3 key numbers means

4 test regions

is (x-1)(x^2+x+1) a product of linear factors

no, simplify the last trinomial by doing quadratic formula to get a product of linear factors

do you put brackets around solutions that make the bottom 0

no

is x^2 + 49 factorable?

yes

x-7i) (x+7i

The theorem that can be used to determine the possible numbers of positive real zeros and negative real zeros of a function is called __________ __________ of ________

Descartes’s Rule, Signs

the formula that relates cost, revenue, and profit is ______

profit=revenue-cost

key numbers are ____

whatever makes the top and bottom 0

how do you find HA if the numerator’s degree=denominator’s degree

y=ratio of leading coefficients

f(x)=2x-5/4-x HA:y=-2

The _______ ________ _______ states that if f(x) is a polynomial of degree n (n>0), then f(x) has precisely n linear factors

linear factorization theorem

what is intermediate form? is it the same as undefined

0/0, no

The _______ _______ of _______ states that if f(x) is a polynomial of degree n (n>0), then f has at least one zero in the complex number system

fundamental, theorem, algebra

All graphs _____ the asymptotes

Follow

Increasing and decreasing intervals use the ___ values

X (watch asymptotes)

what is upper bound

you plug in a higher number and get all positive sums in return

this will be your highest zero

Make sure graphs don’t cross if there aren’t enough X intercepts

..

When finding key points of rational functions ___ first

Simplify

how do you find the vertical asymptote

whatever makes the denominator 0

roots of denominator

x=

be sure to factor function first as asymptotes might cancel

for the rational function f(x)=N(x)/D(x), if the degree of N(x) is exactly one more than the degree of D(x), then the graph of f has a ______ (or oblique) ________

slant asymptote

how do you factor a perfect cube

in the first (), take the cubed root of both variables in the equation

in the second (), first, put in whatever you need to get the leading coefficient, then put in the opposite of the product of the first (), then whatever you need to get the constant

slant asymptotes and parabolic asymptotes all begin with

y=

so do HA

VA begins with x=

If a+bi is a complex zero of a polynomial with real coefficients, then so is its ________, a-bi

conjugate

write x^6-x^7 as a product of linear factors

(x)(x)(x)(x)(x)(x)(1-x)

Why is x^4 + x^2-60 guaranteed two real roots

It’s down 60 going up eternally

Increasing and decreasing intervals use the ___ values

X (watch asymptotes)

A real number b is a ______ bound for the real zeros of f when no real zeros are less than b, and is a ______ bound when no real zeros are greater than b

lower, upper

To solve a polynomial inequality, find the _______numbers of the polynomial, and use these numbers to create __________ ________ for the inequality

critical/key, test intervals

what is DesCartes’s Rule of signs

the number of positive real zeros=the number of sign variations or less than that by an even integer

the number of negative real zeros=the number of sign variations at

f(-x) or less than that by an even integer

when f(x)—>+/- infinity as x—>a from the left or right, x=a is a _______ ________ of the graph of f

vertical asymptote

what does lower bound say

you plug in a negative number and get alternating sums in return,

this will be your lowest zero

Can an answer written as the product of linear factors have imaginary factors?

Yes, they are still linear

what does irreducible over the reals mean

does not reduce into real factors

x^2-2x+10

The test that gives a list of the possible rational zeros of a polynomial function is the ______ _______ Test

rational zero

in an inequality coordinate plane graph, the shaded area are the ______ and the line indicates ________

values that make the equation true, value that makes the equation =

what are the possible root combinations for a cubed equation

all real, 2 imaginary and one real

factor x^3+8

(x+2)(x^2-2x+4)

Can you cross through a HA? VA? SA?

Yes no yes

if 5i is a zero, what else is true

-5i is a zero

when you divide by a - in an inequality, ____ the sign

flip

key values with solutions in the middle are called ____ points

border

What is root 3i times root 3i

3i^2

-3

special cases are also called ______

unusual solution sets

what is the radicand

the polynomial under the root

A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be _______ over the _______

irreducible, reals

the key numbers of a rational expression are its ______ and its ___ ____

zeros, undefined values

what is a rational function

fraction with polynomials

what is the rational zero test

p(constant)/q (leading coefficient

some factors of both of these things will present a zero of the equation

Every polynomial of degree n>0 with real coefficients can be written as the product of _______ and ______ factors with real coefficients, where the ___________ factors have no real zeros

polynomial, linear, quadratic

how can you tell if an equation has 4 test regions

find solutions and make sure there are 3 roots

how do you find the horizontal asymptote if numerator’s degree is < denominator’s degree

y=0

it is the x-axis

a set of ordered pairs

relation

between two consecutive zeros, a polynomial must be entirely ____ or entirely _____

positive, negative

are solutions in interval notation ordered pairs

no

how do you find the HA if you numerator’s degree > denominator’s degree by 1?

by 2?

by 1: y=quotient slant asymptote

by 2: y=quotient parabolic asymptote

when f(x)—>b as x—>+/- infinity, y=b is a ______ ______ of the graph of f

horizontal asymptote

how do you find x intercepts and y intercepts of a rational function

x: roots of numerator (set=0) make sure to factor first in case things cancel

y: plug 0 in for x

NOTE- be sure to write as ordered pairs, (0,0) is not a y or x intercept

functions of the form f(x)=N(x)/D(x), where N(x) and D(x) are polynomials and D(x) is not the zero polynomial are called ______ _______

rational functions

the _______ is divided by the _______ which gives you a _____ with a ________

dividend, divisor, quotient, remainder

describe the behavior of the following graph

f(x)=7x^2-x^7

left up, right down (be careful with highest degree)

how do you find roots of f(x)=x^3-x^2+2

factors of p/ factors of q

+1, -1, +2, -2/+1, -1

test -1,1,2

The graphs of all polynomial functions are _______, which means that the graphs have no breaks, holes, or gaps

continuous

A ______ function is a second degree polynomial function, and its graph is called _________

quadratic, parabola

The _______ _________ Theorem states that if f is a polynomial function such that f(a) does not = f(b), then, in the interval [a,b], f takes on every value between f(a) and f(b)

Intermediate Value

when the graph of a quadratic function opens upward, its leading coefficient is ______ and the vertex is a _______

positive, minimum

what does upper bound mean

when you plug in a number into synthetic division and you get all positive numbers as a result, that will be your highest root

The ________ __________ _____ is used to determine the left-hand and right-hand behavior of the graph of a polynomial function

leading coefficient test

x^2+1 has ____ real solutions

no

what is standard form for complex numbers

a+bi

Critical points include

Relative max and mins and intercepts

imaginary roots come in _______, meaning a cubic function will have 2 roots with possible combinations ____…….

pairs,

3 real, 0 imaginary

1 real, 2 imaginary

NEVER 2 real, 1 imaginary

what is a rational number?

irrational?

- fraction using integers (4/7, 10) –decimal will repeat or terminate
- decimal that will never repeat nor terminate (pi, root2)

a set of ordered pairs

relation

The imaginary unit i is defined as i=_________, where i^2=___________

square root -1, -1

a+bi

if a=0, what number do you have?

pure imaginary

what is the pattern for imaginary numbers starting with i^1 and ending with I^4, before repeating itself over and over….

i (or, root -1), -1, -i, 1

what is the complex conjugate of root 6

root 6

The numbers a + bi and a-bi are called _________ ________ and their product is a real number a^2 + b^2

complex conjugates

The graph of a quadratic function is symmetric about its ____________________.

axis of symmetry

zeros of a polynomial are also called …..

solutions, factors, x-intercepts, roots

In the Division Algorithm, the rational expression f(x)/d(x) is _____ because the degree of f(x) is greater than or equal to the degree of d(x)

improper

how does a quartic graph behave

both sides go up/down

what is the standard quadratic function

f(x)=ax^2+bx+c

a cannot =0, c y intercept

an alternative method to long division of polynomials is calles _______ ______, in which the divisor must be of the form x-k

synthetic division (only when divisor is linear)

how do you rationalize 3+2i/6-i

multiply top and bottom by 6+i

A polynomial function is written in _______ form when its terms are written in descending order of exponents from left to right

standard

what is the position function (meters)

s(t)=-4.9t^2+Vot+So

all polynomial functions are ____ and _____

continuous, curvy

A _____ _______ number has the form a+bi, where a=0 and b does not equal 0

pure imaginary

When the graph of a quadratic function opens downward, its leading coefficient is _________ and the vertex of the graph is a _________

negative, maximum

you can only use synthetic division with __ functions

linear

A polynomial function of x with degree n, must have a __________ ___________ degree n and a _______ x

positive integer, real

what is the standard form for the equation of a parabola

f(x)=a(x-h)^2+k

a cannot =0

vertex (h,k)

When a is a positive real number, the _____ _____ root of -a ois defined as square root -a=square root ai

principal square

how many points of inflection can an equation have

degree-2

if f(a) (y value) <0 and f(b) >0 then in between a and b exists an _____

x-intercept

how do you find the vertex of an equation

-b/2a

plug result in for y part

The ______ Theorem states that a polynomial f(x) has a factor (x-k) iff f(k)=0

Factor

A factor (x-a)^k, k>1, yields a _______ _________ x=a of ______ k

repeated zero, multiplicity

how would you do

i^44+i^150-i^74-i^109+i^61

divide each power by 4, is remainder is 1, result is i, if the remainder is 2, result is -1, if remainder is 3, result is -i, if remainder is 0, result is 1

when writing complex numbers in standard form, what do you ALWAYS do first

convert to imaginary

ex) root -3 times root -12

=root 3i times root 12i

=-6

what is the conjugate of i

-i

describe the graph of g(x)=(3x)^2 +1

horizontal shrink, up 1

dividend=quotient*divisor+remainder

division algorithm

Intermediate value theorem is an ______ theorem

Existence

In the Division Algorithm, the rational expression r(x)/d(x)is _____ because the degree of r(x) is less than the degree of d(x)

proper

what 3 solutions do you test first with synthetic division

-1,1,2

When a real zero of a polynomial function is of even multiplicity, the graph of f _________ the x-axis at x=a, and when it is of odd multiplicity, the graph of f ______ the x-axis at x=a

bounces (touches) crosses

what are points of inflection

where concavity changes

What are extrema and extremum

Relative max and mins

True or False, i is a variable

False- it is not a variable, but treat it like it is when performing operations

when you test a positive number and get all positives in quotient, you have a __________ ____________

Upper bound

The _____ Theorem states that if a polynomial f(x)is divided by x-k, then the remainder is r=f(k)

Remainder

A _________ number has the form a+bi, where a does not equal 0 and b=0

real

how do you complete the square

make sure function is in standard quadratic form

group variable terms

make a=1 or factor out a

add (1/2b)^2

balance function (subtract ^ on outside)

factor trinomial (perfect square trinomial=2 same binomials)

rewrite in standard for for equation of parabola*REMEMBER–WHEN BALANCING, IF YOU FACTORED OUT AN A BEFOREHAND, CONSIDER THE ACTUAL VALUE TO SUBTRACT ON THE OUTSIDE*

equation of a parabola= f(x)=2(x+3/4)^2 -65/8 what is vertex? axis of symmetry? x intercepts? y intercepts? image to y-intercept?

(-3/4, -65/8)

x=-3/4 (remember to include x=)

x intercepts- use quadratic formula or plug in 0 for y (original equation)

y intercepts- plug in 0 for x (original equation)

imagine- (-3/2, -7) (same y value, cross over axis)

dividend/divisor=quotient+remainder/divisor

alternative (division) algorithm

intermediate value theorem is also called ____________ ______________

existence theorem

what is the position function? (feet)

s(t)=-16t^2+Vot+So

- 16 is constant
- V is initial velocity
- S is initial position in feet
- t is time

every complex number has a ______ number (a) and an _____ number (b)

real, imaginary

Linear, constant, and squaring functions are examples of ____ functions

polynomial

A polynomial function of degree n has at most ___real zeroes and at most _____ turning points

n, n-1

are absolute value functions polynomials?

no- they are not curvy

what is the complex conjugate of (3+2i)

(3-2i)

- same exact term with a sign change in the middle*
- things will cancel*

When x=a is a zero of a polynomial function f, the following three statements are true:

a) x=a is a ______ of the polynomial equation f(x)=0

b) _____ is a factor of the polynomial f(x)

c) (a,0) is an _____ of the graph of f

zero, (x-a), x-intercept

A(n) ______ number has the form a+bi, where a does not equal 0 and b does not equal 0

imaginary

quadratic functions-

degree 2 polynomial

graph: parabola

cubic/linear is an example of a _________ rational expression

improper

when writing an imaginary quotient in standard form, what cannot be in the denominator

i, rationalize it

how would you do

2/1+i)-(3/1-i

do (2[1-i])-(3[1+i])

_______________

(1+i)(1-i)

what are the logarithmic models?

y=a+bLnx

y=a+blogx

what would you do with

y=2^-x^2

take the square of the number first then multiply by the implied negative one

the exponential function f(x)=e^x is called the ______ _____ function, and the base e is called the ______ base

natural exponential, natural

what is the product property of logs

goes to addition

log4X*y^2=log4X+log4Y^2

half life equations are decay, meaning k is

negative

logarithmic graphs have a ____ asymptote (__=#) while exponential graphs of a ____ asymptote (__=#)

vertical x

horizontal y

while doing the change-of-base formula, do you use the natural log or the common log

it does not matter as long as you are consistent

you can use the _______ Property to solve simple exponential equations

one-to-one

like terms have the same ____ and same _________

base, exponent

the domain of the natural logarithmic function is the set of __________ ________ __________.

positive real numbers

all real numbers such that x is greater than 0

is e a variable?

no, it is a constant

what is the equation for exponential growth?

decay?

what does each variable stand for

A=Ie^kt and A=le^-kt A is what you have (dependent) I is your initial amount e is 2.71828 (constant/base/multiplier) k is constant of exponentialism (constant coefficient/growth rate) and t is time (independent)

to evaluate a logarithm to any base, use the ______ formula

change-of-base

the inverse properties of logarithms state that logaA^x=x and

a^logaX=X

polynomial functions are examples of ___ functions

algebraic

the logarithmic function f(x)+lnx is called the ______ logarithmic function and has base _________

natural, e

how do you solve exponential or logarithmic (or any, really) equation

use inverse operations

logarithmic and exponential equations are _______

inverses

a logistic growth model has the form

y=a/1+be^-rx

if you have a log that can be taken by reducing the number, what do you do

reduce it, multiply it by the number you reduced it by, separate by addition, and finish

when solving an equation, it is important to check for _____________ by plugging your solutions back into the original equation

extraneous

to find the amount A in an account after t years with principal P and an annual interest rate r compounded continuously, you can use the formula _____________.

A=Pe^rt

in probability and statistics, Gaussian models commonly represent populations that are ___________ _____________.

normally distributed

how would you simplify 3^x-2

3^x3^-2

3^x(1/9)

the inverse function of the exponential function f(x)=a^x is called the ______ function with base a

logarithmic

for like terms with the same base and exponent, when multiplying, add exponents but ______________________ the bases

do not do anything with the bases

a logarithm is an _____

exponent

how you know if an equation is exponential

it has a variable as an exponent

what is the quotient property of logs

to difference

log7 x^3/y=log7X^3-log7Y

how can you make powers roots and roots powers

the cubed root equals ^1/3 and so on

what does e=

2.71821

when simplifying logs, make sure to use parentheses between subtraction and addition and also make sure that the log of a certain number–make sure that number cannot be divided by anything to get a whole number answer. if it can, multiply and add based on rules of expansion

..

what is the power property of logs

logx^4=4*logx

(make sure you only bring it out if it is for the WHOLE THING

exponential and logarithmic functions are examples of nonalgebraic functions, also called _____ functions

transcendental

your equation is done when _ is by itself

x

when solving an exponential equation, take the ___ of both sides

log

to one-to-one property of natural logarithms states that if Inx=Iny, then_______________

x=y

asymptotes begin with _________________________.

x= or y=

An exponential growth model has the form ______, and an exponential decay model has the form_________.

y=ae^bx or A=Ie^kt

y=ae^-bx or A=Ie^-kt

remember to use ________ when you are solving logs to indicate separation

parentheses

to find the amount A in an account after t years with principal P and an annual interest rate r compounded n times per year, you can use the formula______________.

A=P(1+r/n)^nt

what is true of all logarithmic graphs?

how do you restrict the domain?

how do they look?

they all pass vertical line test, always have x-intercept and asymptote

you cannot take the log of 0 or a negative number

boomerang

you can consider logaX to be a constant multiple of logbX; the constant multiplier is ______________

1/logbA

the parent log has an assumed base of ____

the natural log has an assumed base of __

10 (common log)

e

what is the log of 625 to base 5

what is the log of .001

4, -4

when you are told to find an exponential model, make sure you have all variables expect____

the x and y, or the A and t

the common logarithmic function has base

10

the change-of-base formula for base a is given by logaX=__________

logbX/logbA

describe the graph

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

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

what is (f-g)(0)

f(0)-g(0)

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)

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

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

y=x

what is f^-1(x)

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

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)

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

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

composition

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)

f(1)=4 really is

an ordered pair (1,4)

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

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

line of regression

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

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

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

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

initial condition

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

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

what is the composition of functions

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

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

inverse

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

test

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

The inverse function of f is denoted by

f^-1

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

relation

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

example?

nonrigid transformations

dilation

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])

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

a constant function can be horizontal and vertical

true or false

false–ONLY HORIZONTAL

y varies directly as x

y is directly proportional to x

y=kx for some nonzero constant k

direct variation

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

sum, square differences

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

variation, regression

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

what are the 4 types of functions

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

what is a defined function

one that has a domain of all real numbers

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

range, domain

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

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

is (f of g)(x) muliplication

No

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

(4t)^2=16t

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

constant, variation

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

linear

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

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

what kinds of transformations are included

rigid transformation

translations, reflections

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

inverse, direct

what is mathematic modeling

coming up with the equation

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

restrict the domain (x>_ 0)

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

one to one

only __________ have an inverse function

one to one

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

what is k in y=kx?

constant of variation, also the rate

Inverse operations _________ each other

undo

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

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)

what are the three types of transformations

what are the four types of translations

translation, reflection, dilation

up, down, left, right

f(x)=square root x

parent radical function

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

addition, subtraction, multiplication, division

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])

what is the equation for Interest

I=P*r*t

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

To reflect over x axis, make _ values negative

Vice versa

y

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

rigid

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

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

f(x)=x^2

parent quadratic function

f(x)=1/x

parent rational function (reciprocal function)

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

domains

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

inverse

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

(1,4)

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)

inverse variation says as one gets bigger, _________________________

the other gets smaller

what are transcendental functions

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

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

one-to-one

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

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

directly proportional

f(x)= [[x]]

greatest integer function (or, step function)

f(x)=x

identity function

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

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

Horizontal

z=kxy

joint variation (z varies jointly as x and y)

what is a piecewise function

a function with pieces (normally 2 or 3)

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

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

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)

what is a model?

an equation

what would the graph x=2 look like?

y=3?

vertical line through 2

horizontal through 3

what is a piecewise function

a function with multiple equations, each with designated rules

In calculus, one of the basic definitions is that of a ___________, given f(x+h)-f(x)/h, h cannot =0

difference quotient

true or false– if an equation has a y value which produces 2 x values, it is NOT a function

false- although an x value cannot produce two or more y values, a y value can produce two or more x values

The _______ of a function f are the values of x for which f(x)=0

zeroes (roots, x-intercepts)

The ______ ______ _______ _______ between any two points (x1, f(x1)) and (x2, f(x2)) is the slope of the line through the two points, and this lines is called the _____ line

average rate of change, secant

what are points of inflection

where concavity changes

translation means ____

slide

what axis does your independent variable (x) go on? (y)?

x, y

what is slope intercept form

y=mx+b

when you construct and use a table to solve a problem, you are using a ______ approach

graphic

what is a function

a specific relation saying your x value can never produce two or more y values

what is a secant line

a line that intersects two points

When the x-axis and the y-axis have different units of measure, the slope can be interpreted as a ______

rate

which way do you view a graph to tell if it is increasing or decreasing

left to right

An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the ____ plane

cartesian

The ________ ________ ________ is used to determine whether the graph of an equation is a function of y in terms of x

Vertical Line Test

The set of all solution points of an equation is the ___ of the equation

graph

how would you find the equation of a line parallel to y=3x-7 through (4,1)

your slope is 3, then plug in 4 for x, 1 for y, and 3 for m (in slope intercept) and solve for b.

OR–plug (4,1) in point slope form with m as 3

If the domain of the function f is not given, then the set of values of the independent variable for which the expression is defined is called the ___________.

implied domain

what is the equation of a circle

(x-h)^2+(y-k)^2=r^2

center (h,k)

radius r

when writing an equation as a function of another equation, look for

shared variables

A function f is ______ on an interval when, for any x1 and x2 in the interval, x1 is less than x2 implies f(x1) is greater than f(x2)

decreasing

how do you know if an equation’s graph has x-axis symm? y-axis? origin?

x- plug in negative for y, must be the same

y- plug in negative for x, must be the same

origin- plug in - for both, must be the same

how do you find average rate of change

F(new x) - F(old x)/new x-old x

the ___________ is a result derived from the pythagorean theorem

distance formula

height=

length=

top-bottom

right-left

if the graph of a function was a) even or b) odd. what ordered pair could also lie on that graph if (-5/3, -7) did

a) (5/3, -7)

b) (5/3, 7)

The points at which a graph intersects or touches an axis are called the ______ of the graph

intercepts

distance formula is derived from ______

pythagorean theorem

what does { mean?

: ?

set, such that

Two lines are ______ iff their slopes are negative reciprocals of each other

perpendicular

what is percentage increase (or decrease)

amount increase/original amount

an odd exponent (x^3) signifies what

that the graph will got through each intercept (no repeats)

A graph is symmetric with respect to the _____if, whenever (x,y) is on the graph, (-x,y) is also on the graph

y-axis

For an equation that represents y as a function of x, the set of all values taken on by the ________ variable x is the domain, and the set of all values taken on by the ______ variable is the range

independent, dependent

how do you solve a difference quotient

make two large brackets, plug in when they are asking you to. simplify and solve

The prediction method __________ is the method used to estimate a point on a line when the point does not lie between the given points

linear extrapolation

if your function is not odd or even, what is it? what kind of symmetry does this kind of function have

neither–no symmetry

What is point slope form

y2-y1=m(x2-x1)

what is domain

set of all values that the independent variable (usually x) can be

what does an open point signify on a graph

that breaks one of the rules in your piecewise function rules

how would you express that your domain can be all real numbers greater than -2 using interval notation

Domain= {R’s: (-2, [infinity sign])}

Two lines are ______ iff their slopes are equal

parallel

when a graph is constant, you always express the ordered pair with _______

brackets [ ]

Every line has an equation that can be written in __ form

standard

intercepts are written in

ordered pairs

true or false- a function with a square root cannot have a domain that is the set of all real numbers

false– the x value could be raised to an even root underneath the square root

i.e. y=square root of x^2

Polynomial functions are _________ and ___________

continuous and curvy

how do you express that the domain of a function is x greater than or equal to 1 in interval notation. What notation is it in above?

[-1, infinity) inequality

Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the

midpoint formula

A relation that assigns to each element x from a set of inputs, or _________, exactly one element y in a set of outputs, or ______, is called a ______.

domain, range, function

An ordered pair (a,b) is a ______ of an equation in x and y when the substitutions x=a and y=b result in a true statement

solution

The point of intersection of the x axis and the y axis is the ________, and the two axes divide the coordinate plane into four parts called ____

origin, quadrants

what is implied domain

values acceptable for x for a certain function (given)

usually in in piecewise

what does relation mean

set of ordered pairs (mapping, x/y chart, etc.)

what is the variable under the radical called

radicand

what is the range

the set of all values that the dependent variable (usually y) can be

the simplest mathematical model for relating two variables is the ____ equation in two variables y=mx+b

linear

what is (x^2-1) said to be

difference of perfect squares

what is intercept form

x/a + y/b = 1

when a does not equal nor b does not equal 0

what is another name for the coordinate system, where does it come from

cartesian plane, Rene Descarte

what is slope formula

y2-y1/x2-x1

A function f is _____ when, for each x in the domain of f, f(-x)=-f(x)

odd

The _____________ form of the equation of a line with slope m passing through (x1, y1) is y1-y2=m(x2-x1)

point slope

what are the types of transitions

dilation, reflection, rotation

what is f(1)=4 in simpler terms?

(1,4) an ordered pair

What is standard form

Ax+By=C

what is the parent function for absolute value?

y=IxI

How can you tell if a function is even? What kind of symmetry does it have?

Plug in -x for x, and your equation will remain exactly the same {{f(x)}}

Reflection symmetry over the y-axis

For a line, the ratio of the change in y to the change in x is called the __ of the line

slope

how do you know if a function is odd? What kind of symmetry does it have?

Plug in -x for x and your equation will be -f(x) (exact opposite). It has rotational symmetry 180 degrees about the origin

what is the difference between intercepts, roots, and zeroes?

no difference

how do you determine if an equation represents y as a function of x

set it equal to y–determine if one x value could produce two y values

A ________________ is a location on a graph where your line stops increasing and starts decreasing

relative max