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Lesson 4-6 Graphs of Secant and Cosecant

Lesson 4-6 Graphs of Secant and Cosecant. Get out your graphing calculator…. Graph the following y = cos x y = sec x What do you see??. The graph y = sec x , use the identity. y. Properties of y = sec x. 1. domain : all real x. x. 4. vertical asymptotes:.

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Lesson 4-6 Graphs of Secant and Cosecant

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  1. Lesson 4-6Graphs of Secant and Cosecant

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

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

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

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

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

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

  8. Key Steps in Graphing Secant and Cosecant • Identify the key points of your reciprocal graph (sine/cosine), note the original zeros, maximums and minimums • Find the new period (2π/b) • Find the new beginning (bx - c = 0) • Find the new end (bx - c = 2π) • 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 • Adjust the y values of the key points by applying the change in height (a) and the vertical shift (d) • Using the original zeros, draw asymptotes, maximums become minimums, minimums become maximums… • Graph key points and connect the dots based upon known shape

  9. Graphs of Tangent and Cotangent Functions

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

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

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

  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? 13

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

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

  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 16

  17. Key Steps in Graphing Tangent and Cotangent Identify the key points of your basic graph Find the new period (π/b) Find the new beginning (bx - c = 0) Find the new end (bx - c =π) 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 Adjust the y values of the key points by applying the amplitude (a) and the vertical shift (d) Graph key points and connect the dots 17

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