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Exploring Polynomial and Rational Functions in Algebra

Understand polynomial and rational functions, graphing techniques, turning points, intercepts, and asymptotes. Solve polynomial and rational inequalities. Discover zeros of polynomial functions.

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Exploring Polynomial and Rational Functions in Algebra

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  1. College AlgebraCall in Show #8November 13, 2003

  2. Section 5.2 Polynomial Functions and Models

  3. A polynomial function is a function of the form

  4. Determine which of the following are polynomials.For those that are, state the degree. (a) (b) (c)

  5. Common Function: Constant Function

  6. Common Function: Identity Function

  7. Common Function: Cubic Function

  8. Common Function: Square Root Function

  9. Common Function: Reciprocal Function

  10. Common Function: Greatest Integer Function

  11. Common Function: Quadratic Function

  12. If r is a (real) zero of f, then (a) r is an x-intercept of the graph of f. (b) (x - r) is a factor of f.

  13. If r is a Zero or Odd Multiplicity Sign of f(x) changes from one side to the other side of r Graph crosses x-axis at r.

  14. If r is a Zero or Even Multiplicity Graph touches x-axis at r Sign of f(x) does not change from one side to the other side of r.

  15. Theorem If f is a polynomial function of degree n, then f has at most n-1 turning points.

  16. For the polynomial (a) Find the x- and y-interceptsof the graph of f. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept.

  17. For the polynomial (c) Find the power function that the graph of f resembles for large values of x. (d) Determine the maximum number of turning points on the graph of f.

  18. For the polynomial (e) Use the x-intercepts and test numbers to find the intervals on which the graph of f is above the x-axis and the intervals on which the graph is below the x-axis.

  19. For the polynomial (f) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph of f.

  20. An open box with a square base is to be made from a square piece of cardboard 30 inches wide on a side by cutting out a square from each corner and turning up the sides. (a) Express the volume V of the box as a function of the length x of the side of the square cut from each corner.

  21. (b) Express the domain of V(x).

  22. Section 5.3 Rational Functions I

  23. (c) Graph V=V(x). (d) For what value of x is V largest?

  24. V is largest (2000 cubic inches),when x = 5 inches.

  25. A rational function is a function of the form

  26. Find the domain of the following rational functions:

  27. Horizontal Asymptotes

  28. Horizontal Asymptotes

  29. Vertical Asymptotes

  30. Vertical Asymptotes

  31. If an asymptote is neither horizontal nor vertical it is called oblique.

  32. Theorem Locating Vertical Asymptotes A rational function In lowest terms, will have a vertical asymptote x = r, if x - r is a factor of the denominator q.

  33. Find the vertical asymptotes, if any, of the graph of each rational function. a)

  34. Find the vertical asymptotes, if any, of the graph of each rational function. b)

  35. Find the vertical asymptotes, if any, of the graph of each rational function. c)

  36. Consider the rational function 1. If n < m, then y = 0 is a horizontal asymptote. 2. If n = m, then y = an / bmis a horizontal asymptote 3. If n = m + 1, then y = ax + b is an oblique asymptote 4. If n > m + 1, the graph of R has neither a horizontal nor oblique asymptote.

  37. Find the horizontal and oblique asymptotes if any, of the graph of

  38. Section 5.5 Polynomial and Rational Inequalities

  39. Write the inequality in one of the following forms: where f(x) is written as a single quotient. Determine the numbers at which f(x) equals zero and also those numbers at which it is undefined. Steps for Solving Polynomial and Rational Inequalities Algebraically

  40. Section 5.6 The Real Zeros of a Polynomial Function

  41. Factor Theorem • If f(c)=0, then x - c is a factor of f(x). • If x - c is a factor of f(x), then f(c)=0.

  42. Use the Factor Theorem to determine whether thefunction has the factor (a) x + 3 (b) x + 4

  43. Theorem Number of Zeros A polynomial function cannot have more zeros than its degree.

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