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Ch. 9.3 Rational Functions and Their Graphs

Ch. 9.3 Rational Functions and Their Graphs. 3 x 2 – x –12. a. y =. Rational Functions and Their Graphs. ALGEBRA 2 LESSON 9-3. For each rational function, find any points of discontinuity. The function is undefined at values of x for which x 2 – x – 12 = 0.

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Ch. 9.3 Rational Functions and Their Graphs

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  1. Ch. 9.3 Rational Functions and Their Graphs

  2. 3 x2 – x –12 a. y = Rational Functions and Their Graphs ALGEBRA 2 LESSON 9-3 For each rational function, find any points of discontinuity. The function is undefined at values of x for which x2 – x – 12 = 0. x2 – x – 12 = 0 Set the denominator equal to zero. (x – 4)(x + 3) = 0 Solve by factoring or using the Quadratic Formula. x – 4 = 0 or x + 3 = 0 Zero-Product Property x = 4 or x = –3 Solve for x. There are points of discontinuity at x = 4 and x = –3. 9-3

  3. 4 3 x2 = – Solve for x. ± 2i 3 ± –4 3 x = = Since is not a real number, there is no real value for x for which the function y = is undefined. There is no point of discontinuity. ± 2i 3 2x 3x2 + 4 Rational Functions and Their Graphs ALGEBRA 2 LESSON 9-3 (continued) 2x 3x2 + 4 b.y = The function is undefined at values of 3x2 + 4 = 0. 3x2 + 4 = 0 Set the denominator equal to zero. 9-3

  4. Try This For each rational function, find any points of discontinuity x = 4 and x = -4

  5. Try This For each rational function, find any points of discontinuity No point of discontinuity

  6. Try This For each rational function, find any points of discontinuity x = -4 and x = 2

  7. Vertical Asymptotes The rational function has a point of discontinuity for each real zero of Q(x). If P(x) and Q(x) have no common real zeros, then the graph of f(x) has a vertical asymptote at each real zero of Q(x). If P(x) and Q(x) have a common real zero a, then there is a hole in the graph or a vertical asymptote at x = a

  8. x – 7 (x + 1)(x + 5) a.y = (x + 3)x x + 3 b.y = (x – 6)(x + 9) (x + 9)(x + 9)(x – 6) c.y = 1 (x + 9) , The graph of the function is the same as the graph y = which has a vertical asymptote at x = –9, except it has a hole at x = 6. Rational Functions and Their Graphs ALGEBRA 2 LESSON 9-3 Describe the vertical asymptotes and holes for the graph of each rational function. Since –1 and –5 are the zeros of the denominator and neither is a zero of the numerator, x = –1 and x = –5 are vertical asymptotes. –3 is a zero of both the numerator and the denominator. The graph of this function is the same as the graph y = x, except it has a hole at x = –3. 6 is a zero of both the numerator and the denominator. 9-3

  9. Try This Describe the vertical asymptotes and holes for the graph of each rational function. Vertical asymptotes: x = 1 and x = -3

  10. Try This Describe the vertical asymptotes and holes for the graph of each rational function. Hole : x = 2; V.A.: x = -3

  11. Try This Describe the vertical asymptotes and holes for the graph of each rational function. Hole: x = -1; no V.A.

  12. Horizontal Asymptotes • If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is y = 0 (x-axis) Ex. H.A.: y = 0 Degree of 1 Degree of 2 2. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the fraction formed by the coefficients of the terms with the highest degree. Ex. H.A.: Degree of 2 Degree of 2

  13. Horizontal Asymptotes • If the degree of the numerator is greater than the degree of the denominator, then the graph has no horizontal asymptote. Ex. No horizontal asymptote

  14. Rational Functions and Their Graphs ALGEBRA 2 LESSON 9-3 –4x + 3 2x + 1 . Find the horizontal asymptote of y = The degree of the numerator and denominator are equal The horizontal asymptote is y = –2. 9-3

  15. Try this Find the horizontal asymptote No H.A.

  16. Try this Find the horizontal asymptote H.A.: y = 0

  17. Try this Find the horizontal asymptote

  18. Since 3 and –2 are the zeros of the denominator, the vertical asymptotes are at x = 3 and x = –2. Calculate the values of y for values of x near the asymptotes. Plot those points and sketch the graph. Rational Functions and Their Graphs ALGEBRA 2 LESSON 9-3 x + 1 (x – 3)(x + 2) Sketch the graph y = . The degree of the denominator is greater than the degree of the numerator, so the x-axis is the horizontal asymptote. When x > 3, y is positive. So as x increases, the graph approaches the y-axis from above. When x < –2, y is negative. So as x decreases, the graph approaches the y-axis from below. Since –1 is the zero of the numerator, the x-intercept is at –1. 9-3

  19. Try This Sketch the graph of H.A.: V.A.: x-int.: y-int.:

  20. Page 497, Exercises: #2 – 18 e, 26 – 30 all Quiz Ch. 9.1-9.2 next class

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