1EXAM- HPreCalc Flashcards Preview

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Flashcards in 1EXAM- HPreCalc Deck (335):
0

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

1

3 key numbers means

4 test regions

2

do you put brackets around solutions that make the bottom 0

no

3

is x^2 + 49 factorable?

yes
(x-7i) (x+7i)

4

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

5

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

profit=revenue-cost

6

key numbers are ____

whatever makes the top and bottom 0

7

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

8

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

9

what is intermediate form? is it the same as undefined

0/0, no

10

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

11

All graphs _____ the asymptotes

Follow

12

Increasing and decreasing intervals use the ___ values

X (watch asymptotes)

13

what is upper bound

you plug in a higher number and get all positive sums in return
this will be your highest zero

14

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

..

15

When finding key points of rational functions ___ first

Simplify

16

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

17

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

18

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

19

slant asymptotes and parabolic asymptotes all begin with

y=
so do HA
VA begins with x=

20

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

conjugate

21

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

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

22

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

It's down 60 going up eternally

23

Increasing and decreasing intervals use the ___ values

X (watch asymptotes)

24

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

25

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

critical/key, test intervals

26

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

27

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

vertical asymptote

28

what does lower bound say

you plug in a negative number and get alternating sums in return,
this will be your lowest zero

29

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

Yes, they are still linear

30

what does irreducible over the reals mean

does not reduce into real factors
x^2-2x+10

31

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

rational zero

32

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 =

33

what are the possible root combinations for a cubed equation

all real, 2 imaginary and one real

34

factor x^3+8

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

35

Can you cross through a HA? VA? SA?

Yes no yes

36

if 5i is a zero, what else is true

-5i is a zero

37

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

flip

38

key values with solutions in the middle are called ____ points

border

39

What is root 3i times root 3i

3i^2
-3

40

special cases are also called ______

unusual solution sets

41

what is the radicand

the polynomial under the root

42

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

43

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

zeros, undefined values

44

what is a rational function

fraction with polynomials

45

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

46

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

47

how can you tell if an equation has 4 test regions

find solutions and make sure there are 3 roots

48

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

y=0
it is the x-axis

49

a set of ordered pairs

relation

50

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

positive, negative

51

are solutions in interval notation ordered pairs

no

52

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

53

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

horizontal asymptote

54

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

55

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

56

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

dividend, divisor, quotient, remainder

57

describe the behavior of the following graph
f(x)=7x^2-x^7

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

58

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

59

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

continuous

60

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

quadratic, parabola

61

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

62

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

positive, minimum

63

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

64

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

leading coefficient test

65

x^2+1 has ____ real solutions

no

66

what is standard form for complex numbers

a+bi

67

Critical points include

Relative max and mins and intercepts

68

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

69

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)

70

a set of ordered pairs

relation

71

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

square root -1, -1

72

a+bi
if a=0, what number do you have?

pure imaginary

73

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

74

what is the complex conjugate of root 6

root 6

75

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

complex conjugates

76

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

axis of symmetry

77

zeros of a polynomial are also called .....

solutions, factors, x-intercepts, roots

78

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

79

how does a quartic graph behave

both sides go up/down

80

what is the standard quadratic function

f(x)=ax^2+bx+c
a cannot =0, c y intercept

81

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)

82

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

multiply top and bottom by 6+i

83

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

standard

84

what is the position function (meters)

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

85

all polynomial functions are ____ and _____

continuous, curvy

86

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

pure imaginary

87

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

negative, maximum

88

you can only use synthetic division with __ functions

linear

89

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

positive integer, real

90

what is the standard form for the equation of a parabola

f(x)=a(x-h)^2+k
a cannot =0
vertex (h,k)

91

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

principal square

92

how many points of inflection can an equation have

degree-2

93

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

x-intercept

94

how do you find the vertex of an equation

-b/2a
plug result in for y part

95

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

Factor

96

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

repeated zero, multiplicity

97

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

98

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

99

what is the conjugate of i

-i

100

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

horizontal shrink, up 1

101

dividend=quotient*divisor+remainder

division algorithm

102

Intermediate value theorem is an ______ theorem

Existence

103

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

104

what 3 solutions do you test first with synthetic division

-1,1,2

105

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

106

what are points of inflection

where concavity changes

107

What are extrema and extremum

Relative max and mins

108

True or False, i is a variable

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

109

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

Upper bound

110

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

Remainder

111

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

real

112

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*

113

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)

114

dividend/divisor=quotient+remainder/divisor

alternative (division) algorithm

115

intermediate value theorem is also called ____________ ______________

existence theorem

116

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

117

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

real, imaginary

118

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

polynomial

119

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

n, n-1

120

are absolute value functions polynomials?

no- they are not curvy

121

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

(3-2i)
*same exact term with a sign change in the middle*
*things will cancel*

122

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

123

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

imaginary

124

quadratic functions-

degree 2 polynomial
graph: parabola

125

cubic/linear is an example of a _________ rational expression

improper

126

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

i, rationalize it

127

how would you do
(2/1+i)-(3/1-i)

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

128

what are the logarithmic models?

y=a+bLnx
y=a+blogx

129

what would you do with
y=2^-x^2

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

130

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

natural exponential, natural

131

what is the product property of logs

goes to addition
log4X*y^2=log4X+log4Y^2

132

half life equations are decay, meaning k is

negative

133

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

vertical x
horizontal y

134

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

135

you can use the _______ Property to solve simple exponential equations

one-to-one

136

like terms have the same ____ and same _________

base, exponent

137

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

positive real numbers
all real numbers such that x is greater than 0

138

is e a variable?

no, it is a constant

139

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)

140

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

change-of-base

141

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

a^logaX=X

142

polynomial functions are examples of ___ functions

algebraic

143

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

natural, e

144

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

use inverse operations

145

logarithmic and exponential equations are _______

inverses

146

a logistic growth model has the form

y=a/1+be^-rx

147

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

148

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

extraneous

149

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

150

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

normally distributed

151

how would you simplify 3^x-2

3^x3^-2
3^x(1/9)

152

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

logarithmic

153

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

do not do anything with the bases

154

a logarithm is an _____

exponent

155

how you know if an equation is exponential

it has a variable as an exponent

156

what is the quotient property of logs

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

157

how can you make powers roots and roots powers

the cubed root equals ^1/3 and so on

158

what does e=

2.71821

159

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

..

160

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

161

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

transcendental

162

your equation is done when _ is by itself

x

163

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

log

164

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

x=y

165

asymptotes begin with _________________________.

x= or y=

166

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

167

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

parentheses

168

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

169

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

170

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

1/logbA

171

the parent log has an assumed base of ____
the natural log has an assumed base of __

10 (common log)
e

172

what is the log of 625 to base 5
what is the log of .001

4, -4

173

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

174

the common logarithmic function has base

10

175

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

logbX/logbA

176

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

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

177

what is (f-g)(0)

f(0)-g(0)

178

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)

179

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

180

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

y=x

181

what is f^-1(x)

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

182

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)

183

an inverse cannot fail (Horizontal/Vertical) line test

I think just Vertical???

184

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

range, domain

185

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

composition

186

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

187

f(1)=4 really is

an ordered pair (1,4)

188

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

189

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

line of regression

190

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

191

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

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

192

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

initial condition

193

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

194

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

195

what is the composition of functions

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

196

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

inverse

197

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

test

198

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

199

The inverse function of f is denoted by

f^-1

200

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

relation

201

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

nonrigid transformations
dilation

202

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])

203

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

204

a constant function can be horizontal and vertical
true or false

false--ONLY HORIZONTAL

205

y varies directly as x
y is directly proportional to x
y=kx for some nonzero constant k

direct variation

206

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

sum, square differences

207

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

variation, regression

208

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

vertical stretch, vertical shrink

209

what are the 4 types of functions

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

210

what is a defined function

one that has a domain of all real numbers

211

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

range, domain

212

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

213

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

214

is (f of g)(x) muliplication

No

215

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

(4t)^2=16t

216

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

constant, variation

217

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

linear

218

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

219

when the shapes are congruent or unchanged after a transformation, the transformation is a _________
what kinds of transformations are included

rigid transformation
translations, reflections

220

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

inverse, direct

221

what is mathematic modeling

coming up with the equation

222

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

restrict the domain (x>_ 0)

223

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

one to one

224

only __________ have an inverse function

one to one

225

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

226

what is k in y=kx?

constant of variation, also the rate

227

Inverse operations _________ each other

undo

228

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

229

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)

230

what are the three types of transformations
what are the four types of translations

translation, reflection, dilation
up, down, left, right

231

f(x)=square root x

parent radical function

232

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

addition, subtraction, multiplication, division

233

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])

234

what is the equation for Interest

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

235

To reflect over x axis, make _ values negative
(Vice versa)

y

236

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

rigid

237

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

238

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

239

f(x)=x^2

parent quadratic function

240

f(x)=1/x

parent rational function (reciprocal function)

241

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

domains

242

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

inverse

243

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

(1,4)

244

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)

245

inverse variation says as one gets bigger, _________________________

the other gets smaller

246

what are transcendental functions

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

247

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

one-to-one

248

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)

249

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

directly proportional

250

f(x)= [[x]]

greatest integer function (or, step function)

251

f(x)=x

identity function

252

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

253

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

Horizontal

254

z=kxy

joint variation (z varies jointly as x and y)

255

what is a piecewise function

a function with pieces (normally 2 or 3)

256

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

257

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

258

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)

259

what is a model?

an equation

260

what would the graph x=2 look like?
y=3?

vertical line through 2
horizontal through 3

261

what is a piecewise function

a function with multiple equations, each with designated rules

262

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

difference quotient

263

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

264

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

zeroes (roots, x-intercepts)

265

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

266

what are points of inflection

where concavity changes

267

translation means ____

slide

268

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

x, y

269

what is slope intercept form

y=mx+b

270

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

graphic

271

what is a function

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

272

what is a secant line

a line that intersects two points

273

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

rate

274

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

left to right

275

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

cartesian

276

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

Vertical Line Test

277

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

graph

278

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

279

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

280

what is the equation of a circle

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

281

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

shared variables

282

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

283

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

284

how do you find average rate of change

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

285

the ___________ is a result derived from the pythagorean theorem

distance formula

286

height=
length=

top-bottom
right-left

287

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)

288

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

intercepts

289

distance formula is derived from ______

pythagorean theorem

290

what does { mean?
: ?

set, such that

291

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

perpendicular

292

what is percentage increase (or decrease)

amount increase/original amount

293

an odd exponent (x^3) signifies what

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

294

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

295

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

296

how do you solve a difference quotient

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

297

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

298

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

neither--no symmetry

299

What is point slope form

y2-y1=m(x2-x1)

300

what is domain

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

301

what does an open point signify on a graph

that breaks one of the rules in your piecewise function rules

302

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

Domain= {R's: (-2, [infinity sign])}

303

Two lines are ______ iff their slopes are equal

parallel

304

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

brackets [ ]

305

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

standard

306

intercepts are written in

ordered pairs

307

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

308

Polynomial functions are _________ and ___________

continuous and curvy

309

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

310

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

311

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

312

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

313

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

314

what is implied domain

values acceptable for x for a certain function (given)
usually in in piecewise

315

what does relation mean

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

316

what is the variable under the radical called

radicand

317

what is the range

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

318

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

linear

319

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

difference of perfect squares

320

what is intercept form

x/a + y/b = 1
when a does not equal nor b does not equal 0

321

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

cartesian plane, Rene Descarte

322

what is slope formula

y2-y1/x2-x1

323

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

odd

324

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

point slope

325

what are the types of transitions

dilation, reflection, rotation

326

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

(1,4) an ordered pair

327

What is standard form

Ax+By=C

328

what is the parent function for absolute value?

y=IxI

329

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

330

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

slope

331

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

332

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

no difference

333

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

334

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

relative max