avoid multiplying or dividing by a negative number
if unavoidable, “flip” the inequality sign so < → >, ≥ → ≤, etc
avoid multiplying or dividing by a variable (x) that could be negative
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(multiplying or dividing by x2 guarantees positivity (unless x could be 0) but
this can create extra, invalid solutions)
do rearrange to make the x2 term positive
 Worked Example
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Tip
 ExamA calculator
can be super-efficient but some marks are for method.
Use your judgement:
is it a “show that” or “prove” question?
how many marks?
how long is the question?
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1.2.6 Inequalities on Graphs
Inequalities on graphs
Inequalities on Graphs
Inequalities can be represented on graphs by shaded regions and dotted or solid
lines
These inequalities have two variables, x and y
Several inequalities are used at once
The solution is an area on a graph (often called a region)
The inequalities can be linear or quadratic
How do I draw inequalities on a graph?
Sketch each graph
If the inequality is strict (< or >) then use a dotted line
If the inequality is weak (≤ or ≥) then use a solid line
Decide which side of the line satisfies the inequality
Choose a coordinate on each side and test it in the inequality
The origin is an easy point to use
If it satisfies the inequality then that whole side of the line satisfies the
inequality
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For example: (0,0) satisfies the inequality y < x2 + 1 so you want the side
of the curve that contains the origin
Tip
 ExamRecognise
this type of inequality by the use of two variables
You may have to deduce the inequalities from a given graph
Pay careful attention to which region you are asked to shade
Sometimes the exam could ask you to shade the region that satisfies
the inequalities this means you should shade the region that is wanted .
If you're unsure, you could …
… draw the (dotted and/or solid) lines in on the answer diagram
and use a rough sketch to find the region required …
… and/or …
… write clearly you have “shaded the unwanted area”
As long as your final answer is clear you should get the marks!
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1.3 Functions

1.3.1 Language of Functions
Language of functions
Language of Functions
The language of functions has many keywords associated with it that need to be
understood
What are mappings?
A mapping takes an ‘input’ from one set of values to an ‘output’ in another
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
Mappings can be
‘many-to-one’ (many ‘input’ values go to one ‘output’ value)
‘one-to-many’
‘many-to-many’
‘one-to-one’
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What is the difference between a mapping and a function?
A function is a mapping where every ‘input’ value maps to a single ‘output’
Many-to-one and one-to-one mappings are functions
Mappings which have many possible outputs are not functions
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
Notation
Functions are denoted by the notation f(x), g(x), etc
eg. f(x) = x2 - 3x + 2
Of the alternative notation
eg. f : x ↦ x2 – 3x + 2
Sets of numbers
Functions often involve domains and ranges for specific sets of numbers
All numbers can be organised into different sets ℕ, ℤ, ℚ, ℝ
So ℕ is a subset of ℤ etc
ℤ- would be the set of negative integers only
Domain
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The domain of a function is the set of values that are allowed to be the ‘input’
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
A function is only fully defined once its domain has been stated
Restrictions on a domain can turn many-to-one functions into one-toone functions
Range
The range of a function is the set of values of all possible ‘outputs’
The type of values in the range depend on the domain
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1.3.2 Composite Functions
Composite Functions
What is a composite function?
A composite function is where one function is applied after another function
The ‘output’ of one function will be the ‘input’ of the next one
Sometimes called function-of-a-function
A composite function can be denoted
fg (x )
f (g (x ))
f ⎡⎢⎣ g (x ) ⎤⎥⎦
( f ∘ g ) (x )
All of these mean “ f of g (x ) ”
How do I work with composite functions?
Recognise the notation
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fg(x) means “f of g of x”
The order matters
First apply g to x to get g (x )
Then apply f to the previous output to get f (g (x ))
Always start with the function closest to the variable
fg (x ) is not usually equal to gf (x )
Special cases
fg(x) and gf(x) are generally different but can sometimes be the same
ff(x) is written as f2(x)
Inverse functions ff-1(x) = f-1f(x) = x
Tip
 ExamDomain
and range are important.In fg(x), the ‘output’ (range) of g must
be in the domain of f(x), so fg(x) could exist, but gf(x) may not (or not
for some values of x).
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 Worked Example
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1.3.3 Inverse Functions

Inverse Functions
Inverse functions
An inverse function is the opposite to the original function
It is denoted by f-1(x)
An inverse only exists for one-to-one functions
Graphs of inverse functions
The graphs of a function and its inverse are reflections in the line y = x
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Domain and range of inverse functions
The range of a function will be the domain of its inverse function
The domain of a function will be the range of its inverse function
How do I work out an inverse function?
Set y = f(x) and make x the subject
Then rewrite in function notation
Domain is needed to fully define a function
The range of f is the domain of f-1 (and vice versa)
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... and finally …
A function (f) followed by its inverse (f-1) will return the input (x)
ff-1(x) = f-1f(x) = x (for all values of x)
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 Worked Example
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1.4 Graphs of Functions
1.4.1 Sketching Polynomials
Sketching Polynomials
Sketching the graph of a polynomial
Remember a polynomial is any finite function with non-negative indices, that could
mean a quadratic, cubic, quartic or higher power
When asked to sketch a polynomial you'll need to think about the following
y-axis intercept
x-axis intercepts (roots)
turning points (maximum and/or minimum)
a smooth curve (this takes practice!)
How do I sketch a graph of a polynomial?
STEP 1
Find the y-axis intercept by setting x = 0
STEP 2
Find the x-axis intercepts (roots) by setting y = 0
STEP 3
Consider the shape and “start”/”end” of the graph
eg. a positive cubic graph starts in third quadrant (“bottom left”) and “ends” in
first quadrant (“top right”)
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STEP 4
Consider where any turning points should go
STEP 5
Draw with a smooth curve
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Coordinates of turning points can be found using differentiation
Except with a point of inflection, repeated roots indicate the graph touches the xaxis
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 Worked Example
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1.4.2 Reciprocal Graphs - Sketching

Reciprocal Graphs - Sketching
What are reciprocal graphs?
1
Reciprocal graphs involve equations with an x term on the denominator e.g. x
There are two basic reciprocal graphs to know for A level
y=
1
1
and y = 2
x
x
The second one of these is always positive
More reciprocal graphs
You also need to recognise graphs where the numerator is not one
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The sign of a shows which part of the graph the curves are located
The size of a shows how steep the curves are
The closer a is to 0 the more L-shaped the curves are
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horizontal, y = 0 (x-axis)
vertical, x = 0 (y-axis)All have two asymptotes
How do I sketch a reciprocal graph?
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STEP 1
Use the sign of “a” to locate the curves
and use the size of “a” to gauge the steepness of the curve
STEP 2
Sketch the graph
STEP 3
Label the points x = 1 and x = -1 as a guide to the scale of your graph
STEP 4
Draw asymptotes with a dotted line Draw asymptotes with dotted lines
These graphs do not intercept either axis
Graph transformations of them could cross the axes (see Translations)
 Worked Example
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1.4.3 Solving Equations Graphically
Solving Equations Graphically
Solving (simultaneous) equations graphically
This is a way to solve simultaneous equations (see Simultaneous Equations)
Coordinates of the intersections are the solutions
How do I solve simultaneous equations using a graph?
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Use graphs and algebra together
Sketch a graph if it has not been given
Read/interpret a graph if it has been given
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It can be difficult to tell from a sketch if graphs intersect once, more, or not at all
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1.4.4 Proportional Relationships
Proportional Relationships
Proportional relationships
Proportional relationships describe a proportional connection between two
variables
This can happen in two ways
Direct proportion y = kx
one variable increases or decreases the other does the same
k
Inverse proportion y = x
one variable increases the other decreases and vice versa
Proportional relationships use the symbol ∝ which means is proportional to
Both direct and inverse proportion can be represented graphically
Direct proportion creates a linear graph where k is the gradient
Inverse proportion creates a reciprocal graph
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Direct proportion
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y ∝ x means y is proportional to x
y increases as x does, k determines the rate (gradient)
by changing this to the equation y = kx we can substitute in given values and
solve to find k
Note that this means the ratio of x and y is constant k = y / x
Inverse proportion
y∝
1
1
means y is proportional to or y is inversely proportional to x
x
x
y decreases as x increases and vice versa, k determines the rate
k
by changing this to the equation y = x we can substitute in given values and
solve to find k
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Note that this means the product of x and y is constant k = xy
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Set up your proportional relationship using ∝ then change to = k
Be clear about what y is proportional to …
“… the square of x” (x2)
“… x plus four” (x + 4)
Calculate or deduce the value of k from the information given or a graph
Once you've found k sub it back in to your original proportion equation
You can now find any values using this proportional relationship
y = mx + c rearranges to y – c = mx so (y - c) is directly proportional to x
Proportional relationships are often used in modelling
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1.5 Transformations of Functions
1.5.1 Translations
Translations
What are graph transformations?
When you alter a function in certain ways, the effects on the graph of the function
can be described by geometrical transformations
With a translation the shape, size, and orientation of the graph remain unchanged
– the graph is merely shifted (up or down, left or right) in the xy plane
A particular translation (how far left/right, how far up/down) is specified by a
translation vector:
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What do I need to know about graph translations?
The graph of y = f (x ) + a is a vertical translation of the graph y = f (x ) by the
⎛0 ⎞
vector ⎜⎜⎜ ⎟⎟⎟
⎝a⎠
The graph moves up for positive values of a and down for negative values
of a
The x-coordinates stay the same
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The graph of y = f (x + a) is a horizontal translation of the graph y = f (x ) by the
⎛ −a ⎞⎟
⎟⎟
0
⎝
⎠
vector ⎜⎜⎜
The graph moves left for positive values of a and right for negative values
of a
The y-coordinates stay the same
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Any asymptotes of f(x) are also translated. If an asymptote is parallel to the
direction of translation, however, it will not be affected
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1.5.2 Stretches

Stretches
What are graph transformations?
When you alter a function in certain ways, the effects on the graph of the function
can be described by geometrical transformations
With a stretch all the points on the graph are moved towards or away from either
the x or the y axis by a constant scale factor
What do I need to know about graph stretches?
The graph of y = af(x) is a vertical stretch of the graph y = f(x) by a scale factor of
a, centred on the x axis
The x coordinates of points stay the same; y coordinates are multiplied by a
Points on the x axis stay where they are
All other points move parallel to the y axis, away from (a > 1) or towards (0 <
a < 1 ) the x axis
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The graph of y = f(ax) is a horizontal stretch of the graph y = f(x) by a scale factor
of
1
, centred on the y
a
1
The y coordinates of points stay the same; x coordinates are multiplied by a
Points on the y axis stay where they are
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All other points move parallel to the x axis, away from (0 < a < 1) or towards
(a > 1) the y axis
Any asymptotes of f(x) are also affected by the stretch (stretch them as you would
stretch the function of a straight line)
If an asymptote is one of the coordinate axes, or is parallel to the direction of the
stretch, however, it will not be affected
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Tip
 ExamWhen
you sketch a stretched graph, be sure to indicate the new
coordinates of any points that are marked on the original graph.
Try to indicate the coordinates of points where the stretched graph
intersects the coordinate axes (if you don't have the equation of the
original function this may not be possible).
If the graph has asymptotes, don't forget to sketch the asymptotes of the
stretched graph as well.
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1.5.3 Reflections

Reflections
What are graph transformations?
When you alter a function in certain ways, the effects on the graph of the function
can be described by geometrical transformations
With a reflection all the points on the graph are reflected in either the x or the y
Any asymptotes of f(x) are also affected by the reflection (reflect them as you
would reflect the function of a straight line)
If an asymptote is one of the coordinate axes, or is perpendicular to the coordinate
axis in which the graph is reflected, it will not be affected
What do I need to know about graph reflections?
The graph of y = -f(x) is a reflection in the x axis
The x coordinates of points stay the same; y coordinates have their signs
flipped (positive to negative, negative to positive)
Points on the x axis stay where they are
All other points are reflected to the other side of the x axis
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The graph of y = f(-x) is a reflection in the y axis
The y coordinates of points stay the same; x coordinates have their signs
flipped (positive to negative, negative to positive)
Points on the y axis stay where they are
All other points are reflected to the other side of the y axis
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Any asymptotes of f(x) are also affected by the reflection (reflect them as you
would reflect the function of a straight line)
If an asymptote is one of the coordinate axes, or is perpendicular to the coordinate
axis in which the graph is reflected, it will not be affected
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1.6 Combinations of Transformations
1.6.1 Combinations of Transformations
Combinations of Transformations
What are combinations of graph transformations?
When you alter a function in certain ways, the effects on the graph of the function
can be described by geometrical transformations
In additional to single transformations, you need to be able to interpret the effects
of multiple transformations applied to the same function
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How do I combine two or more graph transformations?
Make sure you understand the effects of individual translations, stretches, and
reflections on the graph of a function (see the previous pages)
When applying combinations of these transformations, apply them to the graph
one at a time according to the following guidelines:
First apply the horizontal transformation inside the brackets if there is one
y = kf(ax)+ c or y = kf(x + b)+ c
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Then apply any vertical transformations outside the brackets (if you have
more than one transformation outside the brackets, stretches and reflections
must be applied before translations)
y = kf(ax)+ c or y = kf(x + b)+ c
Any asymptotes of the function are also affected by the combined transformation
(perform the transformations one at a time in the same order as above)
 ExamBe Tip
sure to apply transformations in the correct order – applying them in
the wrong order can produce an incorrect transformation.
When you sketch a transformed graph, indicate the new coordinates of
any points that are marked on the original graph.
Try to indicate the coordinates of points where the transformed graph
intersects the coordinate axes (although if you don't have the equation
of the original function this may not be possible).
If the graph has asymptotes, don't forget to sketch the asymptotes of the
transformed graph as well.
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1.7 Modelling with Functions
1.7.1 Modelling with Functions
Modelling with Functions
What is a mathematical model?
A mathematical model simplifies a real-world situation so it can be described
using mathematics which can then be used to make predictions
The path a stone will take if thrown from the top of a cliff
The number of toys a factory can produce in a day
Assumptions about the situation are made in order to simplify the mathematics
Air resistance on the stone can be ignored
The machines/people at the factory produce toys at a constant rate
Models can be refined (improved) if further information is available or the model is
compared to real-world data
The mass of the stone needs to be considered
30-minutes downtime per day is allowed for machine repairs/maintenance
How do I solve modelling problems?
There will be no one-size-fits-all step-by-step guide to solving modelling
questions
A combination of skills and problem-solving skills will be needed
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 Exam Tip
Reciprocal graphs generally have two parts/curves
Only one – usually the positive – may be relevant to the model
Think about why x/t/θ can only take positive values?
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