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AS Mathematics- Core Mathematics 1
Chapter 1: Algebra and Functions
Rules of indices (powers):

am ´ an = am+n
am ¸ an = am-n
1
a-m = m
a
1
m

a =ma
n
m

a = m an
(a m )n = a mn
a 0 =1
Difference of two squares:

x 2 - y 2 = (x + y)(x - y)
Manipulation of surds:

ab = a ´ b
a
a
=
b
b
Rules of rationalizing surds:

1
, multiply the top and bottom by a
...

a+ b
1
 Fractions in the form
, multiply the top and bottom by a + b
...


f (x) = ax 2 + bx + c
Solving quadratic equations:
-Factorisation
...
e
...
e
...


-Points where curve crosses x- and y-axes
 Put y=0 to find the x-axis crossing points coordinates
...

-Discriminant for general shape:

b2 - 4ac > 0  the curve cuts the x-axis at two points (two different roots)
...


b2 - 4ac < 0  the curve does not cut the x-axis (no real roots)
...

-Substitution
...

Steps to solve a quadratic inequality:
-Long method:
 Solve the corresponding quadratic equation (find the critical values)
...

 Use the sketch to find the required set of values
...


+ bx + c < 0 and ‘a’ is positive then use the values between the two



If ax
lines
...




If ax
lines
...


2

2

2

2

+ bx + c > 0 and ‘a’ is negative then use the values between the two

Chapter 4: Sketching Curves
Sketching cubic curves:
Form of y = ax + bx + cx + d
-Shape:
 If ‘a’ is positive the line will first go up and then down and up
...

 If there are 2 discriminants it will touch the x-axis twice, turning at the point
that is found twice when solving the equation
...

-X-axis and y-axis:
 Find points at x-axis by making y=0
...

Common cubic curves:
3

y = x3

y = x2

y=

1
x

Transformations:

2

f (x + a) is a translation of -a in the x-direction
...

1
1
f (ax) is a stretch of in the x-direction (multiply x-coordinates by )
...

Sketching reciprocal functions:

y=

k
x

-Two types:
If k>0 then:

If k<0 then:

Chapter 5: Coordinate Geometry in the (x,y) Plane
Forms to write the equation of a straight line:

y = mx + c , where ‘m’ is the gradient and (0, c) is is the intercept on the y-axis
...


Finding the gradient:
You can find the gradient ‘m’ of the line joining the point with coordinates
the point with coordinates


m=

y2 - y1
x2 - x1

(x1, y1 ) to

(x2 , y2 ) by using the formula:

Finding the equation of a line with gradient and one point:
-Method 1:
 Put gradient into form of y = mx + c
...

 Solve the equation and put the value of c back to the original
...

 Put gradient into form of y = mx + c
...

 Solve the equation and put the value of c back to the original
...


So if two lines are perpendicular, the product of their gradients is -1: m ´ -

Chapter 6: Arithmetic series
Sequences and terms:

1
= -1
m

-A series of numbers following a set rule is called a sequence
...

Formula for the nth term of sequence:
-Used to find any term in a sequence
...

Recurrence formula:
Un+1 =Un + d when k≥1
...
For example the
sequences 5, 9, 13, 17 can be formed from Un+1 =Un + 4 , U1 = 5
...

-Recurrence relationship of the form:
 Uk+1 =Uk + n , k≥1
...

-The first term is represented by a
...

Formula for the nth term of an arithmetic series:

Un = a + (n -1)d

n= number of terms
a= first term
d= difference between terms
...


Chapter 7: Differentiation
Gradient of a curve:

-The gradient of a curve at a specific point is defined as being the same as the gradient
of the tangent to the curve at that point
...

Un+1 =Un + d when k≥1
...

-If f (x) = x n then f '(x) = nx n-1
...

dy
-So if y = f (x) , then
= f '(x)
...

-You need to expand and simplify polynomial functions in order to differentiate them
...

-This can be written as:
d2y

dx 2
...

Rate of change of a function:
You find the rate of change of a function ‘f’ at a particular point by using f '(x) and
substituting the value of x
...

Ways of saying differentiate:
Un+1 =Un + d when k≥1
...

dx
n +1

-Apply this principle of integration separately to each term of
Recurrence formula:
Un+1 =Un + d when k≥1
...

dx
-Find f(x) when f’(x)=…
-Find ò (
...

dx



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