Precalculus
Graphing Calculator Exploration
Graphing Sine and Cosine
Name____________________
Today we are going to use our graphing calculators to experiment with changing the values of a, b, c and d in
the general equation:
y = a * sin (b(x - c)) + d.
Our goal is to discover how a, b, c, and d affect the graph.
First, use what you learned during the Unwrapping Task (the Spaghetti Lab), and graph y = sin(x)
and y = cos(x) from 0 to 2π below.
y = sin(x)
What are the values of a, b, c, and d?
a=
b=
c=
d=
y = cos(x)
What are the values of a, b, c, and d?
a=
b=
c=
d=
*Before we begin…
To what mode should our calculator be set?
Remember: The general equation is
y = a * sin (b(x - c)) + d
1.) Set your window to view x from 0 to 2π and y from -3 to 3.
Graph
𝑌1 = sin(𝑥)
1
𝑌2 = 2sin(𝑥)
𝑌3 = sin 2 (𝑥)
As the absolute value of a increases, …
As the absolute value of a decreases,…
In the general form of the equation, changes in a impact the _________________________ of the
graph.
2.) Set your window to view x from 0 to 2π and y from -2 to 2.
Graph
𝑌1 = sin(𝑥)
1
𝑌2 = sin(2𝑥)
𝑌3 = sin(2 𝑥)
As b increases……
As b decreases…….
A change in b alters the ________________________ of the graph of the equation.
3.) Set your window to view x from -π to 3π and y from -2 to 2.
Graph
𝑌1 = sin(𝑥)
𝜋
𝑌2 = sin(𝑥 − 2 )
𝜋
𝑌3 = sin(𝑥 + 4 )
When c is positive(x - c), the graph ….
When c is negative(x + c), the graph….
A change in c causes a _______________________
_____________________ in the graph
4.) Set your window to view x from 0 to 2π and y from -6 to 6.
Graph
𝑌1 = sin(𝑥)
𝑌2 = sin(𝑥) + 5
𝑌3 = sin(𝑥) − 3
When d is positive, the graph…
When d is negative, the graph…
A change in d causes a _________________________ ________________ in the graph. This shifts the
___________________________________ up or down.
Now, use what you learned to complete the following.
1.) Determine the amplitude, period, phase shift, and vertical shift for each function:
Then, graph the function. Label the five critical points
a.) f ( x)  5sin( x)  2
𝜋
b.) 𝑓(𝑥) = −2𝑐𝑜𝑠 (𝑥 − 2 ) + 3
Write a sine and cosine equation for each given graph.
In general, when given a graph, how do you determine the values of a and b?
Ferris Wheel Comparison Project
The 1893 Chicago World’s Fair is considered the birthplace of the classic amusement park ride, the Ferris
wheel. The architectural wonder was created by an American engineer named George Ferris. The original
Ferris wheel no longer exists. But, in 1990, a new Ferris wheel was built at Navy Pier in Chicago to resemble
the original. While the Navy Pier Ferris Wheel is a beautiful Chicago landmark, its grandeur actually pales in
comparison to Mr. Ferris’ creation.
The Ferris wheel built for the World’s Fair had a diameter of 250 feet. It stood 14 feet off the ground. It had 36
wooden boxcars that were the size of train cars. Each car could hold 60 people! The wheel would load cars in
such a way that each rider could enjoy a full rotation that lasted about 10 minutes.
The Ferris wheel at Navy Pier has a diameter of 140 feet. It stands 10 feet off the ground. The wheel has 40
gondolas that seat six passengers each. It takes about 6 minutes for the Navy Pier Ferris Wheel to complete one
rotation.
Below is a picture of the first Ferris wheel next to the Ferris wheel at Navy Pier.
Ferris Wheel Comparison Activity
Name____________________
Date_____________ Per_____
In the space below is a diagram of the World’s Fair Ferris wheel and the boarding platform. Fill in the necessary
information.
d=
h=
Now, complete the table below. Let h represent your vertical position (height) at time t where t is given in
minutes. Remember, you need 5 critical t-axis values.
t
h
Before you sketch the cosine and sine graphs, answer each of the following.
1.) What is the amplitude of the graph? _____________________________
a = __________
2.) What is the period of your curve? _______________________________
2
(Remember
b)
period
b = __________
3.) What are your 5 critical t-values? _________________________________
4.) What are the maximum and minimum values Max ________ Min ________
of your curve? (Be careful!)
d = __________
5.) What is the height at time = 0 (y-intercept)? ______________
Write a cosine equation for your curve.
(Remember there are an infinite number of possible answers)
_____________________________________
Graph the equation.
Write a sine equation for your curve.
__________________________________
Graph the equation.
Now, answer the following questions. You may want to graph your equation on your graphing calculator.
1.) What is the circumference of the wheel?
2.) At what speed is the wheel traveling? Please give your answer in feet/second.
3.) If you begin your ride at the base of the wheel, what is your height after…
a.) 1 minute?
b.) 4 minutes?
4.) At what approximate time(s) will you reach the following heights?
a.) 100 ft.
b.) 240 ft.
Now, repeat each step for the Navy Pier Ferris Wheel.
In the space below is a diagram of the Navy Pier Fair Ferris Wheel and the boarding platform. Fill in the
necessary information.
d=
h=
Now, complete the table below. Let h represent your vertical position (height) at time t where t is given in
minutes. Remember, you need 5 critical t-axis values.
t
h
Before you sketch the cosine and sine graphs, answer each of the following.
1.) What is the amplitude of the graph? _____________________________
a = __________
2.) What is the period of your curve? _______________________________
2
(Remember
b)
period
b = __________
3.) What are your 5 critical t-values? _________________________________
4.) What are the maximum and minimum values Max ________ Min ________
of your curve? (Be careful!)
d = __________
5.) What is the height at time = 0 (y-intercept)? ______________
Write a cosine equation for your curve.
(Remember there are an infinite number of possible answers)
_____________________________________
Graph the equation.
Write a sine equation for your curve.
__________________________________
Graph the equation.
Now, answer the following questions. You may want to graph your equation on your graphing calculator.
1.) What is the circumference of the wheel?
2.) At what speed is the wheel traveling? Please give your answer in feet/second.
3.) If you begin your ride at the base of the wheel, what is your height after…
a. 1 minute?
b.) 4 minutes?
4.) At what approximate time(s) will you reach the following heights?
a.) 100 ft.
b.) 240 ft.
Ferris Wheel Extra Credit
1.) Imagine the Navy Pier and the Worlds Fair Ferris Wheel being built beside each other. If both wheels
begin turning at once, over a 20 minute time period, at what times are the wheels at the same height?
2.) What is the length of the arc traveled by the Navy Pier Ferris wheel from the
4 o’clock to the 7 o’clock position?