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Rational Function Discontinuities

Rational Function Discontinuities. Objectives. I can identify Graph Discontinuities Vertical Asymptotes Horizontal Asymptotes Slant Asymptotes Holes I can find “x” and “y” intercepts. Rational Functions.

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Rational Function Discontinuities

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  1. Rational Function Discontinuities

  2. Objectives • I can identify Graph Discontinuities • Vertical Asymptotes • Horizontal Asymptotes • Slant Asymptotes • Holes • I can find “x” and “y” intercepts

  3. Rational Functions • A rational function is any ratio of two polynomials, where denominator cannot be ZERO! • Examples:

  4. Asymptotes • Asymptotes are the boundary lines that a rational function approaches, but never crosses. • We draw these as Dashed Lines on our graphs. • There are three types of asymptotes: • Vertical • Horizontal (Graph can cross these) • Slant

  5. Vertical Asymptotes • Vertical Asymptotes exist where the denominator would be zero. • They are graphed as Vertical Dashed Lines • There can be more than one! • To find them, set the denominator equal to zero and solve for “x”

  6. Example #1 • Find the vertical asymptotes for the following function: • Set the denominator equal to zero • x – 1 = 0, so x = 1 • This graph has a vertical asymptote at x = 1

  7. y-axis 9 8 7 Vertical Asymptote at X = 1 6 5 4 3 2 10 -9 -8 -7 -6 -2 1 0 -5 -4 -3 -1 x-axis 0 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9

  8. Other Examples: • Find the vertical asymptotes for the following functions:

  9. To find Vertical Asymptote(s) 1) Set reduced denominator = 0 • Solve for x = #. • Your answer is written as a line.

  10. Horizontal Asymptotes • Horizontal Asymptotes are also Dashed Lines drawn horizontally to represent another boundary. • To find the horizontal asymptote you compare the degree of the numerator with the degree of the denominator • See next slide:

  11. Horizontal Asymptote (HA) Given Rational Function: Compare DEGREE of Numerator to Denominator If N < D , then y = 0 is the HA If N > D, then the graph has NO HA If N = D, then the HA is

  12. Example #1 • Find the horizontal asymptote for the following function: • Since the degree of numerator is equal to degree of denominator (m = n) • Then HA: y = 1/1 = 1 • This graph has a horizontal asymptote at y = 1

  13. y-axis 9 8 7 Horizontal Asymptote at y = 1 6 5 4 3 2 10 -9 -8 -7 -6 -2 1 0 -5 -4 -3 -1 x-axis 0 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9

  14. Other Examples: • Find the horizontal asymptote for the following functions:

  15. To find Horizontal Asymptote(s) 1) Compare DEGREE of numerator and denominator Num BIGGER then NO HA Num SMALLER then y = 0 Degree is SAME then

  16. Slant Asymptotes (SA) • Slant asymptotes exist when the degree of the numerator is one larger than the denominator. • Cannot have both a HA and SA • To find the SA, divide the Numerator by the Denominator. • The results is a line y = mx + b that is the SA.

  17. Example of SA -2

  18. To find Slant Asymptote(s) 1) DEGREE of Numerator must be ONE bigger than Denominator • Divide with Synthetic or Long Division • Don’t use the Remainder Get y = mx + b

  19. Holes • A hole exists when the same factor exists in both the numerator and denominator of the rational expression and the factor is eliminated when you reduce!

  20. Example of Hole Discontinuity

  21. HOLES • To Find Holes • 1) Factor. • 2) Reduce. • A hole is formed when a factor is eliminated from the denominator. • Set eliminated factor = 0 and solve for x. • 5) Find the y-value of the hole by substituting the x-value into reduced form and solve for y. • 6) Your answer is written as a point. (x, y)

  22. To find x- intercept(s) • Set reduced numerator = 0 2) Solve for x. 3) Answer is written as a point. (#, 0)

  23. To find y- intercept 1) Substitute 0 in for all x’s in reduced form. • Solve for y. • Answer is a point. (0, #)

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