Algebra and functions Flashcards
laws of indices
- when multiplying, add the powers
- when dividing, minus the powers
- when you have a power to a power of something else, multiply the powers
- to the power of a fraction, you can write as roots
- a minus power (A^-m), you can write as 1 over A^m
- to the power of 0 is always 1
rules of surds
- to make the number in the root smaller/simplified or get rid of the fraction use:
- root AB = root A x root B
- root A/B = (root A) / (root B)
simplifying by factorising and cancelling factors
1) look for common factors
2) if there is a fraction in the numerator or denominator, multiply the whole thing by the same factor to cancel it out
3) multiply like any other fraction (multiplying the denominator) then factorise
‘degree’ definition
the number of the highest power of x in a polynomial
‘quotient’ definition
the stuff you get when dividing by a divisor
factor theorem
when you get a remainder of 0 when putting (x-a) into f(x) as a.
Make sure to state that it is a factor if the remainder is zero at the end of the question
partial fractions
splitting a fraction into more then one linear factor
solving quadratic equations
- factorise to solve
- complete the square for a specific form or to find exact solutions (usually involving surds)
- the formula
the discriminant
X-intercepts in a quadratic = b^2 +4ac
if the Discriminant is:
< 0 0 solutions
= 0 1 solutions
> 0 2 solutions
Using this section of the quadratic formula ( which is in the square root sign) you can have a positive negative or a zero. Therefor this tells you if you have two, one or no real roots.
sketching a quadratic
1) - if coefficient of x^2 is positive then it is a ‘u-shaped’
- if coefficient of x^2 is negative then it is a ‘n-shaped’
2) Axis
- to get y intercepts (set x=0)
- to get x intercepts (set y=0)
3) find the maximum or minimum of the curve ( complete the square)
simultaneous equations
- match coefficients
- eliminate to find one variable
- find the variable you eliminated
- or use the substitution method if one is a quadratic
modulus
- doesn’t matter if it is positive or negative
e.g f(x) = -6
|f(x)| = 6
|f(x)| = f(x) when f(x) is greater then or equal to 0
|f(x)| = -f(x) when f(x) is less than 0
modulus graphs
y = |f(x)| - reflect the negative part of the line on the x axis
y = f(|x|) - for negative x values, reflect on the y axis
y = |f(-x)| - reflect the negative part of the line reflect on the x axis
method for solving |f(x)|= n
substitute n for g(x)
1) sketch both y = |f(x)| and y = n on the same x axis
2) work out the ranges for f(x) when f(x) is ‘greater then or equal to’ 0 and when f(x) is ‘less than’ 0
3) use this to write two new equations that are true for the range of x.
4) then just solve each equation
solving modulus algebraically
when solving |f(x)| = |g(x)| you have to think about where f(x) is positive and negative.
start by squaring both sides
if |a| = |b|
- then a^2 = b^2
so if |f(x)|= |g(x)|
-then |f(x)|^2 = |g(x)|^2
the graph of y = kx^2 is a different shape for different k and n values
1) n positive and even
u-shape when k is positive
n-shape when k is negative
2) n positive and odd
get a corner to corner shape
3) n negative and even
you get two bits next to each other
4) n negative and odd
get a graph with two bits opposite each other
graph transformatons
y = f ( x+c ) shift c to the left
y = f ( x-c ) shift c to the right
y = f ( x ) + c shift c down
y = f ( x ) - c shift c up
y = af ( x ) stretch vertically |a|>1
y = 1/a f ( x ) squashed vertically |a|<1
y = f ( ax ) squashed horizontally
(1/a )
|a|>1
direct proportions graphs
are straight lines through the origin
y=kx
inverse proportions graphs
curved
multiplying one variable by any constant is the same as dividing the other by the same constant.
y = k/x
a function is a type of mapping
an operation that takes numbers and maps each on to one only number
if f(x) = 2x is a function as it is one to one as there is one possible value for x
domain
the set of starting numbers in a function, the input
range
the numbers that the function becomes, the out put
many-to-one function
domain has more then one element but the range has one element
one-to-one function
domain has one element and the range has one element
