Flashcards in PreCalc Chapter 2 Test Deck (72):

0

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

### real

1

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

### imaginary

2

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

### pure imaginary

3

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

### square root -1, -1

4

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

### principal square

5

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

### complex conjugates

6

## what is the complex conjugate of root 6

### root 6

7

## what is the conjugate of i

### -i

8

##
how would you do

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

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

_______________

(1+i)(1-i)

9

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

### i, rationalize it

10

##
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

11

## x^2+1 has ____ real solutions

### no

12

## 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

13

## what is standard form for complex numbers

### a+bi

14

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

### real, imaginary

15

##
a+bi

if a=0, what number do you have?

### pure imaginary

16

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

###
(3-2i)

*same exact term with a sign change in the middle*

*things will cancel*

17

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

### multiply top and bottom by 6+i

18

## 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

19

## what 3 solutions do you test first with synthetic division

### -1,1,2

20

## 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

21

## 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

22

## True or False, i is a variable

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

23

## a set of ordered pairs

### relation

24

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

### continuous

25

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

### leading coefficient test

26

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

### n, n-1

27

##
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

28

## 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

29

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

### repeated zero, multiplicity

30

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

### standard

31

## 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

32

## how does a quartic graph behave

### both sides go up/down

33

## all polynomial functions are ____ and _____

### continuous, curvy

34

## are absolute value functions polynomials?

### no- they are not curvy

35

##
describe the behavior of the following graph

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

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

36

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

### solutions, factors, x-intercepts, roots

37

## what are points of inflection

### where concavity changes

38

## how many points of inflection can an equation have

### degree-2

39

## intermediate value theorem is also called ____________ ______________

### existence theorem

40

##
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)

41

## 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

42

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

### x-intercept

43

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

### polynomial

44

## you can only use synthetic division with __ functions

### linear

45

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

### positive integer, real

46

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

### quadratic, parabola

47

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

### axis of symmetry

48

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

### negative, maximum

49

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

### positive, minimum

50

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

### horizontal shrink, up 1

51

## how do you find the vertex of an equation

###
-b/2a

plug result in for y part

52

## quadratic functions-

###
degree 2 polynomial

graph: parabola

53

## what is the standard quadratic function

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

a cannot =0, c y intercept

54

## what is the standard form for the equation of a parabola

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

a cannot =0

vertex (h,k)

55

## 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*

56

##
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)

57

## 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

58

## what is the position function (meters)

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

59

## 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

60

## 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

61

## cubic/linear is an example of a _________ rational expression

### improper

62

## 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)

63

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

### Factor

64

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

### Remainder

65

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

### dividend, divisor, quotient, remainder

66

## dividend=quotient*divisor+remainder

### division algorithm

67

## dividend/divisor=quotient+remainder/divisor

### alternative (division) algorithm

68

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

### Upper bound

69

## What are extrema and extremum

### Relative max and mins

70

## Critical points include

### Relative max and mins and intercepts

71