Unit 2

graphing inequality absolute value equations

1. graph 2. find the intersection points 3. see when one line is higher than the other shade the parts where on the graph the y values fit the inequality answer would be written in the form of x>_ and x<_ or _<x<_

decreasing function

A function is decreasing on the interval (a,b) if f(a)>f(b) AKA →↓ is decreasing AKA a and b are both x coordinates, and intervals always go from small to big, so a<b and so as x gets greater, y gets smaller

increasing function

A function is increasing on the interval (a,b) if f(b)>f(a) AKA →↑ is increasing AKA a and b are both x coordinates, and intervals always go from small to big, so a<b and so as x gets greater, y gets greater

composite function: def, how it is written, where are they commutative

When the output of a function is used as the input of another function e.g. f(g(x)) also written as ( f o g)(x) the point where composite functions are commutative is the point where they intersect

piecewise relation/function

a relation/function that is defined on a sequence of intervals For example, the piecewise function for the absolute value function is f(x)={-x, x<0 {x, x≥0 you graphically represent this by graphing them like they are two dif lines with the domain of the inequality

how to graph an absolute value

basically lwhateverl is as if the transformations are happening to x and _l_l+_ is as if it is a v and then treat the transformations like they are happening to y either start with a v and move or start with a line and fold theory: either look at it as doing transformations to f(x)=lxl, a v, or transformations on f(x)=the linear equation, a line that is folded eg: l3x+1l=-3lxl+5 how to graph l3x+1l if f(x)=lxl then this is f(3x+1) if f(x)=3x+1 then it is lf(x)l how to graph -3lxl+5 if f(x)=lxl then it is -3f(x)+5 if f(x)=-3x+5 then it is f(lxl) but this one doesn't make sense to graph so use the first technique if there is stuff outside of the l l

how to find domain of vertical transformation (vertical transformations)dilation, translation, absolute value

dilation: domain*a translation: domain + a absolute value: if negative, turn range positive, if positive, keep same. HOW ABS VALUE DOMAIN CHANGES RANGE: if domain is absolute valued: if domain is in negative and positive, then the range is unknown, because we don't know which part of the range is being erased and which is being flipped; if domain is just in positive, then you know the range of the thing being flipped, so the range stays how it is

Understand how to compose two or more functions to make new functions

f(g(x)) suppose g(x)=x^2+3x and f(x)=2x to find f(g(x)), you plug in g(x) into the x of f(x) so: f(g(x))=2(x^2+3x) f(g(x))=2x^2+6x

how to find domain of f(g(x))/the values that it is defined

f(g(x))'s domain is the x values that are within g(x)'s domain and where g(x) produces an output that is within f(x)'s domain. way to find domain: 1. make a graph 2. draw g(x) with domain restrictions 3. apply the domain restrictions of f(x) to the output of that graph (define the range of g(x) as whatever the domain of f(x) is) 4. all the possible x values that output that range are the domain of f(g(x)) if f(x) has an infinite domain, then the domain is the domain of g(x), if f(x) has a domain restriction and g(x) doesn't, then you say g(x)=the minimum of the domain and g(x)=the maximum of the domain and then do the equation or graph the intercept (that works because you think of it is f(x) and g(x)=x and so g(x) must be the domain. If both are restricted, you graph it and follow the steps above

Understand function notation

f(x) means "there is a function who's name is f, where x is the allowable input" f(x) could also be seen as y. When using an equation to evaluate f(10), you plug 10 into x. When using a graph, you look at what y equals when x equals ten (could be undefined)

use function notation to be able to rewrite functions in terms of another

for the y values you apply them to the things being applied to y. you always plug the new x value into the x. eg f(x)=2x+1 f(x+3)=2(x+3) +1 or k(x)=m(4x+6) k(2x)=m(4(2x)+6) eg when you are given a list of equations: write k(x) in terms of f(x) g(x)=f(lxl) h(x)=g(x+3) +1 k(x)=3h(2x) you start from end 3h(2x)=3g(2x+3)+3 3g(2x+3)+3=3f(l2x+3l)+3 you could also graph it with transformations and then write each of the transformations into the equation

combining a function

functions being combined to form new functions via the four basic operations (+-x/) and composition

how to solve equations with absolute values

graph first and then look at which lines are intersecting and use algebra to find that point (system of equations with the line, not with the entire absolute value). to algebraically solve absolute value: if x=l3l, x=+-3, so to get rid of l l, you say that whatever it equals is +- (eg lx+1l=l3x+2l, so x+1=+-l3x+2l)

horizontal dilation

horizontally scaling (f(ax))

how to find domain of things applied to x (horizontal transformations) dilation, translation, absolute value

if dilation domain/a or if translation domain-a if absolute value, erase stuff on left, copy right to left (so IF SECOND NUMBER IN DOMAIN IS POSITIVE make that positive and negative as domain (if not positive, undefined domain))

when you compress towards the y axis (f(2x)), how is it moved

it is compressed, but also moved a bit towards the y axis

when an exponential function looks like it is heading towards the x or y axis

it will be next to but not inclusive of 0 (helpful when asked to graph something with a certain domain

vertical translation of functions

moving function up/down with addition/subtraction. Domain stays the same, range changes. e.g. g(x)=f(x)+3

horizontal translation

moving left and right (f(x-a))

how to use a table for transformation

new x, old x, old y, new y start with old x (pick a random one on the graph) and then apply the x transformations to it (eg if old x is 6, 6=[transformations on x]) and this answer is the new x, then to find the old y, use the graph to see what the y value of the old x is, then for the new y apply the stuff being done to f(____) to the old y

which transformations are commutative and order of operations

reflection w dilation, things that apply to x with things that apply to y, (translations not commutative)..........with things applies to y, it's regular gemdas, with things applied to x, it's reverse gemdas with opposite symbols (like you are solving an equation)

vertical dilation of function

scales a function and changes the distance between each point and the x axis. Domain same, range dilated e.g. g(x)=2f(x)

vertical / horizontal reflection of function

vertical: flipping the function over the x axis. domain same, range changed. e.g. g(x)=-f(x) horizontal: flipping the function over the y axis. domain changed, range same. e.g. g(x)=f(-x)

explain dif and similarities between f(2x+4), f(2(x+4)), f(2(x+2))

you can think of it like distributing, but you can also of think of it graphically in order of operations without distributing. the first and last are the same because when you compress by two, it also moves it to the left a bit