PreCalc Chapter 2 Test

Question 0

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

Answer 0

imaginary

Question 1

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

Answer 1

real

Question 2

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

Answer 2

pure imaginary

Question 3

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

Answer 3

square root -1, -1

Question 4

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

Answer 4

principal square

Question 5

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

Answer 5

complex conjugates

Question 6

what is the complex conjugate of root 6

Answer 6

root 6

Question 7

what is the conjugate of i

Answer 7

-i

Question 8

how would you do

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

Answer 8

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

Question 9

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

Answer 9

i, rationalize it

Question 10

how would you do

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

Answer 10

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

Question 11

x^2+1 has ____ real solutions

Answer 11

no

Question 12

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

Answer 12

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

Question 13

what is standard form for complex numbers

Answer 13

a+bi

Question 14

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

Answer 14

real, imaginary

Question 15

a+bi

if a=0, what number do you have?

Answer 15

pure imaginary

Question 16

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

Answer 16

(3-2i)

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

Question 17

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

Answer 17

multiply top and bottom by 6+i

Question 18

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

Answer 18

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

Question 19

what 3 solutions do you test first with synthetic division

Answer 19

-1,1,2

Question 20

what does upper bound mean

Answer 20

when you plug in a number into synthetic division and you get all positive numbers as a result, that will be your highest root

Question 21

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

Answer 21

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

Question 22

True or False, i is a variable

Answer 22

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

Question 23

a set of ordered pairs

Answer 23

relation

Question 24

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

Answer 24

continuous

Question 25

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

Answer 25

leading coefficient test

Question 26

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

Answer 26

n, n-1

Question 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

Answer 27

zero, (x-a), x-intercept

Question 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

Answer 28

bounces (touches) crosses

Question 29

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

Answer 29

repeated zero, multiplicity

Question 30

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

Answer 30

standard

Question 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)

Answer 31

Intermediate Value

Question 32

how does a quartic graph behave

Answer 32

both sides go up/down

Question 33

all polynomial functions are ____ and _____

Answer 33

continuous, curvy

Question 34

are absolute value functions polynomials?

Answer 34

no- they are not curvy

Question 35

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

Answer 35

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

Question 36

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

Answer 36

solutions, factors, x-intercepts, roots

Question 37

what are points of inflection

Answer 37

where concavity changes

Question 38

how many points of inflection can an equation have

Answer 38

degree-2

Question 39

intermediate value theorem is also called ____________ ______________

Answer 39

existence theorem

Question 40

what is a rational number? | irrational?

Answer 40

- fraction using integers (4/7, 10) --decimal will repeat or terminate - decimal that will never repeat nor terminate (pi, root2)

Question 41

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

Answer 41

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

Question 42

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

Answer 42

x-intercept

Question 43

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

Answer 43

polynomial

Question 44

you can only use synthetic division with __ functions

Answer 44

linear

Question 45

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

Answer 45

positive integer, real

Question 46

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

Answer 46

quadratic, parabola

Question 47

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

Answer 47

axis of symmetry

Question 48

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

Answer 48

negative, maximum

Question 49

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

Answer 49

positive, minimum

Question 50

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

Answer 50

horizontal shrink, up 1

Question 51

how do you find the vertex of an equation

Answer 51

-b/2a | plug result in for y part

Question 52

quadratic functions-

Answer 52

degree 2 polynomial | graph: parabola

Question 53

what is the standard quadratic function

Answer 53

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

Question 54

what is the standard form for the equation of a parabola

Answer 54

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

Question 55

how do you complete the square

Answer 55

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*

Question 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? ```

Answer 56

(-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)

Question 57

what is the position function? (feet)

Answer 57

s(t)=-16t^2+Vot+So - 16 is constant - V is initial velocity - S is initial position in feet - t is time

Question 58

what is the position function (meters)

Answer 58

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

Question 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)

Answer 59

proper

Question 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)

Answer 60

improper

Question 61

cubic/linear is an example of a _________ rational expression

Answer 61

improper

Question 62

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

Answer 62

synthetic division (only when divisor is linear)

Question 63

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

Answer 63

Factor

Question 64

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

Answer 64

Remainder

Question 65

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

Answer 65

dividend, divisor, quotient, remainder

Question 66

dividend=quotient*divisor+remainder

Answer 66

division algorithm

Question 67

dividend/divisor=quotient+remainder/divisor

Answer 67

alternative (division) algorithm

Question 68

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

Answer 68

Upper bound

Question 69

What are extrema and extremum

Answer 69

Relative max and mins

Question 70

Critical points include

Answer 70

Relative max and mins and intercepts

Question 71

Intermediate value theorem is an ______ theorem

Answer 71

Existence