1. Graphs of Tangent and Cotangent Functions

2. Plan for the Day Review Homework Graphing Tangent and Cotangent

3. Key Steps in Graphing Sine and Cosine
Identify the key points of your basic graph
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 amplitude (a) and the vertical shift (d)
Graph key points and connect the dots

4. 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 amplitude (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

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

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

7. Graph of the Tangent Function
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. y x
Properties of y = tan x
1. domain : all real x
2. range: (–, +)
3. period: 
4. vertical asymptotes:
period:

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

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

10. Graph of the Cotangent Function 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. y x
Properties of y = cot x
vertical asymptotes
1. domain : all real x
2. range: (–, +)
3. period: 
4. vertical asymptotes:
Cotangent Function

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

12. 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