1 / 11

Graphing Sine Functions (16.3)

Graphing Sine Functions (16.3). From Warmups: y = sinx. 1. /2. . 3/2. 2. -1. I. Terminology. periodic ~. When a graph repeats the same part over and over. cycle ~. The smallest part that repeats itself. period ~. The horizontal distance it takes to complete one cycle. 1.

tovah
Download Presentation

Graphing Sine Functions (16.3)

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. Graphing Sine Functions (16.3) From Warmups: y = sinx 1 /2  3/2 2 -1

  2. I. Terminology • periodic ~ When a graph repeats the same part over and over • cycle ~ The smallest part that repeats itself • period ~ The horizontal distance it takes to complete one cycle 1 /2  3/2 2 -1

  3. II. The Sine Graph • The x-intercepts occur at multiples of  • The range is {y: -1 ≤ y ≤ 1} • The maximum points occur when x is -3/2, /2, 5/2 … /2 + 2k where k is an integer • The minimum points occur when x is -/2, 3/2, 7/2 … 3/2 + 2k where k is an integer • The period is 2 1 /2  3/2 2 -3/2 - -/2 -1

  4. III. Amplitude • The “height” of the graph • the graph’s amount of deviation above and below the axis line. Graph the following on graphing calculators: • y = 3sinx • y = -sinx • y = -3sinx How do the graphs change?

  5. IV. Phase Shift and Vertical Shift A. Vertical Shift Recall: How is the graph of y = x2 different from y = x2 + 2? EXAMPLE: Graph y = sinx + 2 Final Graph S1: Graph the base graph S2: Shift it 2 units up 1  3/2 2 -2 /2 -3/2 - -/2 -1

  6. IV. Phase Shift and Vertical Shift A. Phase Shift Recall: How is the graph of y = x2 different from y = (x + 2)2? EXAMPLE: Graph y = sin(x + /2) S1: Graph the base graph S2: Shift it /2 left (in this case, 1 unit) Final Graph 1  3/2 2 -2 /2 -3/2 - -/2 -1

  7. Changing the Period (16.4) Warmups: Graph y = cosx using only the unit circle 1 -2  2 -3/2 /2 3/2 - -/2 -1

  8. Summary of Sine and Cosine Graphs: In the graph of y = a sin [b(x – c)] + d • The amplitude is equal to |a|. • The period is equal to 2/|b|. • The phase shift, with respect to the graph of y = sinx, is c units to the right (if c > 0) or c units to the left (if c < 0). • The vertical shift, with respect to the graph of y = sinx, is d units up (if d > 0) or d units down (if d < 0)

  9. Example: Graph y = -2sin(2x /4) from 0 ≤ x ≤ 2 • Factor out the 2, so it’s written as y = asin[b(x – c)] + d y = -2sin[2(x /8) • Write important information: amplitude = |a| = |-2| = 2 *** graph flips period = 2/|2| =  ~ half of a unit phase shift = /8 right • Graph base graph (amplitude, period, and flipped) ~ start w/ one cycle Final graph 2 1 3/4  5/4 3/2 7/4 2 /4 /2 -1   1 = ? -2 2 2 your “counting number”

  10. Example: Write two equations, one in terms of the sine function and one in terms of the cosine function. Amplitude: 1 Things to notice:  2/|b| =  Period:   b = 2 Vertical Shift: 1 down SINE GRAPH ** /4 right COSINE GRAPH ** No phase shift y = a sin [b(x – c)] + d y = a cos [b(x – c)] + d y = 1 sin [2(x – /4)] – 1 y = 1 cos [2(x – 0)] – 1 y = sin [2(x - /4)] – 1 y = cos(2x) – 1

  11. Example: Write two equations, one in terms of the sine function and one in terms of the cosine function. Things to notice: Amplitude: 4  2/|b| = 6 Period: 6  b = 1/3 Vertical Shift: none SINE GRAPH COSINE GRAPH ** 3/2 right y = a sin [b(x – c)] + d y = a cos [b(x – c)] + d y = 4 cos [1/3(x – 3/2)] – 0 y = 4 sin [1/3(x – 0)] – 0

More Related