Polynomial & Rational Functions

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Author:
Gymnastxoxo17
ID:
265898
Filename:
Polynomial & Rational Functions
Updated:
2014-06-18 06:43:29
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Chapter Four
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H Pre-Calc
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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
    quadratic, parabola
  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
    y=numerator leading coefficient/denominator leading coefficient; a constant function
  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
  44. Quadratic formula
  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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