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MAT 204 SP09 7.6 Graphs of the Sine and Cosine Functions Phase shift; Sinusoidal Curve Fitting In these sections, we will study the following topics: The graphs of basic sine and cosine functions The amplitude and period of sine and cosine functions Transformations of sine and cosine functions Sinusoidal curve fitting

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MAT 204 SP09 The graph of y = sin x The graph of y = sin x is a cyclical curve that takes on values between –1 and 1. The range of y = sin x is _____________. Each cycle (wave) corresponds to one revolution of the unit circle. The period of y = sin x is _______ or _______. Graphing the sine wave on the x-y axes is like “unwrapping” the values of sine on the unit circle.

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**Take a look at the graph of y = sin x:**

MAT 204 SP09 Take a look at the graph of y = sin x: (one cycle) Some points on the graph of y = sin x

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**More about the graph of y = sin x**

MAT 204 SP09 More about the graph of y = sin x Notice that the sine curve is symmetric about the origin. Therefore, we know that the sine function is an ODD function; that is, for every point (x, y) on the graph, the point (-x, -y) is also on the graph. For example, are on the graph of y = sin x.

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**Using Key Points to Graph the Sine Curve**

MAT 204 SP09 Using Key Points to Graph the Sine Curve Once you know the basic shape of the sine curve, you can use the key points to graph the sine curve by hand. The five key points in each cycle (one period) of the graph are: 3 x-intercepts maximum point minimum point

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MAT 204 SP09 The graph of y = cos x The graph of y = cos x is also a cyclical curve that takes on values between –1 and 1. The range of the cosine curve is ________________. The period of the cosine curve is _______ or _______.

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**Take a look at the graph of y = cos x:**

MAT 204 SP09 Take a look at the graph of y = cos x: (one cycle) Some points on the graph of y = cos x

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**More about the graph of y = cos x**

MAT 204 SP09 More about the graph of y = cos x Notice that the cosine curve is symmetric about the y-axis. Therefore, we know that the cosine function is an EVEN function; that is, for every point (x, y) on the graph, the point (-x, y) is also on the graph. For example, are on the graph of y = cos x.

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**Using Key Points to Graph the Cosine Curve**

MAT 204 SP09 Using Key Points to Graph the Cosine Curve Once you know the basic shape of the cosine curve, you can use the key points to graph the cosine curve by hand. The five key points in each cycle (one period) of the graph are: maximum point 2 x-intercepts minimum point

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**Characteristics of the Graphs of y = sin x and y = cos x**

MAT 204 SP09 Characteristics of the Graphs of y = sin x and y = cos x Domain: ____________ Range: ____________ Amplitude: The amplitude of the sine and cosine functions is half the distance between the maximum and minimum values of the function. The amplitude of both y= sin x and y = cos x is ______. Period: The length of the interval needed to complete one cycle. The period of both y= sin x and y = cos x is ________.

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**Transformations of the graphs of y = sin x and y = cos x**

MAT 204 SP09 Transformations of the graphs of y = sin x and y = cos x Reflections over x-axis Vertical Stretches or Shrinks Horizontal Stretches or Shrinks/Compression Vertical Shifts Phase shifts (Horizontal)

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MAT 204 SP09 I. Reflection in x-axis Example:

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**II. Vertical Stretch or Compression**

MAT 204 SP09 II. Vertical Stretch or Compression (Amplitude change) Example

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**II. Vertical Stretch or Compression**

MAT 204 SP09 II. Vertical Stretch or Compression *Note: If the curve is vertically stretched if the curve is vertically shrunk

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MAT 204 SP09 Example The graph of a function in the form y = A sinx or y = A cosx is shown. Determine the equation of the specific function.

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**III. Horizontal Stretch or Compression (Period change)**

MAT 204 SP09 III. Horizontal Stretch or Compression (Period change) Example

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**III. Horizontal Stretch or Compression (Period change)**

MAT 204 SP09 III. Horizontal Stretch or Compression (Period change) *Note: If the curve is horizontally stretched If the curve is horizontally shrunk

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MAT 204 SP09 Graphs of Examples State the amplitude and period for each function. Then graph each of function using your calculator to verify your answers. (Use radian mode and ZOOM 7:ZTrig)

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MAT 204 SP09

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MAT 204 SP09 x y

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MAT 204 SP09 V. Vertical Shifts Example

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MAT 204 SP09 V. Phase Shifts Example

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MAT 204 SP09 EXAMPLE For , determine the amplitude, period, and phase shift. Then sketch at least one cycle of the function by hand. x y

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MAT 204 SP09 EXAMPLE List all of the transformations that the graph of y = sin x has undergone to obtain the graph of the new function. Graph the function by hand.

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MAT 204 SP09 EXAMPLE (CONTINUED) x y

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MAT 204 SP09 EXAMPLE List all of the transformations that the graph of y = sin x has undergone to obtain the graph of the new function. Graph the function by hand.

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MAT 204 SP09 EXAMPLE (CONTINUED) x y

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MAT 204 SP09 * *NOTE: In 2005, summer solstice was on June 21 (172nd day of the year).

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MAT 204 SP09 Use a graphing calculator to graph the scatterplot of the data in the table below. Then find the sine function of best fit for the data. Graph this function with the scatterplot.

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MAT 204 SP09

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MAT 204 SP09 End of Sections 7.6 & 7.8

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