Vertical Translations of Sine θ and Cosine θ

Transcription

1 Vertical Translations of Sine θ and Cosine θ 1. Complete the following table. You will be plotting 360 of the functions. Use one decimal place. sin(θ) sin(θ) Plot the values of y=sin(θ) and y=sin(θ) + 2 for {0 θ 360 } and draw a smooth curve through the points. Your curve should be smooth! Label your axis and use the full scale! Use colour! 3. Look at your graph and answer the following questions: a) State the amplitude and period of y=sin(θ): amplitude, period b) State the amplitude and period of y=sin(θ) + 2: amplitude, period c) In part b), describe the kind of transformation is occurring 4. Without making a table of values, sketch y=sin(θ) 1 on the axis above. 5. Get the idea? Sketch y=cos(θ) and its translations below. Then fill in the table. Domain y=cos(θ) y=cos(θ) + 2 y=cos(θ) - 1

2 Horizontal Translations of Sine θ and Cosine θ 1. Complete the following table. You will be plotting 360 of the functions. Use one decimal place. sin(θ) sin(θ + 45 ) Using θ and the y-axis, plot the values of y=sin(θ) and y=sin(θ + 45 ) for {0 θ 360 } from the table above and draw a smooth curve through the points. Your curve should be smooth! Label your axis and use the full scale! Use colour! 3. Look at your graph and answer the following questions: a) State the amplitude and period of y=sin(θ): amplitude, period b) State the amplitude and period y=sin(θ + 45 ): amplitude, period c) In part b), describe the kind of transformation is occurring 4. Without making a table of values, sketch y=sin(θ - 45 ) on the axis above. 5. Get the idea? Sketch y=cos(θ) and its translations below. Then fill in the table. Domain Phase shift y=cos(θ) y=cos(θ + 45 ) y=cos(θ - 45 )

3 Vertical Stretches of Sine θ and Cosine θ 1. Complete the following table. You will be plotting 360 of the functions. Use one decimal place. sin(θ) sin(θ) Using θ and the y-axis, plot the values of y=sin(θ) and y=2sin(θ) for {0 θ 360 } from the table above and draw a smooth curve through the points. Your curve should be smooth! Label your axis and use the full scale! Use colour! 3. Look at your graph and then fill in the table for y=sin(θ) and y=2sin(θ). Sketch y= sin(θ) without making a table and fill in the amplitude and period as well. y=sin(θ) y=2sin(θ) y= sin(θ) 4. An invariant point is one where the value doesn t change between functions when they are compared. State the invariant points (as compared to y=sin(θ) ) for y=2sin(θ) and y= sin(θ) functions. 5. Get the idea? Sketch y=cos(θ) and its transformations below. Then fill in the table. Mark and label the invariant points on the graph. Invariant Points y=cos(θ) y=2cos(θ) y= cos(θ)

4 Horizontal Stretches of Sine θ and Cosine θ 1. Complete the following table for sin(θ) values. Use one decimal place. sin(θ) sin(2θ) Using θ and the y-axis, plot the values of y=sin(θ) and y=sin(2θ) for {0 θ 720 } from the table above and draw a smooth curve through the points. Your curve should be smooth! Label your axis and use the full scale! Use colour! 3. Look at your graph and then fill in the table for y=sin(θ) and y=sin(2θ). Sketch y=sin( θ) without making a table and fill in the amplitude and period as well. y=sin(θ) y=sin(2θ) y=sin( θ) 4. State the invariant points (as compared to y=sin(θ) ) of y=sin(2θ) and y=sin( θ) functions. 5. Get the idea? Sketch y=cos(θ) and its translations below. Then fill in the table. Mark and label the invariant points on the graph. Invariant Points y=cos(θ) y=cos(2θ) y=cos( θ)

5 Combinations of Transformations of Sine θ and Cosine θ T/I Complete the following table for y = 3sin [2(θ - 45º)] values. Use one decimal place. 3sin 2(θ - 45º) Predict, from your table above: Phase shift 3. Using θ and the y-axis, plot y = 3sin [2(θ - 45º)] for {0 θ 450 } from the table above and draw a smooth curve through the points. Identify using colour the range and phase shift on your curve. 4. Complete the following table for y = 4cos ( θ + 90º) - 1 values. Note how the brackets are different in this function than in q.1. Use one decimal place. θ (degrees) cos ( θ + 90º) Predict, from your table above: Phase shift 6. Using θ and the y-axis, plot y = 4cos ( θ + 90º) - 1 for {0 θ 900 } from the table above and draw a smooth curve through the points. Identify using colour the range and phase shift on your curve. 7. In y = 4cos ( θ + 90º) - 1, if ( θ + 90º) is changed to (θ + 45º), would there be any change in the function? Test this and report here:

6 Homework Exercises! You Bet! Before you can do the exercises, you should know something about radian measure. It s easy. Radians are an angular measurement. They are multiples of π (pi). π radians are equal to 180º That s all! Complete this table of degree radian equivalents: Reduce to lowest form Equivalent in degrees Reduce to lowest form Equivalent in degrees 0π/12 1π/12 2π/12 3π/12 4π/12 5π/12 6π/12 7π/12 8π/12 9π/12 10π/12 11π/12 12π/12 0π π/4 π/2 3π/4 π 0º 45º 90º 135º 180º 13π/12 14π/12 15π/12 16π/12 17π/12 18π/12 19π/12 20π/12 21π/12 22π/12 23π/12 24π/12 7π/6 4π/3 5π/3 2π Simply convert any radian measure to degrees and away you go! One other thing, in case you didn t read your text... Now try these exercises: Finding the period of y=sin(kθ) or y=cos(kθ): The period is 360º/k # 1 16 and 18 on pages 374/6 and # 1 8, 10, 11, 12, 13 & 16 on pages Use DEGREE measure for ALL exercises!