functions / graph transformations

Graph transformations

F(x-a) [+a to x coordinates]
F(x+a) [ - a to x coordinates ]
F(x)-a [- a from y coordinates]
F(ax) [ divide x coordinates by a ]
F(x/a) [ multiply x coordinates by a
aF(x) [ multiply y coordinates by a ]
F(-x) [ multiply x by -1] so if + would be -
-F(x) [ multiply y by -1]

set notation

{x: the inequality}

interval notation

x∈[a,b)
it is [] if or equal to
it is () if not or equal to or if ∞
use ±∞ if is a single inequality e.g x<3
x∈(-∞,3)

if you have multiple inequalities you cant combine how would you write them in set/interval notation

combine using U symbol so set notation a U set notation b
still separate just have U symbol between them
useful in quadratics where e.g x<3 and x>6 say for example
CAN DO THIS AS MANY TIMES AS POSSIBLE

describing transformations

anything added / subtracted can be in vector [x y ]
anything else needs to be in this form
one / two way stretch with sf .... in the x / y direction
if two way describe both x and y direction

Find horizontal asymptotes of f(x)

set denominator equal to 0 and solve for x is 0 if there is no + or - value
always sub in 0 into equation when drawing to get a point

find vertical asymptotes of f(x)

put x/x then simplify e.g if equation is 2x+1/3x-9
2x/3x = 2/3 but if no x value in numerator asymptote = 0
always sub in 0 into equation when drawing to get a point

what is reflection in x and y value in terms of translations

-x = reflect in x axis
-y = reflect in y axis

create an equation based off of a translation

read translation and create a function so if [3 -1] function is (x-3) -1 sub into equation with x-3 in place for x BUT
-1 at end so do not multiply by anything just stick on end
if reflecting in x axis sub in -x
if reflecting in y axis set y to -y and then get equation by multiplying whole equation by -1 to get rid of -y = -ax^2-bx-c

find the translation given two equations

y translation = difference in c coefficient or whole equation has a sf change has been multiplied/divide you know is y and not x as a coefficient usually lagrer
x translation = factorise a coefficient and find sf or check if e.g (x+2) has been subbed in

describe two transformations of 3^x -> 3^(x-1)

f(x)=(x-1) translation one to right
or
rewrite as 3^x / 3^1
therefore whole thing has been divided by 3 so y stretch sf 1/3

how to write asymptotes on graph

not x≠5 , y=5 with dotted line if not on axis

draw translation f(-x) and -(x)

-x = mirror on y axis
as all y coordinates stay same but x ones are negative so goes left/right as x
-(x) = mirror on x axis
as all x coordinates stay same by all y coordinates negative so goes up/down as y
if in doubt use the rules to work out known coordinates

find y coordinate intercept of translation

factorise do what in brackets to brackets and multiply for c = y intercept
e.g factorised = x(x+1)(x+3) translation f(x+3)
do (x+3)(x+4)(x+7) 84 = c
remember if one of your terms is just x not in bracket c=0 so intercepts at 0,0
also remember to plot you still need to inverse these values