1. Proof and Mathematical Communication (Unit 1):
The ⇒⇒ symbol means that the statement follows from a previous statement
The ⇐⇐ symbol means that the previous statement follows from the subsequent statement
The ⇔⇔ symbol means that a subsequent statement is equivalent to the previous one
In a proof by deduction, you list a series of logical steps to show that your proposition is true.
In a proof by exhaustion, you check the statement is true in every possible case.
Disproof by counter example is when you show a statement is untrue by finding a single counter example –
one case where the statement isn’t true.
Proof by contradiction → Prove by contradiction that log2(3) is irrational.
To prove: there are no positive integers p and q such that p/q = log2(3). Proceed by contradiction.
Assume that log2(3) is rational, so it can be expressed in the form p/q where p and q are integers. If p/q =
log2(3), then p = log2(3q) and 2p = 3q.
But 2p is even for all positive integers p and 3q is odd for all positive integers q - therefore this cannot be true.
Our assumption was incorrect: no such p and q exist, so log2(3) is irrational.
For inequalities, the following interval notation may be used:
• x ∈ (a, b) means x is in the interval a < x < b
• x ∈ [a, b] means x is in the interval a ≤ x ≤ b
• x ∈ [a, b) means x is in the interval a ≤ x < b
• x ∈ (a, b] means x is in the interval a < x ≤ b
Where the ∈ symbol means “is in the set…” or “belongs to the
set…”
If there is no solution to an inequality, then it can be
represented by the empty set, written as x∈∅.
Any even number can be written as 2n, where n is an integer.
Any odd number can be written as 2n+1, where n is an integer.
3. Quadratic Functions:
The graph of y=ax2+bx+c crosses
• the y-axes at the point (0, c)
• the x-axis at the solutions to the equation ax2 + bx + c = 0.
These solutions are called the roots of the equation.
When sketching a quadratic function, completing the square gives you the turning point of the graph.
y=a(x+p)2+q gives the turning point at (−p, q). It also gives the line of symmetry at x= −p
When solving a quadratic inequality:
• Rewrite the inequality with the quadratic in the form ax2 + bx + c on the left of the inequality symbol
and zero on the right. (When rearranging remember that you cannot divide or multiply through by a
negative number without changing the direction of the inequality symbol.)
• Sketch a graph of the quadratic function on the left-hand side.
• Use the graph to find the solution set.
• For quadratics that are > 0 or ≥ 0, the solution set is given by the values of x that give the parts of the
graph that are above the x-axis.
• For quadratics that are < 0 or ≤ 0, the solution set is given by the values of x that give the parts of the
graph that are below the x-axis.
For a quadratic function y = ax2 +bx + c the expression b2−4ac is called the discriminant of the quadratic and
is often symbolised by the Greek letter Δ (uppercase delta).
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, • If Δ > 0 the function has two distinct real roots
• If Δ = 0 the function has two equal roots
• If Δ < 0 the function has no real roots
Look out for equations that can be written in the form a(something)2+b(something)+c=0 where the
‘something’ is a function that can be replaced by a single variable using a substitution to give a quadratic
equation.
(YEAR 2) 2. Functions:
A function is a mapping with only one output for any given input.
A function is one-to-one if each output corresponds to only one input, and many-to-one otherwise.
The set of values allowed as input values for a function is known as the function’s domain. (So basically,
where the x starts from).
The range of a function is the set of all possible output values (So basically where the y starts from).
∊ means “is an element of”
: means “such that”
Z Denotes the set of integers
Q Denotes the set of rational numbers
R Denotes the set of real numbers
R+ Denotes the set of positive real numbers (similarly for rational numbers, integers etc.)
R− Denotes the set of negative real numbers (similarly for rational numbers, integers etc.)
Applying the function g to an input x and the function f to the result is called composing f and g, and is
denoted f(g(x)), fg(x) or f∘g(x).
When finding composite functions be careful to apply functions in the correct order. Remember that the
function fg(x) means “first apply g, then apply f to the result”. fg(x) is rarely the same as gf(x).
f∘f(x) or ff(x) may be written as f2(x).
The inverse function of f(x), where it exists, is defined as a function g(x) such that fg(x) = gf(x) = x. That is,
applying the inverse function to the original function, or vice-versa, does not change the input x.
To find the inverse function f−1(x), assuming it exists, start with y = f(x) and rearrange to get the input x in
terms of the output y. The expression for f−1(x) is then given by replacing the x with f−1(x) and the y’s with x’s.
The graphs of y=f(x) and y=f−1(x) are reflections in the line y=x.
The domain of f−1 (x) is the range of f (x).
The range of f−1 (x) is the domain of f (x).
A function has an inverse if and only if it is one-to-one over its domain.
4. Polynomials:
A polynomial is an expression or function which only contains terms of the form axn, where a is a constant, x is
the only variable and n is an integer ≥ 0. So, a polynomial expression might involve terms in x7, x4, x and a
constant term, but not terms in x, 1x, xy etc.
When you divide a polynomial of degree m by a polynomial of degree n the degree of the resulting polynomial,
called the quotient, will be of degree m−n.
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,The factor theorem states that:
If (x−a) is a factor of f(x), then f(a) = 0 and x = a is a root of the equation f(x) = 0. So, the curve passes through
the x-axis at a. Conversely, if f(a) = 0 then (x−a) is a factor of f(x).
For graphs of polynomial functions:
• Polynomials of order n meet the x-axis at most n times.
• Polynomials of order n have at most n−1 turning points.
There are a number of points to consider when sketching a polynomial graph. They are:
• Deduce the order of the polynomial (highest power of x) and whether it is positive/negative to have an
idea of the general shape of the curve
• Let x = 0 to find where the graph passes through the y-axis
• Factorise the given polynomial, if possible
• Find where it passes through the x-axis from the factorised polynomial
• Determine how the graph meets the x-axis at these points
• Finally, sketch the curve from the above points
5. Using Graphs: Unit 2:
To find the intersection (crossing points) of two graphs, solve them simultaneously, either by:
• Elimination
• Substitution
If one equation is linear and one equation is non-linear, follow these steps:
1. Rearrange the linear equation so it starts with y= or x=
2. Substitute the linear equation into the non-linear equation
3. Solve the resulting equation
4. Substitute all solutions into the linear equation to find each pair of solutions.
To find the number of intersections between a line and a curve, substitute the linear equation into the quadratic
equation and find the discriminant of the resulting equation.
• If b2−4ac>0, the line and curve intersect twice.
• If b2−4ac=0, the line and curve intersect once (i.e. the line is a tangent to the curve).
• If b2−4ac<0, the line and the curve do not intersect.
In general, for the graph of the function y=f(x) and a positive constant a:
• y = f (x+a) is a translation of y = f (x) by the vector (−a, 0) (or a units to the left)
• y = f (x−a) is a translation of y = f (x) by the vector (a, 0) (or a units to the right)
• y = f (x)+a is a translation of y = f (x) by the vector (0, a) (or a units up)
• y = f (x)−a is a translation of y = f (x) by the vector (0, −a) (or a units down)
• y = f (−x) is a reflection of y=f(x) in the y-axis
• y = −f (x) is a reflection of y=f(x) in the x-axis
• y = f (ax) is a horizontal stretch (parallel to the x-axis) of y = f (x) by scale factor 1/a -(all the x-
coordinates are multiplied by 1/a)
• y = af (x) is a vertical stretch (parallel to the y-axis) of y = f (x) by scale factor a - (all the y-
coordinates are multiplied by a)
Translations can be combined so that, in general, for the graph of the function y=f(x) and constants a and b:
y=f(x−a) +b is a translation of y=f(x) by the vector (ab) (or a units horizontally and b units vertically).
Since the functions y=ax and y=ax2 have no values for x=0 and y=0, the x-and y-axis are asymptotes for the
graphs of these functions. An asymptote of a curve is a line which the curve gets infinitely close to, but never
touches. Asymptotes are usually indicated by a dotted line on the graph.
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, In words In symbols As an equation
y is proportional to x y∝x y=kx
y is proportional to xn y∝xn y=kxn
y is inversely proportional to x y∝1/x y=k/x
y is inversely proportional to xn y∝1/xn y=k/xn
3. Further Transformation of Graphs:
When a vertical and a horizontal transformation are combined, the transformations can be carried out in either
order without changing the result.
When two transformations in the same direction are combined, the order in which they are combined generally
changes the outcome.
Vertical transformations follow the order of operations. Horizontal transformations occur in the opposite order.
A possible definition of the modulus function is:
Its domain is all real numbers, and its range is |x|≥0.
To sketch the graph of y=|f(x)|, first draw the graph of y=f(x), then reflect in the x-axis any parts of the graph
below the x-axis.
|x−a| < b is the same as –b < x−a < b
|x−a| > b is the same as x−a > b or x−a <−b.
6. Coordinate Geometry:
To find the number of intersections between a line and a circle, substitute the linear equation into the circle
equation and find the discriminant of the resulting quadratic equation.
• If b2−4ac>0, the line crosses the circle at two distinct points.
• If b2−4ac=0, the line touches the circle at a single point and is a tangent to the circle.
• If b2−4ac<0, the line does not cross the circle.
The tangent of a circle is perpendicular to the radius at the point of contact.
The normal to a curve is the line perpendicular to the tangent at that point.
The radius of the circle therefore coincides with the normal to the tangent at the point of
contact.
1 The distance between the centres is The circles are disjoint.
greater than the sum of the radii. They do not touch and one
d>r1+r2 is outside the other.
2 The distance between the centres is The circles are tangent to
equal to the sum of the radii. each other. They touch
d=1+r2 externally at a single point.
3 The distance between the centres is The circles meet at two
less than the sum of the radii but distinct points.
greater than the difference. r2−r1<d<r1+r2
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The ⇒⇒ symbol means that the statement follows from a previous statement
The ⇐⇐ symbol means that the previous statement follows from the subsequent statement
The ⇔⇔ symbol means that a subsequent statement is equivalent to the previous one
In a proof by deduction, you list a series of logical steps to show that your proposition is true.
In a proof by exhaustion, you check the statement is true in every possible case.
Disproof by counter example is when you show a statement is untrue by finding a single counter example –
one case where the statement isn’t true.
Proof by contradiction → Prove by contradiction that log2(3) is irrational.
To prove: there are no positive integers p and q such that p/q = log2(3). Proceed by contradiction.
Assume that log2(3) is rational, so it can be expressed in the form p/q where p and q are integers. If p/q =
log2(3), then p = log2(3q) and 2p = 3q.
But 2p is even for all positive integers p and 3q is odd for all positive integers q - therefore this cannot be true.
Our assumption was incorrect: no such p and q exist, so log2(3) is irrational.
For inequalities, the following interval notation may be used:
• x ∈ (a, b) means x is in the interval a < x < b
• x ∈ [a, b] means x is in the interval a ≤ x ≤ b
• x ∈ [a, b) means x is in the interval a ≤ x < b
• x ∈ (a, b] means x is in the interval a < x ≤ b
Where the ∈ symbol means “is in the set…” or “belongs to the
set…”
If there is no solution to an inequality, then it can be
represented by the empty set, written as x∈∅.
Any even number can be written as 2n, where n is an integer.
Any odd number can be written as 2n+1, where n is an integer.
3. Quadratic Functions:
The graph of y=ax2+bx+c crosses
• the y-axes at the point (0, c)
• the x-axis at the solutions to the equation ax2 + bx + c = 0.
These solutions are called the roots of the equation.
When sketching a quadratic function, completing the square gives you the turning point of the graph.
y=a(x+p)2+q gives the turning point at (−p, q). It also gives the line of symmetry at x= −p
When solving a quadratic inequality:
• Rewrite the inequality with the quadratic in the form ax2 + bx + c on the left of the inequality symbol
and zero on the right. (When rearranging remember that you cannot divide or multiply through by a
negative number without changing the direction of the inequality symbol.)
• Sketch a graph of the quadratic function on the left-hand side.
• Use the graph to find the solution set.
• For quadratics that are > 0 or ≥ 0, the solution set is given by the values of x that give the parts of the
graph that are above the x-axis.
• For quadratics that are < 0 or ≤ 0, the solution set is given by the values of x that give the parts of the
graph that are below the x-axis.
For a quadratic function y = ax2 +bx + c the expression b2−4ac is called the discriminant of the quadratic and
is often symbolised by the Greek letter Δ (uppercase delta).
1|Page
, • If Δ > 0 the function has two distinct real roots
• If Δ = 0 the function has two equal roots
• If Δ < 0 the function has no real roots
Look out for equations that can be written in the form a(something)2+b(something)+c=0 where the
‘something’ is a function that can be replaced by a single variable using a substitution to give a quadratic
equation.
(YEAR 2) 2. Functions:
A function is a mapping with only one output for any given input.
A function is one-to-one if each output corresponds to only one input, and many-to-one otherwise.
The set of values allowed as input values for a function is known as the function’s domain. (So basically,
where the x starts from).
The range of a function is the set of all possible output values (So basically where the y starts from).
∊ means “is an element of”
: means “such that”
Z Denotes the set of integers
Q Denotes the set of rational numbers
R Denotes the set of real numbers
R+ Denotes the set of positive real numbers (similarly for rational numbers, integers etc.)
R− Denotes the set of negative real numbers (similarly for rational numbers, integers etc.)
Applying the function g to an input x and the function f to the result is called composing f and g, and is
denoted f(g(x)), fg(x) or f∘g(x).
When finding composite functions be careful to apply functions in the correct order. Remember that the
function fg(x) means “first apply g, then apply f to the result”. fg(x) is rarely the same as gf(x).
f∘f(x) or ff(x) may be written as f2(x).
The inverse function of f(x), where it exists, is defined as a function g(x) such that fg(x) = gf(x) = x. That is,
applying the inverse function to the original function, or vice-versa, does not change the input x.
To find the inverse function f−1(x), assuming it exists, start with y = f(x) and rearrange to get the input x in
terms of the output y. The expression for f−1(x) is then given by replacing the x with f−1(x) and the y’s with x’s.
The graphs of y=f(x) and y=f−1(x) are reflections in the line y=x.
The domain of f−1 (x) is the range of f (x).
The range of f−1 (x) is the domain of f (x).
A function has an inverse if and only if it is one-to-one over its domain.
4. Polynomials:
A polynomial is an expression or function which only contains terms of the form axn, where a is a constant, x is
the only variable and n is an integer ≥ 0. So, a polynomial expression might involve terms in x7, x4, x and a
constant term, but not terms in x, 1x, xy etc.
When you divide a polynomial of degree m by a polynomial of degree n the degree of the resulting polynomial,
called the quotient, will be of degree m−n.
2|Page
,The factor theorem states that:
If (x−a) is a factor of f(x), then f(a) = 0 and x = a is a root of the equation f(x) = 0. So, the curve passes through
the x-axis at a. Conversely, if f(a) = 0 then (x−a) is a factor of f(x).
For graphs of polynomial functions:
• Polynomials of order n meet the x-axis at most n times.
• Polynomials of order n have at most n−1 turning points.
There are a number of points to consider when sketching a polynomial graph. They are:
• Deduce the order of the polynomial (highest power of x) and whether it is positive/negative to have an
idea of the general shape of the curve
• Let x = 0 to find where the graph passes through the y-axis
• Factorise the given polynomial, if possible
• Find where it passes through the x-axis from the factorised polynomial
• Determine how the graph meets the x-axis at these points
• Finally, sketch the curve from the above points
5. Using Graphs: Unit 2:
To find the intersection (crossing points) of two graphs, solve them simultaneously, either by:
• Elimination
• Substitution
If one equation is linear and one equation is non-linear, follow these steps:
1. Rearrange the linear equation so it starts with y= or x=
2. Substitute the linear equation into the non-linear equation
3. Solve the resulting equation
4. Substitute all solutions into the linear equation to find each pair of solutions.
To find the number of intersections between a line and a curve, substitute the linear equation into the quadratic
equation and find the discriminant of the resulting equation.
• If b2−4ac>0, the line and curve intersect twice.
• If b2−4ac=0, the line and curve intersect once (i.e. the line is a tangent to the curve).
• If b2−4ac<0, the line and the curve do not intersect.
In general, for the graph of the function y=f(x) and a positive constant a:
• y = f (x+a) is a translation of y = f (x) by the vector (−a, 0) (or a units to the left)
• y = f (x−a) is a translation of y = f (x) by the vector (a, 0) (or a units to the right)
• y = f (x)+a is a translation of y = f (x) by the vector (0, a) (or a units up)
• y = f (x)−a is a translation of y = f (x) by the vector (0, −a) (or a units down)
• y = f (−x) is a reflection of y=f(x) in the y-axis
• y = −f (x) is a reflection of y=f(x) in the x-axis
• y = f (ax) is a horizontal stretch (parallel to the x-axis) of y = f (x) by scale factor 1/a -(all the x-
coordinates are multiplied by 1/a)
• y = af (x) is a vertical stretch (parallel to the y-axis) of y = f (x) by scale factor a - (all the y-
coordinates are multiplied by a)
Translations can be combined so that, in general, for the graph of the function y=f(x) and constants a and b:
y=f(x−a) +b is a translation of y=f(x) by the vector (ab) (or a units horizontally and b units vertically).
Since the functions y=ax and y=ax2 have no values for x=0 and y=0, the x-and y-axis are asymptotes for the
graphs of these functions. An asymptote of a curve is a line which the curve gets infinitely close to, but never
touches. Asymptotes are usually indicated by a dotted line on the graph.
3|Page
, In words In symbols As an equation
y is proportional to x y∝x y=kx
y is proportional to xn y∝xn y=kxn
y is inversely proportional to x y∝1/x y=k/x
y is inversely proportional to xn y∝1/xn y=k/xn
3. Further Transformation of Graphs:
When a vertical and a horizontal transformation are combined, the transformations can be carried out in either
order without changing the result.
When two transformations in the same direction are combined, the order in which they are combined generally
changes the outcome.
Vertical transformations follow the order of operations. Horizontal transformations occur in the opposite order.
A possible definition of the modulus function is:
Its domain is all real numbers, and its range is |x|≥0.
To sketch the graph of y=|f(x)|, first draw the graph of y=f(x), then reflect in the x-axis any parts of the graph
below the x-axis.
|x−a| < b is the same as –b < x−a < b
|x−a| > b is the same as x−a > b or x−a <−b.
6. Coordinate Geometry:
To find the number of intersections between a line and a circle, substitute the linear equation into the circle
equation and find the discriminant of the resulting quadratic equation.
• If b2−4ac>0, the line crosses the circle at two distinct points.
• If b2−4ac=0, the line touches the circle at a single point and is a tangent to the circle.
• If b2−4ac<0, the line does not cross the circle.
The tangent of a circle is perpendicular to the radius at the point of contact.
The normal to a curve is the line perpendicular to the tangent at that point.
The radius of the circle therefore coincides with the normal to the tangent at the point of
contact.
1 The distance between the centres is The circles are disjoint.
greater than the sum of the radii. They do not touch and one
d>r1+r2 is outside the other.
2 The distance between the centres is The circles are tangent to
equal to the sum of the radii. each other. They touch
d=1+r2 externally at a single point.
3 The distance between the centres is The circles meet at two
less than the sum of the radii but distinct points.
greater than the difference. r2−r1<d<r1+r2
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