a constant function can be horizontal and vertical

true or false

false–ONLY HORIZONTAL

f(1)=4 really is

an ordered pair (1,4)

f(x)=x^2

parent quadratic function

f(x)=square root x

parent radical function

f(x)=1/x

parent rational function (reciprocal function)

what is the difference between 1/x and x/1 in terms of functions

1/x is the parent rational function and x/1 is a linear function with slope 1/1

f(x)= [[x]]

greatest integer function (or, step function)

what is a piecewise function

a function with pieces (normally 2 or 3)

the greatest integer function takes the next integer ______

so -3.1 would go to __ and 2.9 would go to ___

-4, 2

it is also called the round-down function by some

f(x)=x

identity function

the constant function (f(x)=c) and the identity function (f(x)=x) are two special types of _____ functions

linear

a piecewise defined function will always have at least one x-intercept or at least one y-intercept

true or false

true- defined means that the domain is all real numbers so it will have a y intercept at least

what is a defined function

one that has a domain of all real numbers

when performing a piecewise function, always ______ your solutions and make sure that your functions work

test

a linear equation will always have an x intercept and a y intercept

true or false

false-constant functions will not have an x intercept

what are the three types of transformations

what are the four types of translations

translation, reflection, dilation

up, down, left, right

describe the translation f(x)+c f(x)-c f(x+c) f(x-c)

c units up

c units down

c units left

c units right

describe the graph

h(x)=(x+2)^3 +1

cubic function shifted two units to the left and shifted up one

when is there a reflection in the y-axis for a function?

x-axis?

y-axis: h(x)=f(-x)

x-axis: h(x)=-f(x)

describe the graph

j(x)=-(x+3)^2 +1

quadratic function reflected over the x-axis, shifted three units to the left and up one

when the shapes are congruent or unchanged after a transformation, the transformation is a _________

what kinds of transformations are included

rigid transformation

translations, reflections

transformations that cause shapes to change (horizontal or vertical stretches) are _____________

example?

nonrigid transformations

dilation

when will a function vertically stretch?

Shrink?

n*f(x) when n<-1(-infinty,-1) or n>1(1, infinity)

n*f(x) when -1<n<1 (-1,1)

g(x)=(x-1)^3 +2

what is the parent function?

use function notation to write g in terms of f

f(x)=x^3

g(x)=f(x-1) +2

***Remember not to include the ^3, as that is implied in f(x)

horizontal shifts, vertical shifts, and reflections are called _____ transformations

rigid

A reflection in the x-axis of y=f(x) is represented by h(x)= ________, while a reflection in the y-axis of y=f(x) is represented by h(x)=__________

-f(x)

f(-x)

A nonrigid transformation of y=f(x) represented by g(x)=cf(x) is a _________ when c<-1 or c>1 and a ______ _______ when -1<c<1

vertical stretch, vertical shrink

in (f of g)(x), what is the domain?

the domain of f of g is the set of all x in the domain of g such that g(x) is in the domain of f

what is the composition of functions

taking one function and plugging it into another function (not commutative)

is composition of functions commutative? what does this mean?

no- this means that you will get the same answer in reverse (think addition, multiplication [4+2=2+4])

is (f of g)(x) muliplication

No

how do you decompose a composite function? decompose h(x)=1/(x-2)^2

first find the simplified function (this is your f(x)), then think, what do i plug in (g(x)), to get what I have now?

f(x)=1/x^2 g(x)=x-2

***Note that the ^2 is outside of the parentheses. This is why you cannot have f(x)=1/x and g(x)=x-2^2–> you could have g(x)=(x-2)^2

what are transcendental functions

mix of two types of functions (1/x^2) quadratic and rational

what are the 4 types of functions

polynomial, rational, radical, trigonometric (i dont think we need to know this just in case though)

two functions f and g can be combined by the arithmetic operations of ________,_________,_________, and ____________to create new functions

addition, subtraction, multiplication, division

The _____ of the function f with g is (f of g)(x)=f(g(x))

composition

f(x)=x^2 +6 g(x) square root (1-x)

divide these

x^2 +6/square root of (1-x) —cannot have square roots on the bottom (multiply top and bottom by square root (1-x))—-

x^2 +6 square root (1-x)/1-x

what is (f-g)(0)

f(0)-g(0)

a set ordered pair (mapping, x/y chart, etc.)

relation

Two techniques for fitting models to data are called direct and iverse _______ and least squares ________

variation, regression

Statisticians use a measure called the ______ of _____ _______to find a model that approximates a set of data most accurately

sum, square differences

The linear model with the least sum of square differences is called the ______ ______ _______ line

line of regression

An r value of a set of data, also called a ________ _________, gives a measure of how well a model fits a set of data.

what is the worst of these? best?

correlation coefficient

0, 1

direct variation models can be described as “y varies directly as x,” or “y is _______ ________ to x”

directly proportional

In direct variation models of the form y=kx, k is called the ____ of ____

constant, variation

The direction variation model y=kx^n can be described as “y varies directly as the nth power of x,” or “y is ____ _____ to the nth power of x”

directly proportional

The mathematical model y=k/x is an example of _____ variation

inverse

The joint variation model z=kxy can be described as “z varies jointly as x and y,” or “z is ________ ________ to x and y.”

directly proportional

what is mathematic modeling

coming up with the equation

what is a model?

an equation

y varies directly as x

y is directly proportional to x

y=kx for some nonzero constant k

direct variation

what is k in y=kx?

constant of variation, also the rate

what is the equation for state income tax

what kind of variation does it have

state income tax=k(gross income) (T=k*g where T is the dependent variable and g is independent)

direct

what must you have in order to find k-the constant of variation

initial condition

inverse variation says as one gets bigger, _________________________

the other gets smaller

y=k/x is ______ variation. It is the opposite of _____

inverse, direct

z=kxy

joint variation (z varies jointly as x and y)

what is the equation for Interest

I=P*r*t

since k is also rate, you could say I=k(P)(t

how do you find a lines equation and graph a scatter plot on your calculator?

lines equation given a lot of points- hit stat, edit, enter x values in L1 and y values in L2, make sure you have the same number of data entries, hit stat again, calc, LinReg, make sure Xlist says L1 and Ylist says L2, calculate

this will give you your equation!!!

to plot a scatter graph- go to y=, graph the line you got above^ (may have to adjust windows) then hit Stat Plot (second y=), plot 1 on, choose type, graph (should get a line with plots)

what kind of variation will these ordered pairs have?

5, -3.5)(10, -7)(15, -10.5)(20, -14)(25, -17.5

Direct—although technically the numbers are getting smaller, the positive values are increasing

z varies directly with the square of x and inversely with y with a constant variation of 2/3

how would you write this?

z=2x^2/3y (separate your fraction)

S=4pi r^2

how would you use variation terminology to say this aloud

The surface area of a sphere varies directly as the square of the radius r (your constant of variation is 4pi [you will never say a number])

how would you use variation terminology to say A=1/2bh

the area of a triangle is jointly proportional to its base and height

Inverse operations _________ each other

undo

what is f^-1(x)

inverse of f (-1 has new mathematical value!)

what does a graph’s inverse do

what kind of symmetry do they have

switches x and y

reflectional symmetry over the line y=x

for a function to have an inverse, it must pass (Horizontal/Vertical) Line Test

BOTH- it must pass vertical to be a legitimate function. It must pass horizontal to have an inverse

an inverse cannot fail (Horizontal/Vertical) line test

I think just Vertical???

The domain of a function = the _____ of its inverse.

the range of a function = the _____ of its inverse

range, domain

how do you prove f(x) and g(x) are inverses of eachother

use composite functions (analytically) plug g(x) into f(x) [f(g(x))] and you will get x

a function is __________ if it passes Horizontal and Vertical line test

one to one

only __________ have an inverse function

one to one

how would you make f(x)=x^2 a function with an inverse

restrict the domain (x>_ 0)

in a piecewise function, the ranges of the starting functions are the _________ of the inverse functions

domains

If the composite functions f(g(x)) and g(f(x)) both equal x, then the function g is the ______ function of f

inverse

The inverse function of f is denoted by

f^-1

The domain of f is the ____ of f^-1 and the, and the ______ of f^-1 is the range of f

range, domain

The graphs of f and f^-1 are reflections of each other in the line ___

y=x

A function is _____ when each value of the dependent variable corresponds to exactly one value of the independent variable

one-to-one

A graphical test for the existence of an inverse function of f is called the _____ Line Test

Horizontal

To reflect over x axis, make _ values negative

Vice versa

y

Be careful with distributing negatives in reflection cases

Square root of (x+6) reflected in both x and y axes is…

-square root of -x-6

Remember when a graph is up 3 and a point is (1,7), it’s technically

(1,4)

If f(x)=x^2 and you plug in 4t, what do you get

(4t)^2=16t