Chapter 9-Polynomial And Rational Functions

Front 1

What is the general form of a power function?

Back 1

y=kx^p

where k and p are any number

Front 2

Are there any restrictions on what p can be for a power function y=kx^p?

Back 2

no, p only has to be a number but can be any number

Front 3

let y=4x^5

Then, in terms of proportionalities,
y is _________ to _______

Back 3

y is proportional to x to the 5th

Front 4

let y=4/(x^10)

Then, in terms of proportionalities,
y is _____________ to _________

Back 4

y is inversely proportional to x to the tenth

Front 5

The graphs of all power functions, y=x^p where p is an even, positive integer resemble....

Back 5

y=x^2 and will be U shaped

Front 6

If y=x^p where p is an odd, positive integer, then what will be the general shape of its graph?

Back 6

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

Front 7

If y=x^p is a power function and p is positive and even, then does y have any symmetry?

Back 7

y is symmetric with respect to the y axis

Front 8

If y=kx^p is a power function where p is positive and odd, then y is symmetric with respect to ...

Back 8

the origin (or the line y=x)

Front 9

What do all graphs of power functions with odd, negative powers resemble?

Back 9

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

Front 10

What function do all power functions with negative, even integers resemble?

Back 10

y=1/(x^2)
You should graph y=1/(x^2) and be sure to know what it looks like

Front 11

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?

Back 11

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

Front 12

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?

Back 12

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

Front 13

A polynomial is a sum of________
whose exponents are ___________

Back 13

power functions
non-negative integers

Front 14

In a polynomial, the term with the highest power is called the _________
and the power is the _________ of the polynomial

Back 14

leading term
degree

Front 15

What degree is the following polynomial?

y = 4(x^20) +x^2 + 7(x^101)

Back 15

The polynomial is of degree 101 since 101 is the highest degree

Front 16

what is the degree of the polynomial?

y=5(x-2)(x-6)(x-4) +3

Back 16

x^3

Front 17

To write a polynomial in standard form, you need to ....

Back 17

arrange its terms from highest power to lowest power going left to right

Front 18

What is the standard form of

y=(x-1)(x+1) +x^3 +7

Back 18

y=x^3 + x^2 + 6

Front 19

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?

Back 19

The highest order degree always wins out. Thus, the graph will resemble y=x^3

Front 20

True or false, on different scales functions that are power functions may not look like power functions at all.

Back 20

True

Front 21

What are the zeros of a polynomial and what are three ways to find them?

Back 21

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

Front 22

If the graph of a polynomial has n bumps, then it is most likely of degree......

Back 22

n+1

Front 23

If the graph of a polynomial has 3 bumps then it may be a polynomial of degree...

Back 23

four

Front 24

If a polynomial is of degree 10, then at most it can have how many bumps?

Back 24

9

Front 25

True or false, an nth degree polynomial can have n+1 zeros.

Back 25

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

Front 26

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?

Back 26

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

Front 27

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?

Back 27

p(x) will pass through the point x=k

Front 28

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?

Back 28

p(x) will bounce off the x-axis at x=k

Front 29

If you are given a graph of a polynomial and told to find an equation for the polynomial, what steps should you take?

Back 29

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.

Front 30

What is a rational function?

Back 30

r(x) = p(x)/q(x)

Where p(x) and q(x) are polynomials and q(x) does not =0

Front 31

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

Back 31

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

Front 32

The graph of y=(x^2-1)/(x-1) is the graph of ....

Back 32

the line y= x+1 with a hole (where it is undefined) at x=1

Front 33

Any _________, ___________ exponential function eventually grows faster than any _________ ________

Back 33

positive, increasing
power function

Front 34

Any ___________, ___________ exponential function eventually approaches the ____________ axis faster than any _________,________ power function

Back 34

positive, decreasing
x-axis
positive, decreasing

Front 35

Eventually, what will grow more rapidly, a positive, increasing power function or log(x)?

Back 35

power func

Front 36

In the long run, does the graph of ln(x) or the graph of a positive increasing power function grow more rapidly?

Back 36

the power function grows more rapidly

Front 37

True or false, all quadratic functions are power functions?

Back 37

True. Quadratic functions are functions of power two and thus are power functions

Front 38

Is the function 3*2^(x) a power function?

Back 38

No, the exponent is a variable, not a number. This function is an exponential function

Front 39

let g(x) = x^p

If p is positive, even integer, then is the graph of g concave up or down?

Back 39

It will look like a parabola so it will be concave up

Front 40

Does the graph of 1/x pass thru the origin?

Back 40

no, the origin is given when x=0. If x=0, this function is undefined.

Front 41

If f(x)= 1/x, then as x gets really big (approaches + infinity) then what do the values of f(x) approach?

Back 41

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.

Front 42

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?

Back 42

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

Front 43

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?

Back 43

f(x) will become a larger and larger number in the negative direction

Front 44

True or false, the function 2^x eventually grows faster than x^b for any b

Back 44

True, exponential functions always outpace power functions in the long run.

Front 45

Which function approaches the x axis faster as x grows very large?

y=1/(x^3) or y=1/(e^x)

Back 45

y=1/(e^x) because exponentials increase faster than power functions so 1/(e^x) will decay faster than
1/(x^3)

Front 46

What are the zeros of the function

f(x) = (8(x^2-4))/(x^2-9)

Back 46

zeros found by setting

8(x^2-4)=0
x^2=4
x=+/- 2

Front 47

What is the horizontal asymptote of the function

f(x) = (8(x^2-4))/(x^2-9)

Back 47

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

Front 48

What are the vertical asymptotes of the function:

f(x) = (8(x^2-4))/(x^2-9)

Back 48

Vertical asymptotes found by setting denominator=0,

x^2-9=0
x=+/- 3 are the vertical asymptotes

Front 49

What is the horizontal asymptote of the function:

f(x)=(x-5)/(x^2 - 6)

Back 49

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

Front 50

True or false, the power of the first term of a polynomial is its degree.

Back 50

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.

Front 51

For x very large, what graph does the function:

f(x)= x^3 +2x -5 resemble?

Back 51

as x gets larger and larger, it will look like the graph of x^3

Front 52

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?

Back 52

x-coordinates that intersect x axis

Front 53

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?

Back 53

no, polynomials of even degree will fail the horizontal line test

Front 54

If p(x) is a polynomial and (x-a) is a factor of p, then what is a zero of p?

Back 54

a is a zero of p

Front 55

A polynomial of degree n cannot have more than __________ zeros

Back 55

n zeros

Front 56

In the long run (as x gets bigger and bigger), the graph of y=1/x will stay close to what line?

Back 56

y=0

Front 57

How can one tell the difference between a polynomial, an exponential function, and a rational function?

Back 57

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

Front 58

Is the following function a rational function, a power function, a polynomial, or an exponential function?

y=1/(x^2)

Back 58

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

Front 59

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)

Back 59

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

Front 60

A fraction is equal to zero if and only if its ____________ equals to zero and its _________ does not equal zero

Back 60

numerator, denominator

Front 61

What are the zeros of the function

y= 12/(x^2-3x-1)

Back 61

Since the numerator can never equal zero, there are no zeros

Front 62

Can rational functions ever cross a horizontal asymptote? How about a vertical asymptote?

Back 62

horizontal-yes
vertical-no

Front 63

What does the graph of y=(x^2+5x+6)/(x+2)

look like?

Back 63

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