1. Graphs of Sine and Cosine Five Point Method

2. 2 Plan for the Day Review Homework –4.5 P 307 3-21 odd, 23-26 all The effects of “b” and “c” together in the equations: y = a (cos (bx – c)) + d y = a (sin (bx – c)) + d Graphing of Sine and Cosine Functions using the 5 “key” points Homework Quiz next time

3. 3 Cosine Function Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1001cos x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = cos x

4. 4 Sine Function Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 0010sin x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = sin x

5. 5 Properties of Sine and Cosine Functions 6. The cycle repeats itself indefinitely in both directions of the x-axis. Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 1. The domain is the set of real numbers. 5. Each function cycles through all the values of the range over an x-interval of. 2. The range is the set of y values such that.

6. Summarizing … Standard form of the equations: y = a (cos (bx – c)) + d y = a (sin (bx – c)) + d “a” - |a| is called the amplitude, like our other functions it is like a stretch it affects “y” or the output If a < 0 it also causes a reflection across the x- axis “d” – vertical shift, it affects “y” or the output “c” – horizontal shift, it affects “x” or “θ” or the input “b” – period change (“squishes” or “stretches out” the graph – horizontal stretch or shrink) to find the new period, 2π/b The combination of “b” and “c” has another effect …

7. Dealing with (bx – c) The “c” causes a shift (opposite the sign) left and right, “b” it changes the frequency of the graph ( 2π / b is the new period), it is a horizontal stretch or shrink. When they are together, you apply the frequency change and then the shift There is a method to complete this… 7

8. You begin by adjusting the reference period Start with the standard “key points” Determine where the new reference period begins and end Set new intervals for the maximums, minimums, and zeros. Adjust the “x” values based upon this information. Adjust the “y” values with the amplitude and vertical shift. Plot your new points and graph! 8

9. The original reference period is 2π and regular intervals of π / 2. If there is a b or c (or both) that can change. 1.The “parent” has a reference period that begins at zero. You need to find the new beginning of the reference period. Find the new beginning, (bx – c = 0), solve for x. x is the new beginning. 2.The original reference period ends at 2π, find the new end (bx – c = 2π), solve for x. x is the new end. 3.The original reference period is 2π and has 4 equal periods of π/2. Find the new period (2π/b ), and divide the new period into 4 equal parts to create the new intervals. 4.Use this information to find new x values in key points 5.Adjust the y values of the key points by applying the amplitude (with sign or a) and the vertical shift (d) 9

10. Example Graph: Begin with our key points. Where do they come from? 10 1001cos x 0x

11. Example Find the new beginning: bx – c = 0, solve for x. x is the new beginning. Find the new end: bx – c = 2π, solve for x. x is the new end. Find the new period: 2π/b 11 1001cos x 0x

12. Example Find the new beginning: π / 8 Find the new end: 9π / 8 Find the new period: π Break the new period into 4 equal intervals: π / 4 12 1001cos x 0x

13. Example Beginning: π / 8 End: 9π / 8 New intervals: π / 4 13 1001cos x 0x 1001cos x x

14. Example 14 3x-1+1=-23x0+1=13x1+1=4cos x x 1 0 0 1cos x x 3x0+1=1 3x1+1=4

15. Example 15 41 -2 14cos x x

16. 16 Calculator Issues Window settings: –Using your reference period to set your window –Setting scale based upon your new intervals

17. 17 Summarizing … How do you put it all together? 1.Identify the key points of your basic graph 2.Find the new period (2π/b) 3.Find the new beginning (bx - c = 0) 4.Find the new end (bx - c = 2π) 5.Divide the new period into 4 equal parts to create new interval to find x values of the key points 6.Adjust the y values of the key points by applying the amplitude (with sign or a) and the vertical shift (d)

18. Homework 25 Page 308 41, 42, 44, 49, 51 18