3.2: Vertical and Phase Shifts
This activity is intended to supplement Trigonometry, Chapter 2, Lesson 4.
ID: 9608
Time required: 45 minutes
Activity Overview
Students explore vertical and phase shifts of sine and cosine functions, after a brief review of period and amplitude. Students will use values in lists to change the values of parameters in trigonometric functions; they will determine the effect that each change has upon the shape of the graph. They will then use this knowledge to write a sine function as a cosine function.
Topic: Trigonometric Functions

Analyze and predict the effects of changes in
A ,B andC on the graphs ofAsin(Bx+C) ,Acos(Bx+C) , andAtan(Bx+C) and interpretA ,B , andC in terms of amplitude, period, and phase shifts.  Approximate the amplitude, frequency, and phase shift of the primary trigonometric functions by graphing.
Teacher Preparation and Notes
 Students should already be familiar with the graphs of the sine and cosine functions, and they should also have some experience in determining the period (in radians) and identifying the amplitude of a trigonometric function.

The graphs of trigonometric functions that students will view on their calculators have been constructed such that the horizontal scale is in multiples of
π2 .  This activity is intended to be introduced in a wholeclass setting and completed by students individually.
 The programs utilize the Transformation Graphing Application.
 To download the Transformation Graphing Application and calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=9608 and select PHSESHFT, TRIGCOMB, and the Transformation Graphing from the Applications header.
Associated Materials
 Student Worksheet: Vertical and Phase Shifts http://www.ck12.org/flexr/chapter/9700, scroll down to the second activity.
 Transformation Graphing Application
 PHSESHFT.8xp and TRIGCOMB.8xp
Before beginning the activity, students should clear out any functions from the o screen, turn off all Stat Plots, and make sure that the calculator is in Radian mode. They will also utilize the Transformation Graphing App. To start this app, press the APPS and select Transfrm from the list and press ENTER twice to activate the application.
Problem 1 – Amplitude
This problem allows students to review the amplitude of a function of the form
Problem 2 – Period
This problem allows students to review the period of a function of the form
After some examination, students should be able to identify the relationship: period
Problem 3 – A simple phase shift
The graph of
Problem 4 – Vertical shift
The parameter
Problem 5 – Combining transformations
While students may be wondering why they cannot declare “phase shift
To establish exactly how
Note: It is easiest to identify this relationship if parameters
Problem 6 – Bringing it all together
Students are asked to summarize their findings from Problems 1–5. Make sure students have correctly completed these answers before proceeding further.
Next, students are asked to return to Problems 1–5 and replace each “\begin{align*}\sin\end{align*}” in the function definition to “\begin{align*}\cos\end{align*}” to verify that the same characteristics hold true.
Finally, students will apply what they have learned about vertical and phase shifts. They are given the equations and graphs of two sine functions and asked to find equations of cosine functions that coincide. Students should observe that the values of \begin{align*}a, b\end{align*}, and \begin{align*}d\end{align*} remain the same for each sine/cosine pair; the only difference occurs in the value of \begin{align*}c\end{align*}. Because these functions are periodic, there are infinitely many equations that satisfy each condition. Be sure to check students’ equations.
Solutions
Problem 1
 students should conclude that the sine curve is vertically stretched by a factor of \begin{align*}a\end{align*}
 if \begin{align*}a\end{align*} is negative, then the curve is reflected over the \begin{align*}x\end{align*}axis
 amplitude \begin{align*}= a\end{align*}
Problem 2
Students will find that the value of \begin{align*}b\end{align*} affects the horizontal stretch of this function and thus changes the period of the function. After some examination, students should be able to identify the relationship: period \begin{align*}= \frac{2\pi}{b}\end{align*}.
Problem 3
Students might predict that a change in \begin{align*}c\end{align*} will result in a horizontal shift a certain number of units (here called a phase shift), but how that number of units relates to \begin{align*}c\end{align*} will not be immediately clear.
Problem 4
Students should confirm that the vertical shift is equal to this parameter. (vertical shift \begin{align*}= d\end{align*}).
Problem 5
While students may be wondering why they cannot declare “phase shift \begin{align*}= c\end{align*},” have them again consider the graph of \begin{align*}y = x^2\end{align*}. Ask them how to obtain from it the graph of \begin{align*}y = (x + 3)^2\end{align*} (translate 3 units to the left). Then display the equations \begin{align*}y = [4(x + 3)]^2\end{align*} and \begin{align*}y = (4x + 3)^2\end{align*}, and elicit from students that only the former represents a shift 3 units to the left. Students should realize that the phase shift of \begin{align*}f(x) =a \sin(bx + c) + d\end{align*}, depends on two parameters: \begin{align*}b\end{align*} and \begin{align*}c\end{align*}.
To establish exactly how \begin{align*}b\end{align*} and \begin{align*}c\end{align*} determine the phase shift, students may are to change the values. Encourage students to conjecture the relationship on their own, but if they need help, have them consider the phase shift when \begin{align*}b = 1\end{align*} and \begin{align*}c = 2\end{align*}, when \begin{align*}b = 2\end{align*} and \begin{align*}c = 1\end{align*}, and when \begin{align*}b = 2.5\end{align*}, and \begin{align*}c = 5\end{align*}. (It is 2, 0.5, and 2, respectively.) Examining these values, students should conclude that phase shift \begin{align*}= \frac{c}{b}\end{align*}.
Note: It is easiest to identify this relationship if parameters \begin{align*}a\end{align*} and \begin{align*}d\end{align*} are left as initially set (\begin{align*}a = 1\end{align*} and \begin{align*}d = 0\end{align*}). After the relationship is determined, dragging the sliders can verify that neither \begin{align*}a\end{align*} nor \begin{align*}d\end{align*} affects the phase shift.
Problem 6
 amplitude \begin{align*}= a\end{align*}, period \begin{align*}= \frac{2\pi}{b}\end{align*}, phase shift \begin{align*}=  \frac{c}{b}\end{align*}, vertical shift \begin{align*}= d\end{align*}
 \begin{align*}f(x) = 1.5 \cos \left (x + \frac{3 \pi}{4} \right ) + 4\end{align*}
 \begin{align*}g(x) = 3 \cos \left (2x  \frac{\pi}{2} \right )  5\end{align*}
Students should observe that the values of \begin{align*}a\end{align*}, \begin{align*}b\end{align*}, and \begin{align*}d\end{align*} remain the same for each sine/cosine pair; the only difference occurs in the value of \begin{align*}c\end{align*}. Since these functions are periodic, there are infinitely many that satisfy each condition. Be sure to check students’ equations.