* *

*Transformations of the Sine and Cosine Graph
– An Exploration*

*By Sharon K. OÕKelley*

This is an exploration for
Advanced Algebra or Precalculus teachers who have introduced their students to
the basic sine and cosine graphs and now want their students to explore how changes
to the equations affect the graphs.
This is an introductory lesson whose purpose is to connect the language
of Algebraic transformations to the more advanced topic of trignonometry. (A key follows the end of the
exploration.)

**1**. Consider the basic sine equation and graph. LetÕs
call it the ** first function**É.

**2**. If the
first function is rewritten asÉ.

then the values of a = 1, b = 1, and c = 0.

LetÕs find out what happens
when those values changeÉ.

**3**. Take a
look at the blue and red graph and their equations. The graph of the first
function remains in black.

Equation of blue graph

Equation of red graph

a.
Describe how the equation of the first function has changed to become
the equation of the graphs in blue and red. This value is called the **amplitude** of the graph.

b.
Describe how the graph of the first function has changed to become the
blue graph and the red graph. Be as specific as possible. Include in your
answer how a specific point on the graph of the first function transforms to
become a point on the blue graph and on the red graph.

c. Take a look at an animation of this
phenomenon. Click hereÉ. (Be
patient. These movies take awhile
to loadÉ.)

**4. **Now
consider the purple and green graph and their equations. The graph of the first
function remains in black.

Equation of purple graph

Equation of green graph

a. Describe how the equation of the first function has changed
to become the equation of the graphs in purple and green.

b. Describe how the graph of the first function has changed to
become the purple graph and the green graph. Be as specific as possible. Include
in your answer how a specific point on the graph of the first function
transforms to become a point on the purple graph and on the green graph.

c. How does ÒbÓ operate differently
from ÒaÓ in the equation? Look deeper than horizontal versus verticalÉ.

d. Take a look at this phenomenonÉ.
Click hereÉ.

**5**. Examine
the purple and green graph and their equations. The graph of the first function
remains in black.

Equation of red graph

Equation of green graph

a.
Describe how the equation of the first function has changed to become
the equation of the graphs in purple and green.

b.
Describe how the graph of the first function has changed to become the
red graph and the green graph. Be as specific as possible. Include in your
answer how a specific point on the graph of the first function transforms to
become a point on the purple graph and on the green graph.

c.
See how it movesÉ. Click hereÉ.

**6**.
Consider the blue and purple graph and their equations. The graph of the
first function remains in black.
This value is called the **phase shift** of the graph.

Equation of blue graph

Equation of purple graph

a. Describe how the equation of the first function has changed
to become the equation of the graphs in blue and purple.

b. Describe how the graph of the first function has changed to
become the blue graph and the purple graph. Be as specific as possible. Include
in your answer how a specific point on the graph of the first function
transforms to become a point on the blue graph and on the purple graph.

c. How does ÒcÓ operate differently
when it is inside or outside the parentheses? Look deeper than horizontal
versus verticalÉ.

d.
Take a lookÉ. Click hereÉ.

**7. **What
about negatives?

a. Examine the following graphsÉ. (The
first function is in black.)

Equation of red graph

Equation of blue graph

b. Compare the graphs.
Describe in detail the transformations you see.

**8.** Throwing
it all togetherÉ.

Consider the graph of É.

(The first function is in black.)

Describe the transformations fully. Be as specific as possible.

(Hint: Look at this problem
as .)

**9. The Cosine Graph**

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** **

** **

a. On a sheet of graph paper, predict what the following graphs
would look like.

**10. LetÕs go a little furtherÉ. Write two different equations for the
same graph below. Use sine in one
and cosine in the other. Verify your answer with graphing software or a
graphing calculator.**

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**11. **Summarize
what you have learned here. Use
definitions and examples. Be
specific.

** **

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*Key to the Exploration*

** **

**3.** a. The
value of ÒaÓ is 2 and .

b. The blue graph is stretched vertically
by a factor of 2. This can be
determined by comparing point (1.57, 1) on the first function to point (1.57,
2) on the blue graph. The red
graph is shrunk vertically by a factor of . This can be determined by comparing point (1.57, 1) on the
first function to point (1.57, ) on the

red graph.

** **

** **

**4. **a.
The value of ÒbÓ is 2 and .

b. The purple graph is shrunk horizontally
by a factor of . This can be
determined by comparing point (1.57, 1) on the first function to point (0.785,
1) on the purple graph. The green
graph is stretched horizontally by a factor of 2. This can be determined by comparing point (1.57, 1) on the
first function to point (3.14,1) on the green graph.

c. The values for ÒaÓ and ÒbÓ seem to be
inverted. In other words when ÒaÓ
is it is a vertical
shrink by . When ÒbÓ is , however, it is a horizontal stretch by a factor of 2. This can easily be explained by the
fact that you are solving, so to speak, for the given variable. For example, is equivalent to
; therefore, when the variable is isolated it seems
straightforward. However, the x
isnÕt isolated in these casesÉ.

**5. **a.
The value of ÒcÓ is 2 and .

b. The red graph is translated up by 2
units. This can be determined by comparing
point (0,0) on the first function to point (0, 2) on the red graph. The green graph is translated down by 2
units. This can be determined by
comparing point (0, 0) on the first function to point (0, ) on the green graph.)

** **

** **

**6. **a. The value of ÒcÓ is 2 and .

b.
The blue graph is translated left
2 units. This can be
determined by comparing point (0,0) on the first function to point on the blue
graph. The purple graph is
translated right 2 units. This can
be determined by comparing point (0, 0) on the first function to point (2,0) on
the purple graph.

c.
Outside the parentheses, positive c is up and negative c is down. Inside, it seems opposite than what you
would assume. Positive c is to the
left and negative c is to the right.

** **

**7. **They
look the same! Why? Because sine
is a periodic function, a vertical reflection and horizontal reflection yield
the same image.

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** **

**8. **The
graph has translated right units. It has
been horizontally shrunk by a factor of and vertically
shrunk by a factor of .

** **

**9. *** (red)

* (blue)

* (purple)

* (green)

* (yellow)

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**10. **Possible
answer: and

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**11**. StudentsÕ answers should cover definitions and
descriptions of the transformations as well as how the generic equation
changes. They also should define
ÒamplitudeÓ and and Òphase shift.Ó

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