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63 Cards in this Set

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What is the general form of a power function?
y=kx^p

where k and p are any number
Are there any restrictions on what p can be for a power function y=kx^p?
no, p only has to be a number but can be any number
let y=4x^5

Then, in terms of proportionalities,
y is _________ to _______
y is proportional to x to the 5th
let y=4/(x^10)

Then, in terms of proportionalities,
y is _____________ to _________
y is inversely proportional to x to the tenth
The graphs of all power functions, y=x^p where p is an even, positive integer resemble....
y=x^2 and will be U shaped
If y=x^p where p is an odd, positive integer, then what will be the general shape of its graph?
It will be "chair" shaped and resemble y=x^3. You should graph y=x^3 and be sure to know what it looks like
If y=x^p is a power function and p is positive and even, then does y have any symmetry?
y is symmetric with respect to the y axis
If y=kx^p is a power function where p is positive and odd, then y is symmetric with respect to ...
the origin (or the line y=x)
What do all graphs of power functions with odd, negative powers resemble?
They all look like the graph of y=1/x
You should graph y=1/x and be sure to know what it looks like
What function do all power functions with negative, even integers resemble?
y=1/(x^2)
You should graph y=1/(x^2) and be sure to know what it looks like
If a power function has a positive fraction as its power and the fraction has a even denominator, what graph will this power function resemble?

Will it be defined for all x?
It will resemble y=x^(0.5)

It will only be defined for x>=0

Be sure to graph y=x^(0.5) and know what it looks like
If a power function has a positive fraction as its power and the fraction has an odd denominator, what graph will this power function resemble?

where will this power function be defined?
Resembles y=x^(1/3)

Be sure to graph y=x^(1/3) to know what it looks like.

y=x^(1/3) will be defined for all real numbers
A polynomial is a sum of________
whose exponents are ___________
power functions
non-negative integers
In a polynomial, the term with the highest power is called the _________
and the power is the _________ of the polynomial
leading term
degree
What degree is the following polynomial?

y = 4(x^20) +x^2 + 7(x^101)
The polynomial is of degree 101 since 101 is the highest degree
what is the degree of the polynomial?

y=5(x-2)(x-6)(x-4) +3
x^3
To write a polynomial in standard form, you need to ....
arrange its terms from highest power to lowest power going left to right
What is the standard form of

y=(x-1)(x+1) +x^3 +7
y=x^3 + x^2 + 6
If you let x get very very large in the positive direction, then after awhile, what will the graph of y= x^3 +x^2 resemble?
The highest order degree always wins out. Thus, the graph will resemble y=x^3
True or false, on different scales functions that are power functions may not look like power functions at all.
True
What are the zeros of a polynomial and what are three ways to find them?
The zeros are the x values that give a y value of zero. They are the x values where the graph intersects the x axis/

1. factoring
2. quadratic formula
3. completing the square
If the graph of a polynomial has n bumps, then it is most likely of degree......
n+1
If the graph of a polynomial has 3 bumps then it may be a polynomial of degree...
four
If a polynomial is of degree 10, then at most it can have how many bumps?
9
True or false, an nth degree polynomial can have n+1 zeros.
False, an nth degree polynomial has AT MOST n zeros:

(x^2)-1 is of degree 2 and has two zeros: x=1, x=-1
If the zeros of a polynomial are x=3, x=2, x=5 and there are 3 bumps in the graph and at the zero x=3, the graph bounces off the x axis, then what is a general formula for the function?
y=a(x-5)(x-2)(x-3)^2

Where the constant a can be determined by plugging in a known point on the graph
if (x-k) is a factor of a polynomial, p(x), and is repeated an odd number of times, then what is the behavior of p(x) at the point x=k?
p(x) will pass through the point x=k
If (x-k) is a factor of a polynomial and is repeated an even number of times, what is the behavior of p(x) at the point x=k?
p(x) will bounce off the x-axis at x=k
If you are given a graph of a polynomial and told to find an equation for the polynomial, what steps should you take?
1. Determine the zeros of the graph
2. Write the function as a constant times its factors.
(ex: if 1 and 3 are zeros then
y=a(x-1)(x-3))
3. Count the number of bumps in the graph to determine the degree of the polynomial
4. Determine the powers of the factors so that they correspond with the behavior at the corresponding zero and all add up to the power of the polynomial.
What is a rational function?
r(x) = p(x)/q(x)

Where p(x) and q(x) are polynomials and q(x) does not =0
If you are given a rational function, r=p(x)/q(x), how do you find:

1. the zeros of r(x)
2. Horizontal asymptotes
3. vertical asymptotes
1. Set the numerator =0 and solve for x, those x's will be your zeros
2. Take the ratio of the highest degree term on the top and bottom, this will give you your horizontal asymptote
3. Set the denominator =0 and solve for x, these are your vertical asymptotes
The graph of y=(x^2-1)/(x-1) is the graph of ....
the line y= x+1 with a hole (where it is undefined) at x=1
Any _________, ___________ exponential function eventually grows faster than any _________ ________
positive, increasing
power function
Any ___________, ___________ exponential function eventually approaches the ____________ axis faster than any _________,________ power function
positive, decreasing
x-axis
positive, decreasing
Eventually, what will grow more rapidly, a positive, increasing power function or log(x)?
power func
In the long run, does the graph of ln(x) or the graph of a positive increasing power function grow more rapidly?
the power function grows more rapidly
True or false, all quadratic functions are power functions?
True. Quadratic functions are functions of power two and thus are power functions
Is the function 3*2^(x) a power function?
No, the exponent is a variable, not a number. This function is an exponential function
let g(x) = x^p

If p is positive, even integer, then is the graph of g concave up or down?
It will look like a parabola so it will be concave up
Does the graph of 1/x pass thru the origin?
no, the origin is given when x=0. If x=0, this function is undefined.
If f(x)= 1/x, then as x gets really big (approaches + infinity) then what do the values of f(x) approach?
as x gets really big, 1/x gets really small (ex: 1/10000000 is a really small number but x=10000000 is a really big number) Thus, f(x) approaches zero.
as x gets smaller and smaller from the right (ex: goes from 0.5 to 0.3 to 0.1 to 0.01, etc) then what happens to the graph of f(x)=1/x?
f(x) will become bigger and bigger in the positive direction. This is because 1 divided by a very small number is actually a really big number. For example 5/(.25) = 20
as x gets smaller and smaller from the left (ex: goes from -0.5 to -0.3 to -0.1 to -0.01, etc) then what happens to the graph of f(x)=1/x?
f(x) will become a larger and larger number in the negative direction
True or false, the function 2^x eventually grows faster than x^b for any b
True, exponential functions always outpace power functions in the long run.
Which function approaches the x axis faster as x grows very large?

y=1/(x^3) or y=1/(e^x)
y=1/(e^x) because exponentials increase faster than power functions so 1/(e^x) will decay faster than
1/(x^3)
What are the zeros of the function

f(x) = (8(x^2-4))/(x^2-9)
zeros found by setting

8(x^2-4)=0
x^2=4
x=+/- 2
What is the horizontal asymptote of the function

f(x) = (8(x^2-4))/(x^2-9)
horizontal asymptotes found by ratio of leading terms:

in long run:
f(x)= (8x^2)/(x^2)=8
Thus the horizontal asymptote is at y=8
What are the vertical asymptotes of the function:

f(x) = (8(x^2-4))/(x^2-9)
Vertical asymptotes found by setting denominator=0,

x^2-9=0
x=+/- 3 are the vertical asymptotes
What is the horizontal asymptote of the function:

f(x)=(x-5)/(x^2 - 6)
Examine ratio of leading terms:

x/(x^2) = 1/x

As x gets bigger and bigger 1/x goes to 0. thus the asymptote is at y=0
True or false, the power of the first term of a polynomial is its degree.
False. The highest power of the terms is the degree of the polynomial. Unless the polynomial is in standard form, the term with the highest degree may not necessarily be the first term.
For x very large, what graph does the function:

f(x)= x^3 +2x -5 resemble?
as x gets larger and larger, it will look like the graph of x^3
Are the zeros of a polynomial the x-coordinates where the graph intersects the x axis or are they the x coordinates where the graph intersects the y axis?
x-coordinates that intersect x axis
If y=f(x) is a polynomial defined for all real numbers and whose degree is even and positive, then does y have an inverse?
no, polynomials of even degree will fail the horizontal line test
If p(x) is a polynomial and (x-a) is a factor of p, then what is a zero of p?
a is a zero of p
A polynomial of degree n cannot have more than __________ zeros
n zeros
In the long run (as x gets bigger and bigger), the graph of y=1/x will stay close to what line?
y=0
How can one tell the difference between a polynomial, an exponential function, and a rational function?
A polynomial will only have powers that are non-negative integers.

An exponential function will have variables as its powers (ex: e^x)

A rational function will be two polynomials divided by eachother
Is the following function a rational function, a power function, a polynomial, or an exponential function?

y=1/(x^2)
y=1/(x^2) is both a rational function and a power function

It is a rational function b/c 1 is a polynomial and x^2 is a polynomial
and a rational function is one polynomial divided by another.

It is a power function because power functions can have negative powers
As x goes thru large positive values, where is the asymptote of the function? For large negative values?

y=(x^3 + 5x + 1)/(4x^3 +4x +3)
For large positive, y looks like y=(x^3)/(4x^3) = 1/4 so there is a horizontal asymptote at y=1/4

For large negative,
y= (-x^3)/(-4x^3)= 1/4 so there is a horizontal asymptote in the negative x direction at y=1/4
A fraction is equal to zero if and only if its ____________ equals to zero and its _________ does not equal zero
numerator, denominator
What are the zeros of the function

y= 12/(x^2-3x-1)
Since the numerator can never equal zero, there are no zeros
Can rational functions ever cross a horizontal asymptote? How about a vertical asymptote?
horizontal-yes
vertical-no
What does the graph of y=(x^2+5x+6)/(x+2)

look like?
Since x^2 +5x +6 = (x+2)(x+3), then y=[(x+2)(x+3)]/(x+2) = x+3

So y is the graph of y=x+3 with a hole at x=-2