1 / 26

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.

kaili
Download Presentation

RATIONAL FUNCTIONS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related