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RATIONAL FUNCTIONS

CURVE SKETCHING. RATIONAL FUNCTIONS. RATIONAL FUNCTIONS EXAMPLE 1 a) Domain Since x 2 + x – 2 ≠ 0 ( x + 2)( x – 1) ≠ 0 x ≠ – 2 x ≠ 1 x e R b) x -intercepts y ­ - intercept c) Tests for symmetry Circle the correct answer.

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RATIONAL FUNCTIONS

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  1. CURVE SKETCHING RATIONAL FUNCTIONS

  2. RATIONAL FUNCTIONS EXAMPLE 1 a) Domain Since x2 + x – 2 ≠ 0 (x + 2)(x – 1) ≠ 0 x ≠ – 2 x ≠ 1 xe R b) x-intercepts y­-intercept c) Tests for symmetry Circle the correct answer. This function is: a) even b) odd c) neither d) Asymptotes f (x) ( = / ≠) f(–x) The graph (is / is not) symmetrical to the y-axis. – f (x) ( = / ≠) f(–x) The graph (is / is not) symmetrical to the y-axis. Vertical asymptote at x = –2 “Hole” at Horizontal asymptote at y = 1

  3. Now let’s look at the one-sided limits around • Label the x and y- intercepts with ordered pairs • Label the “hole” with an ordered pair • Draw in the asymptotes and write their equations • show how the graph behaves left and right of the vertical asymptote

  4. EXAMPLE 1 a) Domain x ≠ 4, xe R b) x-intercepts y­-intercept DNE

  5. c) Tests for symmetry f (x) ( = / ≠) f(–x) The graph (is / is not) symmetrical to the y-axis. – f(x) = – f(x) ( = / ≠) f(–x) The graph (is / is not) symmetrical to the origin. Circle the correct answer. This function is: a) even b) odd c) neither

  6. d) Asymptotes DNE DNE Vertical Asymptote at x = 4 Now lets look at the one-sided limits around x = 4 - ∞ ∞ 0 Horizontal Asymptote at y = 0

  7. e) First Derivative Test: Find intervals of increase and decrease. decreasing This graph is always ____________________ because f '(x) is always ________________ negative

  8. + 4 f) Concavity: This can change when the second derivative is undefined if The graph is concave down over the interval x < 4 The graph is concave up over the interval x > 4

  9. Graph #1 • Using ordered pairs, • label the x and y-intercepts, • draw in any asymptotes and label with their equations • show how the graph behaves left and right of the vertical asymptote ∞ y = 0 - ∞ x = 4

  10. Graph #2 Highlight the increasing sections and / or the decreasing sections decreasing decreasing

  11. Graph #3 Highlight the concave up sections and concave down sections. CU CD

  12. a) Domain x ≠ -2, x e R EXAMPLE 2 b) x-intercepts y­-intercept DNE c) Tests for symmetry f (x) ( = / ≠) f(–x) The graph (is / is not) symmetrical to the y-axis. – f(x) ( = / ≠) f(–x) The graph (is / is not) symmetrical to the origin. Circle the correct answer. This function is: a) even b) odd c) neither

  13. 0 d) Asymptotes DNE DNE Vertical Asymptote at x = -2 Now lets look at the one-sided limits around - ∞ - ∞ Horizontal Asymptote at y = 0

  14. + - 2 e) First Derivative Test: Find intervals of increase and decrease. Interval of increase: x > -2 Interval of decrease: x < -2

  15. - 2 f) Concavity: This can change when the second derivative is undefined if The graph is always ________________ (CU / CD) because f "(x) is always _________________ (positive / negative)

  16. x = -2 y = 0 - ∞ - ∞ • Graph #1 • Using ordered pairs, label the x and y-intercepts, • draw in any asymptotes • show how the graph behaves left and right of the vertical asymptote

  17. Graph #2 Highlight the increasing sections and the decreasing sections decreasing increasing Graph #3 Highlight the concave up sections and concave down sections. CD CD

  18. EXAMPLE Since x2 + 3 ≠ 0 xe R Domain: y­intercept x-intercepts (0, 0) (0, 0)

  19. Tests for symmetry • f(–x) = • f (x) (= / ≠) f(–x) • The graph (is / is not) symmetrical to the y-axis. – f(x) = = – f(x) ( = / ≠) f(–x) The graph (is / is not) symmetrical to the origin.

  20. Asymptotes x2 + 3 ≠ 0 Therefore there are no vertical asymptotes Horizontal asymptote at y = 1

  21. min (0, 0) + 0 Intervals of Increase or Decrease(First Derivative Test) Increase:x > 0 Decrease:x < 0 6x = 0 x = 0

  22. Local Maximum and Minimum Valuesby second derivative test positive so (0, 0) is a minimum

  23. + CD -1 CU 1 CD Concavity and Points of Inflection The graph is concave down over the intervals x < -1 or x > 1 The graph is concave up over the interval -1 < x < 1

  24. Graph #1 Using ordered pairs, label the xand y-intercepts, draw in any asymptotes y = 1 (0, 0)

  25. Graph #2 Label any turning points with ordered pairs Highlight the increasing sections and the decreasing sections increasing decreasing (0, 0)

  26. Graph #3 Label the inflection points with ordered pairs. Highlight the concave up sections and concave down sections. (1, ¼ ) (-1, ¼ ) CD CD CU

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