# Day 3 - Phase Shift and Vertical Shift

## Introduction

Objectives: To find the phase shift and the vertical shift of sine and cosine functions from their equations; to find the phase shift and vertical shift of sine and cosine functions from their graphs; to graph sine and cosine functions with various phase shifts and vertical shifts.

In the previous lesson, students learned that amplitude and period make the graphs of sine and cosine functions get taller, shorter, stretched, and squeezed. All of these changes affected the "SHAPE" of a graph. In this lesson, students will examine how phase shift and vertical shift affect the "LOCATION" of a graph.

## Student Activity 1 - Phase Shift

I. Graphing phase shift in sine functions.

1. Graph y = sinx in Graphing Calculator.

2. Go to the "Math" Menu and choose "New Math Expression". Graph y = sin(x + p/4). How does the graph of y = sin(x + p/4) differ from the graph of y = sinx?

3. Delete the equation y = sin(x + p/4). Graph y = sin(x - p/ 2). How does the graph of y = sin(x - p/ 2) differ from the graph of y = sinx?

4. Delete the equation y = sin(x - p/ 2). Graph y = sin(x + p). How does the graph of y = sin(x + p) differ from the graph of y = sinx? Be careful!!

5. Delete the equations y = sinx and y = sin(x + p). Graph the equation y = sin(x + n), letting n vary from -10 to 10. Animate. What effect does "n" have on the graphs?

II. Graphing phase shift in cosine functions.

Graph y = cos(x + n), letting n vary from -10 to 10. Animate. Does the graph of y = cos(x + n) react to the values of n the same way the graph of y = sin(x + n) did?

III. Making sense of phase shift.

1. Answer the following questions about equations in the forms y = sin(x + C) and y = cos(x + C):

a) What happens to the graph of y = sinx and y = cosx when C is positive?

b) What about when C is negative?

2. Define phase shift.

## Teacher Key for Student Activity 1

I.

1. Here is a sample graph:

2. Here is a sample graph and answer:

The purple graph moved left by p/4.

3. Here is a sample graph and answer:

The purple graph moved right by p/ 2.

4. Here is a sample graph and answer:

The purple graph moved left by p.

III.

a) The graph moves left C units.

b) The graph moves right C units.

2. A phase shift is a horizontal shift (right or left) of the graph.

## Student Activity 2 - Vertical Shift

I. Graphing vertical shift in cosine functions.

1. Graph y = cosx in Graphing Calculator.

2. Go to the "Math" Menu and choose "New Math Expression". Graph y = cosx + 2. How does the graph of y = cosx + 2 differ from the graph of y = cosx?

3. Delete the equation y = cosx + 2. Graph y = cosx - 1. How does the graph of y = cosx - 1 differ from the graph of y = cosx?

4. Delete the equations y = cosx and y = cosx - 1. Graph the equation y = cosx + n, letting n vary from -10 to 10. Animate. What effect does "n" have on the graphs?

II. Graphing vertical shift in sine functions.

Graph y = sinx + n, letting n vary from -10 to 10. Animate. Does the graph of y = sinx + n react to the values of n the same way the graph of y = cosx + n did?

III. Making sense of period.

1. Answer the following questions about equations in the forms y = sinx + D and y = cosx + D:

a) What happens to the graphs of y = sinx and y = cosx when D is positive?

b) What about when D is negative?

2. Define vertical shift.

## Teacher Key for Student Activity 2

I.

1. Here is a sample graph:

2. Here is a sample graph and answer:

The green graph moved up by 2.

3. Here is a sample graph and answer:

The green graph moved down by 1.

III.

a) The graph moves up D units.

b) The graph moves down D units.

2. A vertical shift is when the graph moves up or down.

## Student Practice

1. Determine the amplitude, period, phase shift, and vertical shift of each function.

a) y = cos2x - 5

b) y = 2sin(3x + 3p)

c) y = 3cos0.5x + 4

d) y = -sin(x - p/4) - 2

2. Determine the phase shift and vertical shift of each function. Then write an equation of each graph.

a)

b)

3. Give the phase shift and vertical shift of each function. Then sketch the graph of the function over the given interval.

a) y = sin(x - p/ 2) + 1, [0, 2p]

b) y = cosx - 3, [-2p, 2p]

c) y = cos(x + p) - 2, [0, 3p]

## Teacher Key for Student Practice

1.

a) amp = 1, pd = p, ps = 0, vs = 5 down

b) (First factor out 3: y = 2sin3(x + p)) amp = 2, pd = 2p/3, ps = p left, vs = 0

c) amp = 3, pd = 4p, ps = 0, vs = 4 up

d) amp = 1, pd = 2p, ps = p/4 right, vs = 2 down

2.

a) ps = 0, vs = 2, y = cosx + 2 OR ps = p/2 left, vs = 2, y = sin(x + p/2) + 2

b) ps = p/4 right, vs = 1 down, y = cos(x - p/4) - 1 OR ps = p/2 left, vs = 1 down, y = sin(x + p/2) - 1

3.

a) ps = p/ 2 right, vs = 1 up

b) ps = 0, vs = 3 down

c) ps = p left, vs = 2 down