PreCalc Chapter 2 Test Flashcards Preview

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Flashcards in PreCalc Chapter 2 Test Deck (72)
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0
Q

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

A

imaginary

1
Q

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

A

real

2
Q

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

A

pure imaginary

3
Q

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

A

square root -1, -1

4
Q

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

A

principal square

5
Q

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

A

complex conjugates

6
Q

what is the complex conjugate of root 6

A

root 6

7
Q

what is the conjugate of i

A

-i

8
Q

how would you do

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

A

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

9
Q

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

A

i, rationalize it

10
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

11
Q

x^2+1 has ____ real solutions

A

no

12
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

13
Q

what is standard form for complex numbers

A

a+bi

14
Q

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

A

real, imaginary

15
Q

a+bi

if a=0, what number do you have?

A

pure imaginary

16
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*
17
Q

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

A

multiply top and bottom by 6+i

18
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

19
Q

what 3 solutions do you test first with synthetic division

A

-1,1,2

20
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

21
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

22
Q

True or False, i is a variable

A

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

23
Q

a set of ordered pairs

A

relation

24
Q

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

A

continuous

25
Q

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

A

leading coefficient test

26
Q

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

A

n, n-1

27
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

28
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

29
Q

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

A

repeated zero, multiplicity

30
Q

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

A

standard

31
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

32
Q

how does a quartic graph behave

A

both sides go up/down

33
Q

all polynomial functions are ____ and _____

A

continuous, curvy

34
Q

are absolute value functions polynomials?

A

no- they are not curvy

35
Q

describe the behavior of the following graph

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

A

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

36
Q

zeros of a polynomial are also called …..

A

solutions, factors, x-intercepts, roots

37
Q

what are points of inflection

A

where concavity changes

38
Q

how many points of inflection can an equation have

A

degree-2

39
Q

intermediate value theorem is also called ____________ ______________

A

existence theorem

40
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)
41
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

42
Q

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

A

x-intercept

43
Q

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

A

polynomial

44
Q

you can only use synthetic division with __ functions

A

linear

45
Q

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

A

positive integer, real

46
Q

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

A

quadratic, parabola

47
Q

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

A

axis of symmetry

48
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

49
Q

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

A

positive, minimum

50
Q

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

A

horizontal shrink, up 1

51
Q

how do you find the vertex of an equation

A

-b/2a

plug result in for y part

52
Q

quadratic functions-

A

degree 2 polynomial

graph: parabola

53
Q

what is the standard quadratic function

A

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

a cannot =0, c y intercept

54
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)

55
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

56
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)

57
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
58
Q

what is the position function (meters)

A

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

59
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

60
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

61
Q

cubic/linear is an example of a _________ rational expression

A

improper

62
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)

63
Q

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

A

Factor

64
Q

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

A

Remainder

65
Q

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

A

dividend, divisor, quotient, remainder

66
Q

dividend=quotient*divisor+remainder

A

division algorithm

67
Q

dividend/divisor=quotient+remainder/divisor

A

alternative (division) algorithm

68
Q

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

A

Upper bound

69
Q

What are extrema and extremum

A

Relative max and mins

70
Q

Critical points include

A

Relative max and mins and intercepts

71
Q

Intermediate value theorem is an ______ theorem

A

Existence