1 / 30

Chapter 4

Chapter 4. Graphing and Inverse Functions. 3.5 Angular and Linear Speed. The line from the center sweeps out a central angle  in an amount time t, then the angular velocity, (omega)

scout
Download Presentation

Chapter 4

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. Chapter 4 Graphing and Inverse Functions

  2. 3.5 Angular and Linear Speed • The line from the center sweeps out a central angle  in an amount timet, then the angular velocity, (omega) • Angular Speed: the amount of rotation per unit of time, where  is the angle of rotation and t is the time.

  3. Periodic Function(page 165) • A periodic function is a function f such that f(x) = f(x + np), for every real number x in the domain of f, every integer n, and some positive real number p. The smallest possible positive value of p is the period of the function.

  4. Sine Function f(x) = sin x

  5. Cosine Function f(x) = cos x

  6. x 0 /2  3/2  sin x 0 1 0 1 0 3sin x 0 3 0 3 0 Example: Graph y = 3 sin x compare to y = sin x. • Make a table of values.

  7. Amplitude • The amplitude of a periodic function is half the difference between the maximum and minimum values. • The graph of y = a sin x or y = a cos x, with a 0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except the range will be [|a|, |a|]. The amplitude is |a|.

  8. Example: Graph y = sin 2x • The period is 2/2 = . The graph will complete one period over the interval [0, ]. • The endpoints are 0 and , the three middle points are: • Plot points and join in a smooth curve.

  9. Example: Graph y = sin 2x continued

  10. Period • For b > 0, the graph of y = sin bx will resemble that of y = sin x, but with period 2/b. • The graph of y = cos bx will resemble that of y = cos x, with period 2/b.

  11. x 0 3/4 3/2 9/4 3 2x/3 0 /2  3/2 2 cos 2x/3 1 0 1 0 1 Graph y = cos 2x/3 over one period • The period is 3. • Divide the interval into four equal parts. • Obtain key points for one period.

  12. Graph y = cos 2x/3 over one period continued • The amplitude is 1. • Join the points and connect with a smooth curve.

  13. Guidelines for Sketching Graphs of Sine and Cosine Functions • To graph y = a sin bx or y = a cos bx, with b > 0, follow these steps. • Step 1Find the period, 2/b. Start with 0 on the x-axis, and lay off a distance of 2/b. • Step 2 Divide the interval into four equal parts. • Step 3 Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and x-intercepts.

  14. Guidelines for Sketching Graphs of Sine and Cosine Functions continued • Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude |a|. • Step 5 Draw the graph over additional periods, to the right and to the left, as needed.

  15. x 0 /8 /4 3/8 /2 4x 0 /2  3/2 2 sin 4x 0 1 0 1 0 2 sin 4x 0 2 0 2 0 Graph y = 2 sin 4x • Step 1 Period = 2/4 = /2. The function will be graphed over the interval [0, /2] . • Step 2 Divide the interval into four equal parts. • Step 3 Make a table of values

  16. Graph y = 2 sin 4x continued • Plot the points and join them with a sinusoidal curve with amplitude 2.

  17. Tangent Functions

  18. Cosecant Function

  19. Cotangent Functions

  20. Secant Function

  21. To Graph Use as a Guide y = a csc bx y = a sin bx y = a sec bx y = cos bx Guidelines for Sketching Graphs of Cosecant and Secant Functions • To graph y = csc bx or y = sec bx, with b > 0, follow these steps. • Step 1 Graph the corresponding reciprocal function as a guide, using a dashed curve.

  22. Guidelines for Sketching Graphs of Cosecant and Secant Functions continued • Step 2Sketch the vertical asymptotes. They will have equations of the form x = k, where k is an x-intercept of the graph of the guide function. • Step 3 Sketch the graph of the desired function by drawing the typical U-shapes branches between the adjacent asymptotes. The branches will be above the graph of the guide function when the guide function values are positive and below the graph of the guide function when the guide function values are negative.

  23. Graph y = 2 sec x/2 • Step 1 Graph the corresponding reciprocal function y = 2 cos x/2. • The function has amplitude 2 and one period of the graph lies along the interval that satisfies the inequality • Divide the interval into four equal parts.

  24. Graph y = 2 sec x/2 continued

  25. Graph y = 2 sec x/2 continued • Sketch the vertical asymptotes. These occur at x-values for which the guide function equals 0, such as x = 3, x = 3, x = , x = 3. • Sketch the graph of y = 2 sec x/2 by drawing the typical U-shaped branches, approaching the asymptotes.

  26. Guidelines for Sketching Graphs of Tangent and Cotangent Functions • To graph y = tan bx or y = cot bx, with b > 0, follow these steps. • Step 1 Determine the period, /b. To locate two adjacent vertical asymptotes solve the following equations for x:

  27. Guidelines for Sketching Graphs of Tangent and Cotangent Functions continued • Step 2 Sketch the two vertical asymptotes found in Step 1. • Step 3 Divide the interval formed by the vertical asymptotes into four equal parts. • Step 4 Evaluate the function for the first-quarter point, midpoint, and third-quarter point, using the x-values found in Step 3. • Step 5 Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary.

  28. Graph y = tan 2x • Step 1 The period of the function is /2. The two asymptotes have equations x = /4 and x = /4. • Step 2 Sketch two vertical asymptotes x =  /4. • Step 3 Divide the interval into four equal parts. This gives the following key x-values. • First quarter: /8 • Middle value: 0Third quarter: /8

  29. Graph y = tan 2x continued • Table of values x /8 0 /8 2x /4 0 /4 tan 2x 1 0 1

  30. Every y value for this function will be 2 units more than the corresponding y in y = tan x, causing the graph to be translated 2 units up compared to y = tan x. Graph y = 2 + tan x

More Related