Section 8.4: The Equations of Sinusoidal Functions Stop Sine 1
In this section, we will examine transformations of the sine and cosine function and learn how to read various properties from the equation. Transformation How a graph can be adjusted. The graph of a function may be changed either by shifting, stretching or compressing, or applying a reflection. Transformed sinusoidal equations are written in the following forms: y = a sin b(x c) + d y = a cos b(x c) + d 2
Examining the Impact of the "a" Value on the Graph of a Sinusoidal Function 1. Examine the following. A) y = sin x Period: Amplitude: Midline: B) y = 2 sin x Period: Amplitude: Midline: C) y = ½ sin x Period: Amplitude: Midline: D) y = 2 sin x Period: Amplitude: Midline: 3
Questions: 1. What happens to the amplitude if a > 0? 2. Is the shape of the graph affected by the parameter a? 3. How is the range affected by the parameter a? 4. Will the value of a affect the cosine graph in the same way that it affects the sine graph? Summary of the "a" Value The "a" value stretches or compresses a graph vertically. It equals the amplitude of the function. Making a negative will cause a reflection in the x axis. Maximums will become Minimums and vice versa. A = amplitude 4
Example: Determine the "a" value and state the amplitude for each equation. (A) y = 2 sin 3(x 90 o ) + 1 (B) y = 0.75 cos 2(x + 45 o ) +3 5
Examining the Impact of the "d" Value on the Graph of a Sinusoidal Function Example 2. Examine the following graphs A) y = sin x Period: (or ) Amplitude: 1 Midline: y = 0 (B) y = sin x + 2 Period: Amplitude: Midline: (C) y = sin x 3 Period: Amplitude: Midline: 6
Questions 1. How does each graph change when compared to y = sinx? 2. How is the value of d related to the equation of the midline? 3. Is the shape of the graph or the location of the graph affected by the parameter d? 4. Is the period affected by changing the value of d? 5. Will the value of d affect the cosine graph in the same way that it affects the sine graph? Summary of the "d" Value The "d" value gives us the equation of the midline of a sinusoidal function. Equation of Midline: y = d 7
Examples: Identify the equation of the midline for each equation. (A) y = 2 sin 3(x 90) + 1 (B) y = 0.75 cos 2(x + 45) 3 8
Examining the Impact of the "b" Value on the Graph of a Sinusoidal Function (A) y = sin x Period: (or ) Amplitude: 1 Midline: y = 0 (B) y = sin ½x Period: Amplitude: Midline: (C) y = sin 3x Period: Amplitude: Midline: 9
QUESTIONS 1. How does each graph change when compared to y = sinx? 2. How is the value of b related to the period? Write an equation that relates period to the b value. Summary of the "b" Value The "b" value gives us the period of a sinusoidal function. In degrees, In radians, 10
Examples: Determine the period for each equation. (A) y = 2 sin 3(x 90 ) + 1 (B) y = 0.75 cos 2(x + 45 ) 3 (C) (D) 11
Examining the Impact of the "c" Value on the Graph of a Sinusoidal Function Use your calculator to graph each function. (A) y = sin x Period: (or ) Amplitude: 1 Midline: y = 0 Key point (0,0) (B) y = sin (x 60) Period: Amplitude: Midline: Key Point: (C) y = sin (x + 30) Period: Amplitude: Midline: Key Point: (D) y = cos(x 60) Period: Amplitude: Midline: Key Point: (E) y = cos (x + 30) Period: Amplitude: Midline: Key Point: 12
QUESTIONS 1. How does each graph change when compared to the original? 2. How is the value of c related to starting point? 3. How can we determine the "phase shift" from the equation of the sinusoidal function? 4. Will the value of c affect the cosine graph in the same way that it affects the sine graph? More on the "c" Value... The "c" value shifts a graph horizontally. The shift is obtained by taking the opposite sign of the number after x in the equation. Horizontal Shift/Translation = c 13
Examples: Determine the horizontal shift for each equation. (A) y = 2 sin3(x 90 ) + 1 (B) y = 0.6 sin2(x + 45 ) 3 14
Maximum/Minimum Points Midline can be read from a graph, or it can be calculated if we are given an equation. Maximum Point = midline + amplitude Minimum Point = midline amplitude Recall... So, amplitude = a and the equation of the midline is y = d Maximum point = d + a Minimum point = d a 15
Domain and Range of a Sinusoidal Equation For a sinusoidal function the domain is as follows: This may change in application problems in which the x values are restricted. For a sinusoidal function the range is as follows: 16
Summary of the effects of the parameters a, b, c, and d on the graphs of y = sin x and y = cos x or 1) a changes the amplitude (stretches the graph vertically) 2) b affects the period of the base graph. It either shortens or lengthens it (speeds it up or slows it down) 3) c i) (x + c) Horizontal Translation: HT = c i.e. the base graph is shifted c units left. ii) (x c) Horizontal Translation: HT = c i.e. the base graph is shifted c units right. 4) d i) causes the base graph to move up or down vertically (shift in y) ii) also y = d is the new MIDLINE 17
Ex) For y = 4 cos 2(x 45 ) + 7 a = b = c = d = Amplitude = Period = Horizontal Translation: Vertical Translation: New Midline: Range: 18
Ex) For a = b = c = d = Amplitude = Period = Horizontal Translation: Vertical Translation: New Midline: Range: 19
Ex) For a = b = c = d = Amplitude = Period = Horizontal Translation: Vertical Translation: New Midline: Range: 20
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Ex) Use the sinusoidal function shown below to answer the questions that follow. (a)determine the amplitude, period, equation of midline and the range. (b)use the information from part (i) to determine a function that represents the graph in the form y = a cos b(x c) + d. Questions 1 9 and 12 14, 15, 17 page 528 529 22
Graph at the table of values using a graphing calculator. Have a look 23
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