Lesson 1 High Noon and Sunset Shadows Develop Understanding

Learning Focus

Shift the graphs of trigonometric functions horizontally.

How can I represent the horizontal and vertical motion of a rider on a Ferris wheel if her starting position varies? When should I use a sine equation, and when should I use a cosine equation, or does it matter?

Open Up the Math: Launch, Explore, Discuss

In this task, we revisit the amusement park Ferris wheel that caused Carlos so much anxiety. Recall the following facts from previous tasks:

The Ferris wheel has a radius of $25 feet$ .
The center of the Ferris wheel is $30 feet$ above the ground.
The Ferris wheel makes one complete rotation counterclockwise every $20 seconds$ .

The amusement park Ferris wheel is located next to a high-rise office complex. At sunset, the moving carts cast a shadow on the exterior wall of the high-rise building. As the Ferris wheel turns, you can watch the shadow of a rider rise and fall along the surface of the building. In fact, you know an equation that would describe the motion of this “sunset shadow.”

1.

Write the equation of this “sunset shadow.”

At noon, when the sun is directly overhead, a rider casts a shadow that moves left and right along the ground as the Ferris wheel turns. In fact, you know an equation that would describe the motion of this “high noon shadow.”

2.

Write the equation of this “high noon shadow.”

3.

Based on your previous work, you probably wrote these equations in terms of the angle of rotation being measured in degrees. Revise your equations so the angle of rotation is measured in radians.

a.

The “sunset shadow” equation in terms of radians:

b.

The “high noon shadow” equation in terms of radians:

4.

In the equations you wrote in problem 3, where on the Ferris wheel was the rider located at time $t = 0$ ? (We will refer to the position as the rider’s initial position on the wheel.)

5.

Revise your equations from problem 3 so that the rider’s initial position at $t = 0$ is at the top of the wheel.

a.

The “sunset shadow” equation, initial position at the top of the wheel:

b.

b.

The “high noon shadow” equation for a radius twice as large and the center twice as high:

11.

Carlos wrote his “sunset equation” for the height of the rider in problem 5 as $h (t) = 25 \sin (\frac{π}{10} t + \frac{π}{2}) + 30$ .

Clarita wrote her equation for the same problem as $h (t) = 25 \sin (\frac{π}{10} (t + 5)) + 30$ .

a.

Are both of these equations equivalent? How do you know?

b.

Carlos says his equation represents starting the rider at an initial position at the top of the wheel. What does Clarita’s equation represent?

Ready for More?

How do the types of numbers used for the parameters $b$ and $c$ affect the graphs of equations of the form in $y (t) = \sin (b t + c)$ ? Experiment by selecting values for $b$ and $c$ , that are integers (e.g., $- 2$ , $- 1$ , $1$ , $2$ , $3$ , $4$ ) or fractions with $π$ in the numerator (e.g., $\frac{π}{2}$ , $\frac{π}{3},$ $\frac{π}{4},$ $\frac{2 π}{3},$ $\frac{3 π}{4}$ ). Decide which values produced easier graphs than others, for example: (1) when both $b$ and $c$ are integers, (2) when both are fractions involving $π$ , or (3) when one is an integer and the other is a fraction involving $π$ .

Takeaways

The horizontal shift, or phase shift, of a trigonometric function can be represented in the following two ways:

Form 1:

Form 2:

Form 2 reveals

In terms of the circular motion of the Ferris wheel,

Form 1 is easier to write if we focus on

Form 2 is easier to write if we focus on

Vocabulary

phase shift
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we examined the horizontal shift of a trigonometric function, which is also referred to as a phase shift. Different forms of the equation representing the horizontal shift led to two different interpretations in the context of the Ferris wheel: shifting the position of the rider on the wheel at time $t = 0$ , or shifting the time when we start timing the position of the rider on the wheel.

Retrieval

1.

a.

Sketch the inverse of the function on the same set of axes. Include the $y = x$ line.

b.

Is the function even, odd, or neither? Recall that an even function is symmetric with the $y$ -axis, while an odd function has $180 °$ rotational symmetry about the origin.

A.

even

B.

odd

C.

neither

c.

Is the inverse of the function even, odd, or neither?

A.

even

B.

odd

C.

neither

2.

Name two angles of rotation for $θ$ that have the given trig value. $0 < θ \leq 2 π$ .

a.

$\sin θ = \frac{1}{2}$

b.

$\cos θ = 0$