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

3 key numbers means

A

4 test regions

1
Q

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

A

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

2
Q

do you put brackets around solutions that make the bottom 0

A

no

3
Q

is x^2 + 49 factorable?

A

yes

x-7i) (x+7i

4
Q

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 ________

A

Descartes’s Rule, Signs

5
Q

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

A

profit=revenue-cost

6
Q

key numbers are ____

A

whatever makes the top and bottom 0

7
Q

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

A

y=ratio of leading coefficients

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

8
Q

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

A

linear factorization theorem

9
Q

what is intermediate form? is it the same as undefined

A

0/0, no

10
Q

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

A

fundamental, theorem, algebra

11
Q

All graphs _____ the asymptotes

A

Follow

12
Q

Increasing and decreasing intervals use the ___ values

A

X (watch asymptotes)

13
Q

what is upper bound

A

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

14
Q

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

A

..

15
Q

When finding key points of rational functions ___ first

A

Simplify

16
Q

how do you find the vertical asymptote

A

whatever makes the denominator 0
roots of denominator
x=
be sure to factor function first as asymptotes might cancel

17
Q

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) ________

A

slant asymptote

18
Q

how do you factor a perfect cube

A

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
Q

slant asymptotes and parabolic asymptotes all begin with

A

y=
so do HA
VA begins with x=

20
Q

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

A

conjugate

21
Q

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

A

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

22
Q

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

A

It’s down 60 going up eternally

23
Q

Increasing and decreasing intervals use the ___ values

A

X (watch asymptotes)

24
Q

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

A

lower, upper

25
Q

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

A

critical/key, test intervals

26
Q

what is DesCartes’s Rule of signs

A

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
Q

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

A

vertical asymptote

28
Q

what does lower bound say

A

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

29
Q

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

A

Yes, they are still linear

30
Q

what does irreducible over the reals mean

A

does not reduce into real factors

x^2-2x+10

31
Q

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

A

rational zero

32
Q

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

A

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

33
Q

what are the possible root combinations for a cubed equation

A

all real, 2 imaginary and one real

34
Q

factor x^3+8

A

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

35
Q

Can you cross through a HA? VA? SA?

A

Yes no yes

36
Q

if 5i is a zero, what else is true

A

-5i is a zero

37
Q

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

A

flip

38
Q

key values with solutions in the middle are called ____ points

A

border

39
Q

What is root 3i times root 3i

A

3i^2

-3

40
Q

special cases are also called ______

A

unusual solution sets

41
Q

what is the radicand

A

the polynomial under the root

42
Q

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

A

irreducible, reals

43
Q

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

A

zeros, undefined values

44
Q

what is a rational function

A

fraction with polynomials

45
Q

what is the rational zero test

A

p(constant)/q (leading coefficient

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

46
Q

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

A

polynomial, linear, quadratic

47
Q

how can you tell if an equation has 4 test regions

A

find solutions and make sure there are 3 roots

48
Q

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

A

y=0

it is the x-axis

49
Q

a set of ordered pairs

A

relation

50
Q

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

A

positive, negative

51
Q

are solutions in interval notation ordered pairs

A

no

52
Q

how do you find the HA if you numerator’s degree > denominator’s degree by 1?
by 2?

A

by 1: y=quotient slant asymptote

by 2: y=quotient parabolic asymptote

53
Q

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

A

horizontal asymptote

54
Q

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

A

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
Q

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 ______ _______

A

rational functions

56
Q

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

A

dividend, divisor, quotient, remainder

57
Q

describe the behavior of the following graph

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

A

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

58
Q

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

A

factors of p/ factors of q
+1, -1, +2, -2/+1, -1

test -1,1,2

59
Q

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

A

continuous

60
Q

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

A

quadratic, parabola

61
Q

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)

A

Intermediate Value

62
Q

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

A

positive, minimum

63
Q

what does upper bound mean

A

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
Q

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

A

leading coefficient test

65
Q

x^2+1 has ____ real solutions

A

no

66
Q

what is standard form for complex numbers

A

a+bi

67
Q

Critical points include

A

Relative max and mins and intercepts

68
Q

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

A

pairs,
3 real, 0 imaginary
1 real, 2 imaginary
NEVER 2 real, 1 imaginary

69
Q

what is a rational number?

irrational?

A
  • fraction using integers (4/7, 10) –decimal will repeat or terminate
  • decimal that will never repeat nor terminate (pi, root2)
70
Q

a set of ordered pairs

A

relation

71
Q

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

A

square root -1, -1

72
Q

a+bi

if a=0, what number do you have?

A

pure imaginary

73
Q

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

A

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

74
Q

what is the complex conjugate of root 6

A

root 6

75
Q

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

A

complex conjugates

76
Q

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

A

axis of symmetry

77
Q

zeros of a polynomial are also called …..

A

solutions, factors, x-intercepts, roots

78
Q

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)

A

improper

79
Q

how does a quartic graph behave

A

both sides go up/down

80
Q

what is the standard quadratic function

A

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

a cannot =0, c y intercept

81
Q

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

A

synthetic division (only when divisor is linear)

82
Q

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

A

multiply top and bottom by 6+i

83
Q

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

A

standard

84
Q

what is the position function (meters)

A

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

85
Q

all polynomial functions are ____ and _____

A

continuous, curvy

86
Q

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

A

pure imaginary

87
Q

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

A

negative, maximum

88
Q

you can only use synthetic division with __ functions

A

linear

89
Q

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

A

positive integer, real

90
Q

what is the standard form for the equation of a parabola

A

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

91
Q

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

A

principal square

92
Q

how many points of inflection can an equation have

A

degree-2

93
Q

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

A

x-intercept

94
Q

how do you find the vertex of an equation

A

-b/2a

plug result in for y part

95
Q

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

A

Factor

96
Q

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

A

repeated zero, multiplicity

97
Q

how would you do

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

A

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
Q

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

A

convert to imaginary
ex) root -3 times root -12
=root 3i times root 12i
=-6

99
Q

what is the conjugate of i

A

-i

100
Q

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

A

horizontal shrink, up 1

101
Q

dividend=quotient*divisor+remainder

A

division algorithm

102
Q

Intermediate value theorem is an ______ theorem

A

Existence

103
Q

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)

A

proper

104
Q

what 3 solutions do you test first with synthetic division

A

-1,1,2

105
Q

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

A

bounces (touches) crosses

106
Q

what are points of inflection

A

where concavity changes

107
Q

What are extrema and extremum

A

Relative max and mins

108
Q

True or False, i is a variable

A

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

109
Q

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

A

Upper bound

110
Q

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

A

Remainder

111
Q

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

A

real

112
Q

how do you complete the square

A

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
Q
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?
A

(-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
Q

dividend/divisor=quotient+remainder/divisor

A

alternative (division) algorithm

115
Q

intermediate value theorem is also called ____________ ______________

A

existence theorem

116
Q

what is the position function? (feet)

A

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

  • 16 is constant
  • V is initial velocity
  • S is initial position in feet
  • t is time
117
Q

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

A

real, imaginary

118
Q

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

A

polynomial

119
Q

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

A

n, n-1

120
Q

are absolute value functions polynomials?

A

no- they are not curvy

121
Q

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

A

(3-2i)

  • same exact term with a sign change in the middle*
  • things will cancel*
122
Q

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

A

zero, (x-a), x-intercept

123
Q

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

A

imaginary

124
Q

quadratic functions-

A

degree 2 polynomial

graph: parabola

125
Q

cubic/linear is an example of a _________ rational expression

A

improper

126
Q

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

A

i, rationalize it

127
Q

how would you do

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

A

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

128
Q

what are the logarithmic models?

A

y=a+bLnx

y=a+blogx

129
Q

what would you do with

y=2^-x^2

A

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

130
Q

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

A

natural exponential, natural

131
Q

what is the product property of logs

A

goes to addition

log4X*y^2=log4X+log4Y^2

132
Q

half life equations are decay, meaning k is

A

negative

133
Q

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

A

vertical x

horizontal y

134
Q

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

A

it does not matter as long as you are consistent

135
Q

you can use the _______ Property to solve simple exponential equations

A

one-to-one

136
Q

like terms have the same ____ and same _________

A

base, exponent

137
Q

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

A

positive real numbers

all real numbers such that x is greater than 0

138
Q

is e a variable?

A

no, it is a constant

139
Q

what is the equation for exponential growth?
decay?
what does each variable stand for

A
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
Q

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

A

change-of-base

141
Q

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

A

a^logaX=X

142
Q

polynomial functions are examples of ___ functions

A

algebraic

143
Q

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

A

natural, e

144
Q

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

A

use inverse operations

145
Q

logarithmic and exponential equations are _______

A

inverses

146
Q

a logistic growth model has the form

A

y=a/1+be^-rx

147
Q

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

A

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

148
Q

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

A

extraneous

149
Q

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

A=Pe^rt

150
Q

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

A

normally distributed

151
Q

how would you simplify 3^x-2

A

3^x3^-2

3^x(1/9)

152
Q

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

A

logarithmic

153
Q

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

A

do not do anything with the bases

154
Q

a logarithm is an _____

A

exponent

155
Q

how you know if an equation is exponential

A

it has a variable as an exponent

156
Q

what is the quotient property of logs

A

to difference

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

157
Q

how can you make powers roots and roots powers

A

the cubed root equals ^1/3 and so on

158
Q

what does e=

A

2.71821

159
Q

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

A

..

160
Q

what is the power property of logs

A

logx^4=4*logx

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

161
Q

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

A

transcendental

162
Q

your equation is done when _ is by itself

A

x

163
Q

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

A

log

164
Q

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

A

x=y

165
Q

asymptotes begin with _________________________.

A

x= or y=

166
Q

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

A

y=ae^bx or A=Ie^kt

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

167
Q

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

A

parentheses

168
Q

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

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

169
Q

what is true of all logarithmic graphs?
how do you restrict the domain?
how do they look?

A

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
Q

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

A

1/logbA

171
Q

the parent log has an assumed base of ____

the natural log has an assumed base of __

A

10 (common log)

e

172
Q

what is the log of 625 to base 5

what is the log of .001

A

4, -4

173
Q

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

A

the x and y, or the A and t

174
Q

the common logarithmic function has base

A

10

175
Q

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

A

logbX/logbA

176
Q

describe the graph

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

A

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

177
Q

what is (f-g)(0)

A

f(0)-g(0)

178
Q

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

x-axis?

A

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

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

179
Q

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

A

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

180
Q

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

A

y=x

181
Q

what is f^-1(x)

A

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

182
Q

g(x)=(x-1)^3 +2
what is the parent function?
use function notation to write g in terms of f

A

f(x)=x^3
g(x)=f(x-1) +2
***Remember not to include the ^3, as that is implied in f(x)

183
Q

an inverse cannot fail (Horizontal/Vertical) line test

A

I think just Vertical???

184
Q

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

the range of a function = the _____ of its inverse

A

range, domain

185
Q

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

A

composition

186
Q

when will a function vertically stretch?

Shrink?

A

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

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

187
Q

f(1)=4 really is

A

an ordered pair (1,4)

188
Q

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

divide these

A

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
Q

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

A

line of regression

190
Q
how do you decompose a composite function?
decompose h(x)=1/(x-2)^2
A

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
Q

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

A

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

192
Q

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

A

initial condition

193
Q

what does a graph’s inverse do

what kind of symmetry do they have

A

switches x and y

reflectional symmetry over the line y=x

194
Q

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

A
use composite functions (analytically)
plug g(x) into f(x)  [f(g(x))] and you will get x
195
Q

what is the composition of functions

A

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

196
Q

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

A

inverse

197
Q

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

A

test

198
Q

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?

A

correlation coefficient

0, 1

199
Q

The inverse function of f is denoted by

A

f^-1

200
Q

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

A

relation

201
Q

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

A

nonrigid transformations

dilation

202
Q

is composition of functions commutative? what does this mean?

A

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

203
Q

what is the equation for state income tax

what kind of variation does it have

A

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

204
Q

a constant function can be horizontal and vertical

true or false

A

false–ONLY HORIZONTAL

205
Q

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

A

direct variation

206
Q

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

A

sum, square differences

207
Q

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

A

variation, regression

208
Q

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

A

vertical stretch, vertical shrink

209
Q

what are the 4 types of functions

A

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

210
Q

what is a defined function

A

one that has a domain of all real numbers

211
Q

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

A

range, domain

212
Q

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

A

directly proportional

213
Q

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

A

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

214
Q

is (f of g)(x) muliplication

A

No

215
Q

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

A

(4t)^2=16t

216
Q

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

A

constant, variation

217
Q

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

A

linear

218
Q
describe the translation
f(x)+c
f(x)-c
f(x+c)
f(x-c)
A

c units up
c units down
c units left
c units right

219
Q

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

A

rigid transformation

translations, reflections

220
Q

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

A

inverse, direct

221
Q

what is mathematic modeling

A

coming up with the equation

222
Q

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

A

restrict the domain (x>_ 0)

223
Q

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

A

one to one

224
Q

only __________ have an inverse function

A

one to one

225
Q

describe the graph

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

A

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

226
Q

what is k in y=kx?

A

constant of variation, also the rate

227
Q

Inverse operations _________ each other

A

undo

228
Q

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

A

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
Q

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)=__________

A

-f(x)

f(-x)

230
Q

what are the three types of transformations

what are the four types of translations

A

translation, reflection, dilation

up, down, left, right

231
Q

f(x)=square root x

A

parent radical function

232
Q

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

A

addition, subtraction, multiplication, division

233
Q

S=4pi r^2

how would you use variation terminology to say this aloud

A

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
Q

what is the equation for Interest

A

I=Prt

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

235
Q

To reflect over x axis, make _ values negative

Vice versa

A

y

236
Q

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

A

rigid

237
Q

Be careful with distributing negatives in reflection cases

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

A

-square root of -x-6

238
Q

what kind of variation will these ordered pairs have?

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

A

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

239
Q

f(x)=x^2

A

parent quadratic function

240
Q

f(x)=1/x

A

parent rational function (reciprocal function)

241
Q

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

A

domains

242
Q

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

A

inverse

243
Q

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

A

(1,4)

244
Q

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

A

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
Q

inverse variation says as one gets bigger, _________________________

A

the other gets smaller

246
Q

what are transcendental functions

A

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

247
Q

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

A

one-to-one

248
Q

the greatest integer function takes the next integer ______

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

A

-4, 2

it is also called the round-down function by some

249
Q

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

A

directly proportional

250
Q

f(x)= [[x]]

A

greatest integer function (or, step function)

251
Q

f(x)=x

A

identity function

252
Q

a linear equation will always have an x intercept and a y intercept
true or false

A

false-constant functions will not have an x intercept

253
Q

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

A

Horizontal

254
Q

z=kxy

A

joint variation (z varies jointly as x and y)

255
Q

what is a piecewise function

A

a function with pieces (normally 2 or 3)

256
Q

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”

A

directly proportional

257
Q

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

A

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

258
Q

z varies directly with the square of x and inversely with y with a constant variation of 2/3
how would you write this?

A

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

259
Q

what is a model?

A

an equation

260
Q

what would the graph x=2 look like?

y=3?

A

vertical line through 2

horizontal through 3

261
Q

what is a piecewise function

A

a function with multiple equations, each with designated rules

262
Q

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

A

difference quotient

263
Q

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

A

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

264
Q

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

A

zeroes (roots, x-intercepts)

265
Q

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

A

average rate of change, secant

266
Q

what are points of inflection

A

where concavity changes

267
Q

translation means ____

A

slide

268
Q

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

A

x, y

269
Q

what is slope intercept form

A

y=mx+b

270
Q

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

A

graphic

271
Q

what is a function

A

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

272
Q

what is a secant line

A

a line that intersects two points

273
Q

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

A

rate

274
Q

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

A

left to right

275
Q

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

A

cartesian

276
Q

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

A

Vertical Line Test

277
Q

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

A

graph

278
Q

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

A

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
Q

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 ___________.

A

implied domain

280
Q

what is the equation of a circle

A

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

281
Q

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

A

shared variables

282
Q

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)

A

decreasing

283
Q

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

A

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
Q

how do you find average rate of change

A

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

285
Q

the ___________ is a result derived from the pythagorean theorem

A

distance formula

286
Q

height=

length=

A

top-bottom

right-left

287
Q

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

a) (5/3, -7)

b) (5/3, 7)

288
Q

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

A

intercepts

289
Q

distance formula is derived from ______

A

pythagorean theorem

290
Q

what does { mean?

: ?

A

set, such that

291
Q

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

A

perpendicular

292
Q

what is percentage increase (or decrease)

A

amount increase/original amount

293
Q

an odd exponent (x^3) signifies what

A

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

294
Q

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

A

y-axis

295
Q

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

A

independent, dependent

296
Q

how do you solve a difference quotient

A

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

297
Q

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

A

linear extrapolation

298
Q

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

A

neither–no symmetry

299
Q

What is point slope form

A

y2-y1=m(x2-x1)

300
Q

what is domain

A

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

301
Q

what does an open point signify on a graph

A

that breaks one of the rules in your piecewise function rules

302
Q

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

A

Domain= {R’s: (-2, [infinity sign])}

303
Q

Two lines are ______ iff their slopes are equal

A

parallel

304
Q

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

A

brackets [ ]

305
Q

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

A

standard

306
Q

intercepts are written in

A

ordered pairs

307
Q

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

A

false– the x value could be raised to an even root underneath the square root
i.e. y=square root of x^2

308
Q

Polynomial functions are _________ and ___________

A

continuous and curvy

309
Q

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?

A

[-1, infinity) inequality

310
Q

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

A

midpoint formula

311
Q

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 ______.

A

domain, range, function

312
Q

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

A

solution

313
Q

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 ____

A

origin, quadrants

314
Q

what is implied domain

A

values acceptable for x for a certain function (given)

usually in in piecewise

315
Q

what does relation mean

A

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

316
Q

what is the variable under the radical called

A

radicand

317
Q

what is the range

A

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

318
Q

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

A

linear

319
Q

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

A

difference of perfect squares

320
Q

what is intercept form

A

x/a + y/b = 1

when a does not equal nor b does not equal 0

321
Q

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

A

cartesian plane, Rene Descarte

322
Q

what is slope formula

A

y2-y1/x2-x1

323
Q

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

A

odd

324
Q

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

A

point slope

325
Q

what are the types of transitions

A

dilation, reflection, rotation

326
Q

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

A

(1,4) an ordered pair

327
Q

What is standard form

A

Ax+By=C

328
Q

what is the parent function for absolute value?

A

y=IxI

329
Q

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

A

Plug in -x for x, and your equation will remain exactly the same {{f(x)}}
Reflection symmetry over the y-axis

330
Q

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

A

slope

331
Q

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

A

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

332
Q

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

A

no difference

333
Q

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

A

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

334
Q

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

A

relative max