SECTION 2- ALGEBRA AND FUNCTIONS Flashcards
INDICES RULE - multiplication
if you multiply the number, you add the powers
- bases MUST be the same e.g a^m x a^n = a^m+n
INDICES RULE - division
if divide two numbers, you subtract their powers
- bases must be the same e.g. a^m / a^n = a^m-n
INDICES RULES - brackets
if you have a power to the power of something else – multiply the powers together e.g. (a^m)^n = a^mn
INDICES RULE - negative power
- a negative power means it’s on the bottom line of a fraction e.g. a^-m = 1 / a^m
INDICES RULE - power of 0
- any number to the power of 0 is equal to 1 e.g. (a+b)^0 =1
INDICES RULE - a^1 / m
- roots can be written as powers so: a^1/ m = m√a
INDICES RULE - a^m/ n
- a power that’s a fraction is the root of a power - or the power of a root so: a^m/ n = n√a^m = (n√a)^m
what does it mean to ‘rationalise the denominator’
- means getting rid of the surds from the bottom of the fraction
how do you ‘simplify a surd’
- to simplify a surd, make the number in the √ sign smaller, or get rid of a fraction in the √ sign
what are the 3 rules of surds
- √ab = √a√b
- √a/b = √a /√b
- (√a)^2 = √a√a
what is the difference of two squares
- can be applied to surds also
- (a+b) (a-b) = a^2 - b^2
how to simplify algebraic fractions
factorising
- look for common factors in the numerator and denominator – factorise top and bottom and see if there’s anything you can cancel
- if there’s a fraction in the numerator or denominator –multiply top and bottom by by the same factor
how to add and subtract fractions by finding a common denominator
- find the common denominator
- put each fraction over the common denominator
- combine into one fraction
what does the term ‘degree’ mean
- the highest power of x in the polynomial
what does the term ‘divisor’ mean
- the thing that you are dividing by
what does the term ‘quotient’
- the stuff that you get when you divide by the divisor (not including the remainder)
what does the term ‘remainder’ mean
- the bit that’s left over at the end (known in A-Levels as the constant)
what is the general quadratic equation
ax^2 + bx + c
what is the quadratic formula
x= −b±√b²−4ac /2a
what is the discriminant
b²-4ac
what happens when the discriminant is
a) =0
b) <0
c) >0
a) when it’s equal to zero, it has one repeated root
b) when it’s less than zero, it has no roots (neither of the solutions are real numbers)
c) when it’s greater than zero, it has two distinct roots
what is the factor theorem
- if you get a remainder of zero when you divide the polynomial f(x) by (x - a), then (x - a) must be a factor of f(x)
how to use the factor theorem when there’s a coefficient of x
- to test if (ax - b) is a factor. substitute f(b/a) and if it =0, it’s a factor
- so if you know the roots, you know the factors – and vice versa
what does transformation y = f(x + c) mean
f(x) is shifted c to the left
- graph is moved in opposite direction e.g. f(x + c) shifts to the left and f(x - c) is shifted to the right
what does transformation y = f(x) + c mean
- movement on y - axis, so will shift c upwards and if it’s f(x)- c, it will shift downwards in y-axis
what does transformation y = af(x) mean
- is stretched in y direction w/ scale factor of a
- if |a| > 1 the graph of af(x) is f(x) stretched vertically by a factor of a
- if 0< |a| < 1 the graphed is squashed vertically
- if a a < 0 the graph is also reflected in the x-axis
what does transformation y = f(ax) mean
- is stretched in x direction w/ scale factor of 1/a
- if |a| > 1, the graph of f(ax) is f(x) squashed horizontally by a factor of a
- if 0 < |a| < 1, the graph is stretched horizontally
- if a < 0, the graph is also reflected in the y-axis
what does transformation y = -f(x)
- a reflection in x-axis
what does transformation y = f(-x)
- a reflection in y-axis
