Math 3Name______

5-9Transforming Trigonometric Functions

  • I can define sine and cosine as functions of real numbers and analyze the resulting periodic graphs
  • I can use the sine and cosine functions to model periodic patterns of change in various physical phenomena
  1. Complete the Nspire document that your teacher sends to your calculator called ‘5-9 Transformation of Trig Function Introduction’ and take notes on the vocabulary words below as you work.

1a.Define periodic motion:

1b. Define periodic function:

1c. Define period of a periodic function:

1d. Define amplitude of a periodic function:

2. Find the period and amplitude of the trigonometry functions below.

  1. b.

3.Below is a graph of a trigonometric function .

3a.What is the equation of ?

3b.Thinking back to when we transformed functions, describe how the function is related to. Sketch on the above axis using a colored pencil.

Description:

3c.Write an equation for in terms of x.

3d. Complete the chart below and compare the period and amplitude of the functions.

Period / Amplitude

3e. Thinking back to when we transformed functions, describe how the function is related to. Sketch on the above axis in a different color.

Description:

3f.Write an equation for in terms of x.

3g. Complete the chart below and compare the period and amplitude of the functions.

Period / Amplitude

4.Graph two periods ofon the graph below.

4a.Thinking back to when we transformed functions, describe how the function is related to. Sketch two periods of on the above axis using a colored pencil.

Description:

4b.Write an equation for in terms of x.=

4c.Complete the chart below and compare the period and amplitude of the functions.

Period / Amplitude

Class Notes:

Amplitude:
Period:
Vertical Shift/
y-displacement:

Notes about x-intercepts and max/mins:

5.Find the amplitude, vertical shift (also called y-displacement), y-intercept, and period of the following functions. Also sketch one period of the graph of each. Label the x-axis!

5a.5b.

amplitude: ______amplitude: ______

vertical shift:______vertical shift:______

y-intercept:______y-intercept:______

period:______period:______

Work:Work:

5c.5d.

amplitude: ______amplitude: ______

vertical shift:______vertical shift:______

y-intercept:______y-intercept:______

period:______period:______

Work:Work:

The Ferris wheel was invented in 1893 as an attraction at the world Columbian Exhibition in Chicago, and it remains a popular ride at carnivals and amusement parks around the world. The wheels provide a great context for the study of circular motion.

Ferris wheels are circular and rotate about the center. The spokes of the wheel are radii, and the seats are like points on the circle. The wheel has horizontal and vertical lines of symmetry through the center of rotation. This suggests a natural coordinate system for describing the circular motion.

To aid your thinking about positions on a rotating circle, the Ferris wheel to the right has a radius of 33 feet, has its center at point C, and your seat is at point A when the wheel begins to turn counterclockwise.

Tracking the location of seats on a spinning Ferris wheel shows how the cosine and sine functions can be used to describe rotation of circular objects.

6a.If an X-Y plane is imposed on the Ferris Wheel such that point C is at (0,0) (as in the given picture), what is the location of point A?

6b.Assuming that the Ferris wheel takes exactly minutes to make one revolution, write an equation for a function that models the height of the Ferris wheel above the x-axis at any time t minutes.

6c.The owners of the above Ferris wheel decided that it is moving too slow. They would like it to make one full revolution in minutes. Write an equation that will model the motion of the Ferris wheel.

7.When riding a Ferris wheel, customers are probably more nervous about their height above the ground than the distance from the horizontal line of symmetry. Suppose a large wheel has radius 25 meters, the center of the wheel is located 30 meters above the ground, and the wheel starts in motion when the seat S is at the “3 o’clock” position.

7a.Assuming that the Ferris wheel takes exactly minutes to make one revolution Fill in the table below for the height of the Ferris wheel above the ground at the given times

Time / 0 / / / / / / /
Height above ground (m)


7b.Graph the data in the table on the below axis. Label the x-axis with the same units as in the table. Label the y-axis appropriately. Use the entire x- and y-axis for your graph!

7c.Assuming that the Ferris wheel takes exactly minutes to make one revolution, write an equation for a function that models the height of the Ferris wheel above the x-axis at any time t minutes. That is, write an equation that models the graph in problem (7b).

7d.The owners of the above Ferris wheel decided that it is moving too slow. They would like it to make one full revolution in 2 minutes. Write an equation that will model the motion of the Ferris wheel.

7e.Explain what the numeric values in your equation above tell you about the Ferris wheel.

8.Pendulums are among the simplest but most useful examples of periodic motion. Once set in motion, the arm of the pendulum swings left and right of a vertical axis. The angle of displacement from vertical is a periodic function of time that depends on both the length of the pendulum and its initial release point.

Suppose that the function gives the displacement from vertical, in degrees, of the tire swing pictured below as a function of time (in seconds). Note that the y-variable is measured in degrees.

8a.What are the amplitude and period of ? What does each tell about the motion of the swing?

8b.If the motion of a different swing is modeled by , what are the amplitude and period of ? What does each tell you about the motion of this swing.

8c.Why does it make sense to use the cosine function to model pendulum motion?

8d.What function would model the motion of a pendulum that is released from a displacement of right of vertical and travels of a swing in one second?

9.At every location on Earth, the number of hours of daylight varies with the seasons in a predictable way. One convenient way to model that pattern of change is to measure time in days, beginning with the spring equinox (about March 21) as . With that frame of reference, the number of daylight hours in Cleveland, Ohio is given by

9a.What are the amplitude and period of ? What do those values tell about the pattern of change in daylight during a year in Cleveland?

9b.What are the maximum and the minimum numbers of hours of daylight in Cleveland? At approximately what day of the year do they occur?

9c.Alaska is on the far north of the Earth and because of this at some times during the year it has days that are very long (almost 24 hours of daylight) and at other times during the year the days are very short (only 4-6 hours of daylight). What part(s) of the function would need to be changed to model the daylight in Alaska? How would those values need to be changed? Explain.

10.Find the amplitude, period and vertical shift of the following functions. Also write the equation for the function.

10a.10b.