# Polynomial & Rational Functions

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 Author: Gymnastxoxo17 ID: 265898 Filename: Polynomial & Rational Functions Updated: 2014-06-18 06:43:29 Tags: Chapter Four Folders: H Pre-Calc Description: l Show Answers:

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1. characteristics of polynomial function
• coefficients all real numbers
• degrees are positive integers
• has to have a degree (can be zero/constant function)
2. domain of a polynomial
all real numbers
3. degree
largest power of x that appears
4. name of no degree function
zero function; x-axis
5. name of 0 degree function
constant function; horizontal line
6. name of 1 degree function
linear, straight line not horizontal or vertical
7. name of 2 degree function
8. parabola opens up/down rule
up if a is positive, down if a is negative
9. the graph of every polynomial is ?
• smooth: no sharp corners or cusps
• continuous: no gaps or holes
10. power function
a single monomial taken from a polynomial, like -5x2
11. important properties of power functions to even integer degrees
• symmetric to y-axis (even)
• domain=all real numbers; range=all nonnegative real numbers
12. important properties of power functions to odd integer degrees
• symmetric to origin (odd)
• domain and range=all real numbers
13. theorem you use to find the x-intercepts based off factored polynomial
zero product property
14. x-r set to zero represents
x-intercepts of the graph; the solutions
15. multiplicity
exponent to which a factor is raised
16. sum of the multiplicities =
degree
17. zeros with even multiplicities ? the x-axis
touch
18. zeros with odd multiplicities ? the x-axis
cross
19. turning points
the minima and maxima on a graph, humps and dips
20. highest number of turning points
degree - 1
21. least degree of a polynomial
turning points + 1
22. end behavior
behavior of the graph for large values of x, tells you the power function the graph will represent
23. end behavior of positive leading coefficient and even degree
like a parabola pointing upward, goes to positive infinity on both sides
24. end behavior of negative leading coefficient and even degree
like a parabola pointing downward, goes to negative infinity on both sides
25. end behavior of positive leading coefficient and odd degree
like a cube function, towards positive infinity it goes towards positive infinity and towards negative infinity it goes towards negative infinity
26. end behavior of negative leading coefficient and odd degree
like a cube function flipped over the y-axis; towards positive infinity it goes towards negative infinity and towards negative infinity it goes towards positive infinity
27. rational numbers
ratios of integers
28. rational functions
ratios of polynomials, has the form
29. domain and range of rational functions
• all real numbers except those where the denominator is zero
• to find the domain, set the denominator to zero and solve for the real numbers, these are excluded in the domain
30. lowest terms
in a rational function, factor out both numerator and denominator and cross out the common factors
31. how to find intercepts of rational functions
• zeros of the numerator in lowest terms are x intercepts
• solve for r(0) / substitute x for 0 and solve
32. horizontal asymptote
• the y= line that the values for f(x) approach as the graph goes far to the left and right of the graph (positive/negative infinity)
• sometimes intersects
33. vertical asymptote
• the x= line that x approaches as the absolute values for r(x) go towards positive infinity
• never intersects
34. how to find the vertical asymptote of a rational function
in lowest terms, solve for the real zeros of the denominator q(x)
35. proper
• rational function where the degree of the numerator is less than the degree of the denominator
• e.g. x/x2
36. improper
rational function where the degree of the numerator is greater than or equal to the degree of the denominator
37. horizontal asymptote when the rational function is proper (degree of numerator is less than degree of denominator)
y=0
38. horizontal asymptotes when the rational function has the same degree in numerator as denominator
39. horizontal asymptotes when the rational function has a numerator with a greater degree than the denominator
use long division to solve for an oblique asymptote, but cannot be anything higher than a linear function
40. test for symmetry
• solve r(-x)
• if the result is the opposite of the original, it has origin symmetry (or, replace x and y with -x and -y)
• if the result is the same as the original, it has y-axis symmetry (or, replace x with -x)
41. how to find if the graph intersects the horizontal asymptote
Set the function to the constant and solve. The number you get is the intersection as an x intercept coordinate point.
42. Should the rational function be in lowest terms when you are looking for the domain?
No; exclude holes from the domain as well
43. How to write the equation of a rational function based off a graph
• Use the x intercepts to get the factors of the numerator, note also if the graph touches (even) or crosses (odd)
• Look at how the graph looks around the vertical asymptotes, determine these and put in the denominator as (x-#)
• If it looks like a reciprocal function (the up/down ends of the graph don't go in the same direction), the factor is odd.
• If the two ends go in the same direction (1/x2), then the factor is even.
• use the horizontal asymptote to determine the correct ratio of degrees and any coefficients
45. (a3+b3)
(a+b)(a2-ab+b2)
46. (a3-b3)
(a-b)(a2+ab+b2)
47. (a+b)2
a2+2ab+b2
48. (a-b)2
a2-2ab+b2
49. how to solve a polynomial inequality using its graph
• solutions include the numbers that make the function positive
• look for where the graph is above the x-axis
• use intercepts to determine where the number line is cut up, then show the intervals between these for where the function is positive in interval notation
50. how to solve a polynomial inequality algebraically
• rewrite so that the polynomial expression is on the left and zero is on the right
• find the real zeros by factoring, use these to divide the number line into intervals
• chose a number in each interval and do a test by plugging it into the function, and see if it comes out as a positive or negative value
• whether or not it is included in the solution set depends on the sign of the inequality
51. how to solve a rational inequality using its graph
• use the vertical asymptotes and the x-intercepts to divide the number line into intervals
• intervals that are a solution to the function show the graph above the x-axis (positive y values)
52. how to solve a rational inequality algebraically
• rewrite so that f is on the left and zero is on the right
• find the x intercepts and vertical asymptotes by finding the solutions to the numerator (intercepts) and denominator (asymptotes)
• use these to divide the number line into intervals, chose a point within each interval and plug it in for x, then take note of whether the solution is positive or negative
• refer to the sign of the inequality
• in the solution set, do not include values of the vertical asymptotes or holes
53. should you check for symmetry on the original equation or lowest terms?
original
54. how to identify a hole
• in lowest terms, sometimes a factor crosses out
• set this to zero and this gives you the x coordinate
• plug x coordinate into the lowest terms expression and solve for y coordinate
55. y=f(x)+k
raise by k units
56. y=f(x)-k
lower by k units
57. y=f(x+h)
left shift h units
58. y=f(x-h)
right shift h units
59. vertical compression/stretch function transformation
• f(x) --> af(x); multiply the function by a
• |a|<1 means compression
• |a|>1 means stretch
60. changes to the coordinate pairs for a vertical compression/stretch
multiply the y-coordinate by a
61. graphically, what does a vertical compression/stretch look like
• stretch: pulling it up
• compression: pushing it down
62. horizontal compression/stretch function transformation
• f(x) --> f(ax); replace x with ax
• |a|<1 means stretch
• |a|>1 means compression
63. changes to the coordinate pairs for horizontal compression/stretch
multiply the x-coordinate by 1/a
64. graphically, what does a horizontal compression/stretch look like
• stretch: pulling the sides out
• compression: pushing the sides together
65. x-axis reflection transformation
• flipped up or below horizontal axis
• f(x) --> -f(x); multiply the function by -1
66. y-axis reflection transformation
• flipped to left or right over the vertical axis
• f(x) --> f(-x); replace x with -x in the function

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