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Section 3.4

Section 3.4. Rational Functions. Rational Function. A rational function is a function of the form Where P and Q are polynomials and P(x) and Q(x) have no factor in common and Q(x) is not equal to zero. Consider . Find the domain and graph f . Solution:

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Section 3.4

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  1. Section 3.4 Rational Functions

  2. Rational Function A rational function is a function of the form Where P and Q are polynomials and P(x) and Q(x) have no factor in common and Q(x) is not equal to zero.

  3. Consider . Find the domain and graph f. Solution: When the denominator x + 4 = 0, we have x = 4, so the only input that results in a denominator of 0 is 4. Thus the domain is {x|x  4} or (, 4)  (4, ). The graph of the function is the graph of y = 1/x translated to the left 4 units. Example

  4. Rational Functions Different from other functions because they have asymptotes. Asymptotes- a line that the graph of a function gets closer and closer to as one travels along that line in either direction.

  5. Vertical Asymptote Occurs where the function is undefined, denominator is equal to zero. Form: x = a where a is the zero of the denominator **Graph never crosses ** Types of Asymptotes

  6. Determine the vertical asymptotes of the function. Factor to find the zeros of the denominator: x2 4 = (x + 2)(x  2) Thus the vertical asymptotes are the lines x = 2 and x = 2. Example

  7. Types of Asymptotes Horizontal Asymptote Determined by the degrees of the numerator and denominator. Form: y = a (next slide has rules) ** Graph can cross **

  8. Horizontal Asymptotes Look at the rational function • If degree of P(x) < degree of Q(x), horizontal asymptote y = 0. • If degree of P(x) = degree of Q(x), horizontal asymptote y = • If degree of P(x) > degree of Q(x), no horizontal asymptote

  9. Find the horizontal asymptote: The numerator and denominator have the same degree. The ratio of the leading coefficients is 6/9, so the line y = 2/3 is the horizontal asymptote. Example

  10. True Statements • The graph of a rational function never crosses a vertical asymptote. • The graph of a rational function might cross a horizontal asymptote but does not necessarily do so.

  11. Oblique Asymptote Degree of P(x) > degree of Q(x) To find oblique asymptote: • Divide numerator by denominator • Disregard remainder • Set quotient equal to y (this gives the equation of the asymptote)

  12. Occurrence of Lines as Asymptotes • For a rational function f(x) = p(x)/q(x), where p(x) and q(x) have no common factors other than constants: • Vertical asymptotes occur at any x-values that make the denominator 0. • The x-axis is the horizontal asymptote when the degree of the numerator is less than the degree of the denominator. • A horizontal asymptote other than the x-axis occurs when the numerator and the denominator have the same degree.

  13. Occurrence of Lines as Asymptotes continued • An oblique asymptote occurs when the degree of the numerator is 1 greater than the degree of the denominator. • There can be only one horizontal asymptote or one oblique asymptote and never both. • An asymptote is not part of the graph of the function.

  14. To Graph a Rational Function • Find vertical asymptotes. (Set denominator equal to zero) • Find horizontal asymptotes (compare degrees) • Find the x-intercepts. (Set y = 0) ** Rational function is zero if numerator is zero.** • Find the y-intercept. (Set x = 0) • Find any necessary additional points to determine behavior between and near vertical asymptotes. • If deg. of P(x) < deg. of Q(x), horiz. asymptote y = 0 • If deg. of P(x) = deg. of Q(x), horiz. asymptote y = • If deg. of P(x) > deg. of Q(x), no horiz. asymptote

  15. Example Graph . 1. Find the zeros by solving: The graph has vertical asymptotes at x = 3 and x = 1/2. We sketch these with dashed lines. 2. Because the degree of the numerator is less than the degree of the denominator, the x-axis, y = 0, is the horizontal asymptote. The zeros are 1/2 and 3, thus the domain excludes these values.

  16. Example continued 3. To find the zeros of the numerator, we solve x + 3 = 0 and get x = 3. Thus, 3 is the zero of the function, and the pair (3, 0) is the x-intercept. 4. We find f(0): Thus (0, 1) is the y-intercept.

  17. x f(x) 1 1/2 1  2/3 2 1 4 7/9 Example continued 5. We find other function values to determine the general shape of the graph and then draw the graph.

  18. More Examples • Graph the following functions. a) b)

  19. Graph a • Vertical Asymptote x = 2 • Horizontal Asymptote y = 1 • x-intercept (3, 0) • y-intercept (0, 3/2)

  20. Vertical Asymptote x = 3, x = 3 Horizontal Asymptote y = 1 x-intercepts (2.828, 0) y-intercept (0, 8/9) Graph b

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