1. Lesson 4-6 Graphs of Secant and Cosecant

2. 2 Get out your graphing calculator… Graph the following y = cos x y = sec x What do you see??

3. 3 y x Secant Function Graph of the Secant Function 2. range: (– ,–1]  [1, +  ) 3. period: 2  4. vertical asymptotes: 1. domain : all real x The graph y = sec x, use the identity. Properties of y = sec x At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes.

4. 4 First graph: y = 2cos (2x – π) + 1 Then try: y = 2sec (2x – π) + 1

5. 5 Graph Graph the following y = sin x y = csc x What do you see??

6. 6 x y Cosecant Function Graph of the Cosecant Function 2. range: (– ,–1]  [1, +  ) 3. period: 2  where sine is zero. 4. vertical asymptotes: 1. domain : all real x To graph y = csc x, use the identity. Properties of y = csc x At values of x for which sin x = 0, the cosecant function is undefined and its graph has vertical asymptotes.

7. 7 First graph: y = -3 sin (½x + π/2) – 1 Then try: y = -3 csc (½x + π/2) – 1

8. 8 Key Steps in Graphing Secant and Cosecant 1.Identify the key points of your reciprocal graph (sine/cosine), note the original zeros, maximums and minimums 2.Find the new period (2π/b) 3.Find the new beginning (bx - c = 0) 4.Find the new end (bx - c = 2π) 5.Find the new interval (new period / 4) to divide the new reference period into 4 equal parts to create new x values for the key points 6.Adjust the y values of the key points by applying the change in height (a) and the vertical shift (d) 7.Using the original zeros, draw asymptotes, maximums become minimums, minimums become maximums… 8.Graph key points and connect the dots based upon known shape

9. Graphs of Tangent and Cotangent Functions

10. 10 Tangent and Cotangent Look at:  Shape  Key points  Key features  Transformations

11. Graph Set window Domain: -2π to 2π x-intervals: π/2 (leave y range) Graph y = tan x 11

12. 12 y x Graph of the Tangent Function 2. range: (– , +  ) 3. period:  4. vertical asymptotes: 1. domain : all real x Properties of y = tan x period: To graph y = tan x, use the identity. At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes.

13. 13 Graph y = tan x and y = 4tan x in the same window What do you notice? y = tan x and y = tan 2x What do you notice? y = tan x and y = -tan x What do you notice?

14. Graph Set window Domain: 0 to 2π x-intervals: π/2 (leave y range) Graph y = cot x 14

15. 15 Cotangent Function Graph of the Cotangent Function 2. range: (– , +  ) 3. period:  4. vertical asymptotes: 1. domain : all real x Properties of y = cot x y x vertical asymptotes To graph y = cot x, use the identity. At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes.

16. 16 Graph Cotangent y = cot x and y = 4cot x in the same window What do you notice? y = cot x and y = cot 2x What do you notice? y = cot x and y = -cot x What do you notice? y= cot x and y = -tan x

17. 17 Key Steps in Graphing Tangent and Cotangent Identify the key points of your basic graph 1.Find the new period (π/b) 2.Find the new beginning (bx - c = 0) 3.Find the new end (bx - c = π) 4.Find the new interval (new period / 2) to divide the new reference period into 2 equal parts to create new x values for the key points 5.Adjust the y values of the key points by applying the amplitude (a) and the vertical shift (d) 6.Graph key points and connect the dots