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Section 5.3 – The Graph of a Rational Function

Section 5.3 – The Graph of a Rational Function. General Steps to Graph a Rational Function. 1) Factor the numerator and the denominator. 2) State the domain and the location of any holes in the graph. 3) Simplify the function to lowest terms.

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Section 5.3 – The Graph of a Rational Function

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  1. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function 1) Factor the numerator and the denominator • 2) State the domain and the location of any holes in the graph • 3) Simplify the function to lowest terms • 4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0) • 5) Identify any existing asymptotes (vertical, horizontal, or oblique

  2. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 6) Identify any points intersecting a horizontal or oblique asymptote. • 7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis 8) Analyze the behavior of the graph on each side of an asymptote • 9) Sketch the graph

  3. Section 5.3 – The Graph of a Rational Function Example 1) Factor the numerator and the denominator • 2) State the domain and the location of any holes in the graph • Domain: • No holes • 3) Simplify the function to lowest terms

  4. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0) • y-intercept (x = 0) • x-intercept(s) (y = 0) • Use numerator factors

  5. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 5) Identify any existing asymptotes (vertical, horizontal, or oblique • Horiz. Or Oblique Asymptotes • Vertical Asymptotes • Use denominator factors • Examine the largest exponents • Same Horiz. - use coefficients

  6. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 6) Identify any points intersecting a horizontal or oblique asymptote.

  7. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis 3 -4 -2 2

  8. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis 3 -4 -2 2

  9. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function 8) Analyze the behavior of the graph on each side of an asymptote 3 -4 -2 2

  10. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function 8) Analyze the behavior of the graph on each side of an asymptote 3 -4 -2 2

  11. Section 5.3 – The Graph of a Rational Function • 9) Sketch the graph

  12. Section 5.3 – The Graph of a Rational Function Example 1) Factor the numerator and the denominator • 2) State the domain and the location of any holes in the graph • Domain: • Hole in the graph at • 3) Simplify the function to lowest terms

  13. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0) • y-intercept (x = 0) • x-intercept(s) (y = 0) • Use numerator factors

  14. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 5) Identify any existing asymptotes (vertical, horizontal, or oblique • Horiz. Or Oblique Asymptotes • Vertical Asymptotes • Use denominator factors • Examine the largest exponents • Same Horiz. - use coefficients

  15. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 6) Identify any points intersecting a horizontal or oblique asymptote.

  16. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis 2 -3

  17. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function 8) Analyze the behavior of the graph on each side of an asymptote 2 -3

  18. Section 5.3 – The Graph of a Rational Function • 9) Sketch the graph

  19. Section 5.3 – The Graph of a Rational Function Example 1) Factor the numerator and the denominator • 2) State the domain and the location of any holes in the graph • Domain: • No holes • 3) Simplify the function to lowest terms

  20. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0) • y-intercept (x = 0) • x-intercept(s) (y = 0) • Use numerator factors

  21. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 5) Identify any existing asymptotes (vertical, horizontal, or oblique • Vertical Asymptotes • Horiz. or Oblique Asymptotes • Use denominator factors • Examine the largest exponents • Oblique: Use long division

  22. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 6) Identify any points intersecting a horizontal or oblique asymptote.

  23. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function • 7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis 1 -2 -1

  24. Section 5.3 – The Graph of a Rational Function General Steps to Graph a Rational Function 8) Analyze the behavior of the graph on each side of an asymptote 1

  25. Section 5.3 – The Graph of a Rational Function • 9) Sketch the graph

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