Math 1050 Ch 2-4

Front 1

A function can have 2 inputs with the same output? (T or F)

Back 1

True

Front 2

A function can have 2 outputs for one input?

(T or F)

Back 2

False

Front 3

What is the domain of a function?

Back 3

The set of all acceptable (X) Inputs

Front 4

How do you find the domain of a rational function?

Back 4

1.) Factor denominator (if factor-able)

2.) Set denominator = 0 and solve

3.) Solutions are numbers that X cannot be

Front 5

What is the natural domain?

Back 5

(-∞,∞) (All real numbers)

Front 6

How do you find the domain of a radical function?

Back 6

1.) See if the index is odd or even

2.) If index is odd the domain is R

3.) If index is even then set the radicand greater than or equal to 0

4.) Solve and write answer in interval notation

Front 7

How do you find the domain of a rational function with a radical denominator having an even index?

Back 7

1.) Set radicand greater than but NOT EQUAL TO 0

2.) Solve and write answer in interval notation

Front 8

Express the function in words. f(x)=2x+3

Back 8

Multiply by 2 then add 3

Front 9

How do you solve a piecewise defined function?

Back 9

Determine which rule X falls into on the "if" side and use the corresponding function to determine the output

Front 10

What is the rule for calculating net change?

Back 10

The difference from A to B where A < or = B

is calculated by: f(B) - f(A) = Dif

Front 11

How do you graph a function?

Back 11

Plot the points (x,f(x)) in other words (x,y). Whose X coordinate is the input and whose y coordinate is the output.

Front 12

What is a linear function?

Back 12

A function of the form f(x)=ax+b with a graph that is a straight line

Front 13

How do you graph power and root functions?

Back 13

Plot points by using a table of x y coordinates. Test various x coordinates and find the output (y coordinate)

Front 14

What type of functions are these?

Back 14

Power Functions

Front 15

What type of function is this?

Back 15

a root function

Front 16

What type of function is this?

Back 16

Absolute value function

Front 17

The graph of an absolute value can be flat?

(T or F)

Back 17

True

Front 18

What does a SOLID dot mean in the graph of a piecewise defined function?

Back 18

The point is included in that line (≤ or ≥)

Front 19

What does a HOLLOW dot mean in the graph of a piecwise defined function?

Back 19

The point is NOT included in that line (>or<)

Front 20

How do you determine if the graph is the graph of a function?

Back 20

The vertical line test

Front 21

Does this graph pass the vertical line test?

Back 21

No

Front 22

What is |-10| equal to?

Back 22

10

Front 23

What is the rule for converting a rational exponent into a radical?

Back 23

Front 24

The solutions of the equation f(x)=0 are the __ intercepts

Back 24

X (what is x when y is 0?)

Front 25

How do you find the domain of a function from it's graph?

Back 25

[ Leftmost point , rightmost point ]

Front 26

How do you find the range of a function from its graph?

Back 26

[ lowest point , highest point ]

Front 27

What will this inequality find?: h(x)≤3

Back 27

the domain for all parts of the graph below or equal to 3

Front 28

What is a local maximum?

Back 28

A point which all points 'near' to are less than

Front 29

A _____ _____ is a point where all points 'near' it are higher than?

Back 29

Local minimum

Front 30

Explain this function in words: f(x)=x²+1

Back 30

The graph of x² shifted up one

Front 31

What is the function for the graph of x² shifted to the left three?

Back 31

f(x)= (x+3)²

Front 32

What is the function for the graph of x² shifted to the right five?

Back 32

f(x)= (x-5)²

Front 33

How do you reflect a graph on the x axis?

Back 33

y=-f(x)

(minus the function of x)

Front 34

How do you reflect a graph on the y axis?

Back 34

y=f(-x)

(the function of negative x)

Front 35

How do you STRETCH a graph vertically?

Back 35

If c>1 then stretch the graph vertically (steepen) by a factor of c

By graphing y=cf(x)

Front 36

How do you SHRINK a graph vertically?

Back 36

if 0

Then graph y=cf(x)

ex:

f(x)=x²

h(x)=1/2x²

Front 37

How do you SHRINK a graph horizontally?

Back 37

if c>1 then:

By graphing y=f(cx)

Front 38

How do you STRETCH a graph horizontally?

Back 38

if 0

Then graph y=f(cx)

Front 39

What is a 1:1 function?

Back 39

A function that has a unique output for every x input (2 inputs cannot have the same output)

Front 40

How do you determine if a function is 1:1

Back 40

Horizontal line test

Front 41

How do you find the INVERSE of a function?

Back 41

1.) Replace f(x) with y

2.) Solve for x

3.) Swap x and y

Front 42

How do you find the inverse of a RATIONAL function?

Back 42

1.) multiply both sides by denominator

2.) distribute

3.) move x terms to LHS

4.) factor x from LHS

5.) divide both sides by the factors of x to solve

6.) swap x and y

Front 43

Average rate of change is the _____ of the ______ line between two points.

Back 43

Slope, Secant

Front 44

The domain of f(g(x)) is:

Back 44

The set of acceptable inputs that f and g both share

Front 45

How do you put a quadratic into standard form?

Back 45

1.) Solve -b/2a to identify x coordinate of vertex

2.) Put solution for x back into original equation to find y coordinate of vertex

3.) put into form f(x)=a(x-h)²+k where h=x coordinate and k = y coordinate of the vertex

Front 46

Find the __ Intercept(s) of a quadratic by solving for f(0)

Back 46

Y

Front 47

Find the x intercepts of a quadratic by finding the ________ of the quadratic. Either by the __ factor theorem, or by the _________ formula.

Back 47

Zeros, 0, Quadratic

Front 48

If the y intercept of a quadratic is 0 then one x intercept MUST be ___ ?

Back 48

0

Front 49

What is the best thing to do after putting a quadratic into standard form, and why?

Back 49

Plot the vertex and determine if the graph opens upwards or downwards. Because this will tell you if there are x intercepts or not.

Front 50

if a>0 then the parabola opens ________

Back 50

Upwards

Front 51

if a<0 then the parabola opens __________

Back 51

Downwards

Front 52

When solving a quadratic you should always solve for the __________ BEFORE simplifying the fraction

Back 52

Discriminant

Front 53

A negative discriminant means that there are __ real solutions for x

Back 53

0 (no x intercepts)

Front 54

What is the degree of a polynomial?

Back 54

The highest power

Front 55

Describe the end behaviors of x²

Back 55

y goes to ∞ as x goes to -∞

y goes to ∞ as x goes to ∞

Front 56

How do you determine the end behaviors of a polynomial using the first term?

Back 56

If the degree is odd, y will go the same direction (+ or -) as x goes.

if the leading coefficient is negative, switch the signs of the direction that y goes

Front 57

If a polynomial has the factors or "zeros" x=-1, 3, and 5. This means that it intercepts the __ axis at ___, ___, and ___

Back 57

-1,3,5

Front 58

How do you find whether the graph of a polynomial is above or below 0 between the zeros?

Back 58

1.) make a number line with each zero plotted

2.) choose test points for each region

3.) test each point using the polynomial

4.) if x>0 then it is above, if x<0 then it is below

Front 59

If you factor a polynomial into a quadratic, you can use the ______ ___________ to find the zeros

Back 59

Quadratic formula

Front 60

What is the Quadratic formula?

Back 60

Front 61

If a rational function has a radical numerator, then to determine its domain you must account for this by _________________.

Back 61

Following the same steps as for a radical denominator.

Front 62

What are the zeros of this polynomial?

P(x) = (x-2)(x-3)(x+4)

Back 62

2 , 3 , -4

Front 63

How do you find the zeros of a polynomial?

Back 63

1.) factor into irreducible factors

2.) set each factor equal to zero

3.) solve for variable

Front 64

How do you find the zeros of a polynomial that is not easily factorable?

Back 64

Use the rational zeros theorem

Front 65

How do you use the rational zeros theorem?

Back 65

1.) list all possible rational zeros by dividing every factor of the constant term by every factor of the leading coefficient

2.) use synthetic division to test each rational zero that you obtain in step one. when the remainder is zero x - rational zero is a factor

3.) repeat steps one and two for the quotient, stop when you reach a quotient that factors easily or is quadratic.

Front 66

Use ______ division to test rational zeros

Back 66

synthetic

Front 67

when using synthetic division what do you do if you are dividing a polynomial that does not have a variable with every descending exponent from degree to 0?

Back 67

put a 0 in its place

Front 68

What is the zero x = 4/3 in factored form?

Back 68

(3x-4)

Front 69

Does the quadratic formula output zeros or factors?

Back 69

Zeros

Front 70

According to descares rule of signs, the number of positive real zeros is equal to the number of ____________ or is less than that by an ______ whole number

Back 70

Even, whole

Front 71

According to descares rule of signs, the number of negative real zeros is equal to the number of sign variations in P(__) or is ________ that by an even whole number

Back 71

P(-x), less than

Front 72

According to descares rule of signs if 2 is a possible amount of zeros then __ is also possible because it is neither _______ or ________

Back 72

0, positive , negative

Front 73

The upper bounds theorem states that if you divide P(x) by x-b when b>0 using synthetic division and the row containing the quotient has _____________ then it is an upper bound.

Back 73

No negative entry

Front 74

the lower bounds theorem states that if you divide P(x) by x-b when x<0 using synthetic division and the row containing the quotient has entries alternating negative and positive then it is a _____________.

Back 74

Lower Bound

Front 75

What do you do when things gets weird?

Back 75

Try another method

Front 76

0 is neither ______ or ______

Back 76

Positive , negative

Front 77

What is the square root of -4?

Back 77

2i

Front 78

What is the pattern for the exponents of imaginary numbers?

Back 78

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

Front 79

a zero with an i number is considered _______.

Back 79

complex

Front 80

If a zero is factored out of a polynomial 3 times then it has a ________ of 3.

Back 80

multiplicity

Front 81

every polynomial of degree n > or = 1 has exactly __ zeros, provided that a zero of _____ k is counted k times.

Back 81

n , multiplicity

Front 82

What does the conjugate zeros theorem state?

Back 82

If a polynomial P has all real coefficients and the complex number z is a zero of p then its complex conjugate is also a zero of p

Front 83

How do you convert the complex zero 1 + i into a factor, and then find its conjugate?

Back 83

1.) convert into a factor

x - 1 + i

2.) convert into conjugate

x - 1 - i

Front 84

to adjust the constant term you must _______________ all terms _______________.

Back 84

Multiply all terms by the number that you multiplied the constant term by.

Front 85

evaluate √-3

Back 85

i√3

Front 86

A complex zero is always ___ because of the conjugate zeros theorem.

Back 86

±

Front 87

If a polynomial has complex coefficients then what theorem is no longer true?

Back 87

conjugate zeros theorem

Front 88

The domain of a rational function includes all numbers except those which ____________________________________

Back 88

Make the denominator zero

Front 89

How do you find the vertical asymptote(s) of a rational function?

Back 89

The zeros of the denominator are the vertical asymptotes

Front 90

The horizontal asymptote is the constant term of the numerator / constant term of the denominator. (T or F)

Back 90

False

Front 91

The horizontal asymptote is the leading coefficient of numerator / leading coefficient of denominator. (T or F)

Back 91

True

Front 92

What are the steps to graphing a rational function?

Back 92

1.) factor numerator and denominator

2.) Find X and Y intercepts

3.) Find vertical asymptotes

4.) Find horizontal asymptotes

5.) plot these and test points, then sketch graph

Front 93

The x intercepts of a rational function are the ___________________.

Back 93

Zeros of the numerator

Front 94

How do you determine the y intercept of a rational function?

Back 94

It is the output of P(0) - or - constant of numerator / constant of denominator

Front 95

The graph of a rational function never touches the _________

Back 95

asymptote

Front 96

the horizontal asymptote is y = 0 when:

Back 96

The degree of the numerator is less than the degree of the denominator by 1

Front 97

there is no horizontal asymptote if:

Back 97

The degree of the numerator is greater than the degree of the denominator

Front 98

If there is no constant in a 2nd degree polynomial then you must use the __________ instead of the rational zeros theorem

Back 98

Quadratic fomula

Front 99

When can there be a "hole" in the asymptote?

Back 99

If there is a common factor between numerator and denominator then the zero of that factor is a hole.

Front 100

If (x-1) is the only factor of the denominator but is also a factor of the numerator, then there is ______________________.

Back 100

No vertical asymptote

Front 101

To determine the range of a rational function with common factors plug the ________________ into the function.

Back 101

zero of the common factor

Front 102

If 0 is a vertical asymptote then ___ is the y intercept.

Back 102

There is no y intercept

Front 103

What are the steps for solving a polynomial inequality?

Back 103

1.) Move all terms to one side

2.) Factor the polynomial

3.) Get the zeros of each factor

4.) plot each zero on a number line to determine intervals

5.) create a sign diagram to determine which intervals satisfy the inequality and write answer in interval notation. using brackets for endpoints if inequality uses ≤ or ≥

Front 104

What are the steps for solving a rational polynomial inequality>

Back 104

1.) Move all terms to one side

2.) Factor the num and denom

3.) Get the zeros of each factor

4.) plot each zero on a number line to determine intervals

5.) create a sign diagram to determine which intervals satisfy the inequality and write answer in interval notation. using brackets for endpoints if inequality uses ≤ or ≥ BUT ONLY FOR ZEROS FROM FACTORS OF THE NUMERATOR, IF ZERO OF DENOMINATOR ALWAYS USE PARENTHESES

Front 105

How do you use a sign diagram?

Back 105

create a grid with ZEROS ON TOP and FACTORS TO THE LEFT. then use imaginary test points between each zero for every factor and put the sign results in the grid. after completed use rules of multiplication going and signs (going downwards) to determine which intervals satisfy the inequality.

Front 106

How do you find the intervals for which the graph of f is greater than the graph of g

Back 106

1.) set f(x)>g(x)

2.) move all terms to LHS and solve like normal

Front 107

What is an exponential function?

Back 107

A function where the independent variable is an exponent

Front 108

what is the rule for negative exponents?

Back 108

Front 109

A rational exponent can also be written as a ______.

Back 109

radical

Front 110

What is the rule for converting a radical into a rational exponent?

Back 110

Front 111

Explain the transformation of -3^x

Back 111

the graph of 3^x reflected on the x axis

Front 112

explain the graph of 2-(1/3)^x

Back 112

The graph of (1/3)^x reflected on the x axis, shifted up 2

Front 113

What is the order of transformations?

Back 113

1.) (H)orizontal shift

2.) (S)tretching/Shrinking

3.) (R)eflections

4.) (V)ertical shift

Front 114

What is the formula for CONSTANT compounding interest?

Back 114

Front 115

The log is the _________ of an exponential function

Back 115

Inverse

Front 116

Log base 10 is the same as _____

Back 116

Log

Front 117

ln means ____ of _. or natural log.

Back 117

Log base e

Front 118

What is the exponential form of this log?

Back 118

Front 119

Explain the reason for this logarithmic property.

Back 119

We must raise b to the power of 0 to get 1

Front 120

Explain the reason for this logarithmic property.

Back 120

We must raise b to the power of 1 to get b

Front 121

Explain the reason for this logarithmic property.

Back 121

we must raise b to the power of x to get b^x

Front 122

The exponent of a log goes _________.

Back 122

In front

Front 123

an exponent of 1/2 is the same as ___________.

Back 123

The square root

Front 124

Laws of logarithms state that this is = to ____.

Back 124

Front 125

Laws of logarithms state that this is = to ____.

Back 125

Front 126

Laws of logarithms state that this is = to ____.

Back 126

Front 127

(T or F) log(A+B) is the same as log A + log B

Back 127

False

Front 128

(T or F) log A + log B is the same as log AB

Back 128

True

Front 129

(T or F) log A/B is the same as log A - log B

Back 129

True

Front 130

(T or F) log A/B is the same as log B - log A

Back 130

False (First term must be the numerator)

Front 131

(T or F) when adding two logs together you should do exponentiation last.

Back 131

False, you should do it first and ONLY to the term the exponent came from.

Front 132

Explain the change of base formula

Back 132

Front 133

In the change of base formula, the numerator and denominator share the log __ or ___. the base of the original log goes to the ______.

Back 133

ln or common log, denominator

Front 134

Solutions for e^x must be _______.

Back 134

positive