Pre-Calculus Unit 6 Lesson 9: The Tangent and Cotangent Functions
Pre-Calculus Honors
Book Reference 4.4
Unit 6 Lesson 9: Discovering the Tangent & Cotangent Function
Objective: ______
1. Discovering the Tangent Function
Unit Circle / Numerical Representation/ Do Now: Complete the table of values without a calculator.
x / / / 0 / /
tanx
/ x / / / /
tanx
Graphical Representation
Graph f(x) = tanx without using a graphing calculator.
Communication
Directions: Answer the following questions below using the graph and table above.
1. What is the domain and range of the function f(x) = tanx in your own words?
______
______
2. What is the period of the function f(x) = tanx? Draw the left side of this function based on these observations.
______
3. What are the vertical asymptotes of the function f(x) = tanx from ?
______
4. What are the zeros of the function f(x) = tanx from ?
______
2. Discovering the Cotangent Function
Unit Circle / Numerical Representation/ Complete the table of values without a calculator.
x / / / 0 / /
cotx
/ x / / / /
cotx
Graphical Representation Graphical Representation Graphical Representation Graphical Representation
Graph f(x) = cotx without using a graphing calculator.
Communication
Directions: Answer the following questions below using the graph and table above.
1. What is the domain and range of the function f(x) = cotx in your own words?
______
______
2. What is the period of the function f(x) = cotx? Draw the left side of this function based on these observations.
______
3. What are the vertical asymptotes of the function f(x) = cotx from ?
______
4. What are the zeros of the function f(x) = cotx from ?
______
5. What do you notice about the functions y = cotx compared to the function y = tanx?
______
Pg 439 #(18, 20, 21, 23, 24, 25, 27, 39, 41, 43, 63, 64)
(Day 2)
1. Do Now: Refer to the following questions from the communication box on yesterday’s classwork. Discuss your answers within your groups.
Write a general formula for where the asymptotes occur for the function y = tanx.
______= ______where ______.
Write a general formula for where the asymptotes occur for the function y = cotx.
______= ______where ______.
What do you notice about properties (intercepts, asymptotes, period, domain, and range) of the function f(x) = cotx compared to the function f(x) = tanx?
______
Based on the graph and table you completed last class, what are the three formulas for the functions y = tanx and y = cotx?
Amplitude: ______
Phase Shift: ______
Period: ______
Pre-Calculus Unit 6 Lesson 9: The Tangent and Cotangent Functions
Pre-Calculus Unit 6 Lesson 9: The Tangent and Cotangent Functions
2. Look up Unwrapping the Unit circle for the tangent function. Write down the three basic points for the trig function and where the two vertical asymptotes occur in the interval [-π2,π2].
Pre-Calculus Unit 6 Lesson 9: The Tangent and Cotangent Functions
Xy = tanx
Vertical Asymptote #1 / Vertical Asymptote #2
Pre-Calculus Unit 6 Lesson 9: The Tangent and Cotangent Functions
3. Guided Practice: Shifting the Tangent Function
Example 1: Sketch and analyze the graph of
a.) Find the two corresponding asymptotes of the function after the transformations.
Asymptote #1 / Asymptote #2b.) Write the three basic points of the parent function, then find the three corresponding points.
Pre-Calculus Unit 6 Lesson 9: The Tangent and Cotangent Functions
xy = tanx
x
Pre-Calculus Unit 6 Lesson 9: The Tangent and Cotangent Functions
c.) Graph 2 periods of the function on the coordinate plane below.
4. Group Practice: Shifting the Tangent Function
Example 2: Sketch and analyze the graph of
a.) Find the two corresponding asymptotes of the function after transformations.
Asymptote #1 / Asymptote #2b.) Write the three basic points of the parent function, then find the three corresponding points.
Pre-Calculus Unit 6 Lesson 9: The Tangent and Cotangent Functions
xy = tanx
x
Pre-Calculus Unit 6 Lesson 9: The Tangent and Cotangent Functions
c.) Graph 2 periods of the function on the coordinate plane below.
Homework: PG 439 #55-58 (same book page handout as last class)