Math Ch. 5 Cheat Sheet

5.1

• Polynomial Degrees: Largest power of X that appears

• Polynomial Functions: Smooth and continuous graphs. Equations are such that domain is all real numbers and exponents are always nonnegative integers.

• Power Functions: f(x) = x^something

• Multiplicities: (x – number)^multiplicity

• Behavior Near a Zero: Choose numbers near a solution. i.e.: If a solution is 1, choose 0.5 1.5, 1.25, 0.75, etc...

• Turning Points: Functions have “degree – 1” turning points.

• End Behavior: Choose a high value for X (ie: 100), the solution should be a power function that determines how the graph behaves at extreme values.

5.2

• Rational Functions: f(x) = poly. func./poly. func.

• y = 1/x^2: Similar to y = 1/x, except the “boomerang” is mirrored across the y-axis.

• Asymptotes:

  • 3 kinds: horizontal, vertical and oblique.

  • Graphs cannot intersect vertical asymptotes, but can intercept horizontal and oblique asymptotes.

  • Finding Vertical Asymptotes: When function is in lowest terms, all zeros of the denominator are vert. asymptotes in the form x = solution.

  • Finding Horizontal Asymptotes: If the degree is proper (degree of numerator is less than degree of denominator), hor. asymptote is y = 0. If improper, use long division (divide den. into num.) If the result is y = number, it is horizontal.

  • Finding Oblique Asymptotes: The degree must be improper. Use the method listed above. If the result is y = ax +b, the asymptote is oblique.

5.3

• Graphing Rational Functions

  • Step 1: Factor numerator and denominator, find domain. If the domain is 0, find y-int by finding R(0) and plot.

  • Step 2: Write fraction in lowest terms. The x-int. are the real zeros of the numerator. Plot and find behavior near the intercepts.

  • Step 3: The vertical asymptotes of the equation are the zeros of the denominator in lowest terms.

  • Step 4: Find hor. or obl. asymptotes using the method in 5.2. Plug in the values to determine if the graph intersects them.

  • Step 5: Find the behavior of the graph between negative infinity, any x-int. and vert. asymp. and positive infinity. Plot these points.

  • Step 6: Determine behavior near asymptotes. Graph.

  • Step 7: Put information together and graph.

  • Quick reference for information about the graph

    • Numerator: x-ints.

    • Denominator: restrictions on domain, vert. asymp.

    • Quotient of Numerator and Denominator: hor. or obl. asymp.;

    • Use chart to find behavior

    • Find behavior near asymptotes

    • Graph.

    • WINRAR.

5.4

• Solving Polynomial and Rational Inequalities:

  • Step 1: Make it such that f(x) (sign: “>”, “<” etc.) 0

  • Step 2: Determine zeros and undefined numbers for the function

  • Step 3: Divide a number line into segments using the zeros and undefineds.

  • Step 4: Evaluate the graph within each interval, determine the solution.

5.5

• Remainder Thm: If f(x) is divisible by x – c, the remainder is f(c).

  • Application: Finding remainders of poly. equations

• Factor Thm: x – c is a factor of f(x) if and only if f(c) = 0

  • Application: Seeing if f(x) has a factor x – c.

• Descartes' Rule of Signs:

  • Number of positive real zeros in an equation f(x): number of variation in signs in the equation or the number of variations – an even integer

  • Number of negative real zeros in an equation: use f(-x). number of variation in signs in the equation or the number of variations – an even integer

• Rational Zeros Thm: When rational eq. is in lowest terms, p is a factor of the constant (last number) and q is a factor of the coefficient of the first number (2x^2, q = 2). Using this, p/q = a potential rational zero of the equation

• Finding Real Zeros of a Poly. Func.:

  • Step 1: Determine maximum number of real zeros (degree)

  • Step 2: Use Descartes' Rule of Signs to find number of pos. and neg. zeros.

  • Step 3: Either use the Rational Zeros Thm or synthetic/long division to find working solutions.

  • Step 4: Reduce and factor, if possible.

• Bounds on Zeros: Bound of a poly. func. is the smaller of the two numbers Max {1, sum of
  |coefficients|} (where max means choose the greatest value) or 1+ Max {sum of |coefficients|}

• Intermediate Value Thm: If a < b and if f(a) and f(b) have opposite signs, there is at least 1 real zero between a and b.

5.6

• Fundamental Theorem of Algebra: Every function of degree > or = 1 has at least one complex zero.

• Conjugate Pairs Theorem: For every r = a + bi that is a solution of a poly. func., its conjugate, r = a – bi, is a solution.

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