PreCalc Chapter 2 Test Flashcards
A(n) ______ number has the form a+bi, where a does not equal 0 and b does not equal 0
imaginary
A _________ number has the form a+bi, where a does not equal 0 and b=0
real
A _____ _______ number has the form a+bi, where a=0 and b does not equal 0
pure imaginary
The imaginary unit i is defined as i=_________, where i^2=___________
square root -1, -1
When a is a positive real number, the _____ _____ root of -a ois defined as square root -a=square root ai
principal square
The numbers a + bi and a-bi are called _________ ________ and their product is a real number a^2 + b^2
complex conjugates
what is the complex conjugate of root 6
root 6
what is the conjugate of i
-i
how would you do
2/1+i)-(3/1-i
do (2[1-i])-(3[1+i])
_______________
(1+i)(1-i)
when writing an imaginary quotient in standard form, what cannot be in the denominator
i, rationalize it
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
x^2+1 has ____ real solutions
no
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
what is standard form for complex numbers
a+bi
every complex number has a ______ number (a) and an _____ number (b)
real, imaginary
a+bi
if a=0, what number do you have?
pure imaginary
what is the complex conjugate of (3+2i)
(3-2i)
- same exact term with a sign change in the middle*
- things will cancel*
how do you rationalize 3+2i/6-i
multiply top and bottom by 6+i
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
what 3 solutions do you test first with synthetic division
-1,1,2
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
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
True or False, i is a variable
False- it is not a variable, but treat it like it is when performing operations
a set of ordered pairs
relation
The graphs of all polynomial functions are _______, which means that the graphs have no breaks, holes, or gaps
continuous
The ________ __________ _____ is used to determine the left-hand and right-hand behavior of the graph of a polynomial function
leading coefficient test
A polynomial function of degree n has at most ___real zeroes and at most _____ turning points
n, n-1
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
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
A factor (x-a)^k, k>1, yields a _______ _________ x=a of ______ k
repeated zero, multiplicity
A polynomial function is written in _______ form when its terms are written in descending order of exponents from left to right
standard
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
how does a quartic graph behave
both sides go up/down
all polynomial functions are ____ and _____
continuous, curvy
are absolute value functions polynomials?
no- they are not curvy
describe the behavior of the following graph
f(x)=7x^2-x^7
left up, right down (be careful with highest degree)
zeros of a polynomial are also called …..
solutions, factors, x-intercepts, roots
what are points of inflection
where concavity changes
how many points of inflection can an equation have
degree-2
intermediate value theorem is also called ____________ ______________
existence theorem
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)
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
if f(a) (y value) <0 and f(b) >0 then in between a and b exists an _____
x-intercept
Linear, constant, and squaring functions are examples of ____ functions
polynomial
you can only use synthetic division with __ functions
linear
A polynomial function of x with degree n, must have a __________ ___________ degree n and a _______ x
positive integer, real
A ______ function is a second degree polynomial function, and its graph is called _________
quadratic, parabola
The graph of a quadratic function is symmetric about its ____________________.
axis of symmetry
When the graph of a quadratic function opens downward, its leading coefficient is _________ and the vertex of the graph is a _________
negative, maximum
when the graph of a quadratic function opens upward, its leading coefficient is ______ and the vertex is a _______
positive, minimum
describe the graph of g(x)=(3x)^2 +1
horizontal shrink, up 1
how do you find the vertex of an equation
-b/2a
plug result in for y part
quadratic functions-
degree 2 polynomial
graph: parabola
what is the standard quadratic function
f(x)=ax^2+bx+c
a cannot =0, c y intercept
what is the standard form for the equation of a parabola
f(x)=a(x-h)^2+k
a cannot =0
vertex (h,k)
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
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)
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
what is the position function (meters)
s(t)=-4.9t^2+Vot+So
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
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
cubic/linear is an example of a _________ rational expression
improper
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)
The ______ Theorem states that a polynomial f(x) has a factor (x-k) iff f(k)=0
Factor
The _____ Theorem states that if a polynomial f(x)is divided by x-k, then the remainder is r=f(k)
Remainder
the _______ is divided by the _______ which gives you a _____ with a ________
dividend, divisor, quotient, remainder
dividend=quotient*divisor+remainder
division algorithm
dividend/divisor=quotient+remainder/divisor
alternative (division) algorithm
when you test a positive number and get all positives in quotient, you have a __________ ____________
Upper bound
What are extrema and extremum
Relative max and mins
Critical points include
Relative max and mins and intercepts
Intermediate value theorem is an ______ theorem
Existence
