1. Periodic Functions. A periodic function is a function f such the f(x) = f(x + np) for every real number x in the domain of f, every integer n, and some positive real number p. The least possible positive value of p is the period of the function.

2. Graphing Sine and Cosine functions.

4. How to graph by hand. 1. Plot the 5 quadrant values for 1 period. 2. Repeat to the left. 3. For better accuracy, consider plotting when sin (x) = ½. These points are one tick mark left and right of the x intercept. y = sin(x) is an odd function. f(-x) = – f(x)

5. How to graph by hand. 1. Plot the 5 quadrant values for 1 period. 2. Repeat to the left. 3. For better accuracy, consider plotting when sin (x) = ½. These points are one tick mark left and right of the x intercept.

6. y = sin (x) y = cos (x) y = 1 y = -1 The Amplitude of a Sine or Cosine curve is half the distance from the relative minimum value and relative maximum value. The distance between -1 to 1 is 2 and half of 2 is 1. The amplitude is 1. The Amplitude is located in the equation. 1 1 The Amplitude of a Sine or Cosine curve is | A |.

7. Graph by hand… The negative on the 3 will make the Sine curve flip over the x – axis.

8. Changing the Period Length. This is a Horizontal Stretch or Shrink. We need to multiply a constant to the x in the function. Graph.

9. Phase Shift. This is a Horizontal Shift left or right. Set Bx – C = 0 and solve for x. x = +, shift Right. x = –, shift Left. Graph. Left pi units.

10. Vertical Shift. +D = shift Up. – D = shift Down. Graph. Up 2 units.

11. Graph. Plot the first 5 points according to the Amplitude. Shift the 5 points up 1 unit. Draw in the Cosine curve for one period. Fill the graph with as many Periods as needed.

12. Determine all the Transformations. Amplitude ________________ Period ___________________ Phase Shift _______________ Vertical Shift _____________ Amplitude ________________ Period ___________________ Phase Shift _______________ Vertical Shift _____________

13. Graphing Tangent and Cotangent functions.

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15. How to graph by hand. 1. Plot vertical asymptotes at every odd. 2. x – intercepts are at every. k is an integer. (halfway between V.A.) 3. Halfway between the asymptotes and x – int. the y values are -1 and 1 from left to right. ( Quarter points) Make what looks like a cubic curve and don’t touch the asymptotes. Domain: Range:

16. How to graph by hand. 1. Plot vertical asymptotes at every. k is an integer. 2. x – intercepts at every odd. (halfway between V.A.) 3. Halfway between the asymptotes and x – int. the y values are 1 and -1 from left to right at the quarter points. Make what looks like a cubic curve and don’t touch the asymptotes. Remember, tangent is increasing and cotangent is decreasing. Domain: Range:

17. How to graph by hand. 1. Plot cos(x) as a reference. 2. V.A. Through the x – int. of the cosine curve. 3. Plot the points at the relative maxs. and mins. of cos(x) curve. 4. Plot the reciprocal value of the y – values of ½ of cos(x) as a 2 for sec(x) curve. 5. Draw the curves as wide parabolas near the vertex. Domain: Range:

18. How to graph by hand. 1. Plot sin(x) as a reference. 2. V.A. Through the x – int. of the sine curve. 3. Plot the points at the relative maxs. and mins. of sin(x) curve. 4. Plot the reciprocal value of the y – values of ½ of sin(x) as a 2 for csc(x) curve. 5. Draw the curves as wide parabolas near the vertex. Domain: Range:

19. How to graph y = Atan(Bx – C) + D by hand. 1. Determine the Vertical Asymptotes. Solve for x. 2. The x – intercept is halfway between the vertical asymptotes. 3. The quarter points have y – values of – A and + A, from left to right. There is no amplitude. If you have – A in the equation, flip over x – axis. **If you have – B in the equation, flip over y – axis. 4. D is still used for the Vertical Shift. Bx – C = 0 is still used to find the Phase Shift. Add this value to the V.A. equations.

20. How to graph y = Acot(Bx – C) + D by hand. 1. Determine the Vertical Asymptotes. Solve for x. 2. The x – intercept is halfway between the vertical asymptotes. 3. The quarter points have y – values of + A and + – A, from left to right. There is no amplitude. If you have – A in the equation, flip over x – axis. **If you have – B in the equation, flip over y – axis. Bx – C = 0 is still used to find the Phase Shift. Add this value to the V.A. equations. 4. D is still used for the Vertical Shift.

21. How to graph by hand. 1. Determine the Vertical Asymptotes. Solve for x. **If you have – B in the equation, flip over y – axis. 2. The vertex is halfway between the vertical asymptotes and is located at A and - A. If A is negative, then flip over x – axis. 3. The reciprocal points should be multiplied by A. Bx – C = 0 is still used to find the Phase Shift. Add this value to the V.A. equations. 4. D is still used for the Vertical Shift. I recommend that we make the changes to the Cosine curve 1 st and then draw in the Secant curve in between the asymptotes.

22. How to graph by hand. 1. Determine the Vertical Asymptotes. Solve for x. **If you have – B in the equation, flip over y – axis. 2. The vertex is halfway between the vertical asymptotes and is located at A and - A. If A is negative, then flip over x – axis. 3. The reciprocal points should be multiplied by A. Bx – C = 0 is stilled used to find the Phase Shift. Add this value to the V.A. equations. 4. D is still used for the Vertical Shift. I recommend that we make the changes to the Sine curve 1 st and then draw in the Cosecant curve in between the asymptotes.

23. Graph Shift the curve down 1 unit. Find the location of the VA’s Draw the tangent curve. The x-intercept is halfway between the VA’s Copy the graph to the right and left. The quarter points are at half of the halves with y coordinates of 1 and -1.

24. Graph Draw one period of y = tan(x). Find the period length. Find the shift. Shift to the left. Double the radians for the points. Down 1. Copy the graph to the right and left.

25. Graph Multiplying 2 to every y-coordinate. Vertical shift up 1 unit. Copy to the right and left. Find the location of the VA’s The x-intercept is halfway between the VA’s The quarter points are at half of the halves with y coordinates of 1(A) and -1(A). A = 2, so 2 & -2. Draw the cotangent curve.

26. Graph Draw one period of y = cot(x). Find the period length. No change. Find the shift. Shift to the right. Multiplying 2 to every y-coordinate. Vertical shift up 1 unit.Copy to the right and left.

27. Graph No changes inside the secant function. Start with the basic curve of Cosine from Quadrant 4 to Quadrant 3 Vertical Asymptotes at the x-intercepts. A = ½, multiply all y-coordinates by ½. D = 1, vertical shift up one unit. Draw in the Secant Curve…remember that the y-coordinates of ½ for Cosine are now flipped and the value is 2 for the y-coordinate. Copy to the right and left.

28. Graph 1. V.A. Solve for x. Draw the Sine Curve for reference. Draw in the Cosecant Curve. Copy the curve into the rest of the graph.

29. Change the MODE to Degrees Enter the equation into Y = ZOOM 7 to activate the Trig window