Lesson 2 You Are the Imagineer Develop Understanding

Learning Focus

Predict and examine the graphical effects of adding or multiplying functions from two different function families.

How can we combine function types to describe more complicated real-world behavior?

Open Up the Math: Launch, Explore, Discuss

You are excited to get to vote on the plans for a proposed new thrill ride at a local theme park. The engineers want public input on the design for the new ride. You are one of ten teenagers who have been selected to review the plans.

As your excitement mounts, the engineers begin their presentation. To your dismay, there are no models or illustrations of the proposed rides—each ride is described only with equations. The equations represent the path a rider would follow through the course of the ride.

Unfortunately, your cell phone—which contains a graphing calculator app—is completely discharged due to too much texting and surfing the internet. So, you are trying hard to keep up with the presentation by trying to imagine what the graphs of each of these equations would look like. While each equation consists of functions you are familiar with, the combination of functions in each equation has you wondering about their combined effects.

For each of the following proposed thrill rides, use your imagination and best reasoning about the individual functions involved to sketch a graph of the path of the rider. Let $y$ represent the height of the rider above the ground and $x$ represent the distance from the start of the ride. Explain your reasoning about the shape of the graph. (Note: Use radians for trigonometric functions.)

1.

Proposal #1: “The Mountain Climb”

The equation: $y = 2 x + 5 \sin (x)$

a.

My graph:

b.

My explanation:

2.

Proposal #2: “The Periodic Bump”

The equation: $y = | 10 \sin (x) |$

a.

My graph:

b.

My explanation:

3.

Proposal #3: “The Amplifier”

The equation: $y = x \cdot \sin (x)$

a.

My graph:

b.

My explanation:

4.

Proposal #4: “The Gentle Wave”

The equation: $y = 10 {(0.9)}^{x} \cdot \sin (x)$

a.

My graph:

b.

My explanation:

5.

Proposal #5: “The Spinning High Dive”

The equation: $y = 100 - x^{2} + 5 \sin (4 x)$

a.

My graph:

b.

My explanation:

When you got home, your friends were all anxiously waiting to hear about the proposed new rides. After you explained the situation, your friends all pulled out their calculators and they began comparing your imagined images with the actual graphs.

Some of your friends’ graphs differed from the others because of their window settings. Some window settings revealed the features of the graphs you were expecting to see, while other window settings obscured those features.

Examine the actual graphs of each of the thrill ride proposals. Select a window setting that will reveal as many of the features of the graphs as possible. Explain any differences between your imagined graphs and the actual graphs. What features did you get right? What features did you miss?

After each problem, respond to the writing prompts to explain how each of the two functions and the operation that combines them affects the graph.

6.

Proposal #1: “The Mountain Climb”

The Equation: $y = 2 x + 5 \sin (x)$

a.

Sketch of Actual Graph:

b.

What features I got right and what I missed:

c.

What I noticed:

The function affects .
The function alters .
The result of the combining operation is .

7.

Proposal #2: “The Periodic Bump”

The Equation: $y = | 10 \sin (x) |$

a.

Sketch of Actual Graph:

b.

What features I got right and what I missed:

c.

What I noticed:

The function affects .
The function alters .
The result of the combining operation is .

8.

Proposal #3: “The Amplifier”

The Equation: $y = x \cdot \sin (x)$

a.

Sketch of Actual Graph:

b.

What features I got right and what I missed:

c.

What I noticed:

The function affects .
The function alters .
The result of the combining operation is .

9.

Proposal #4: “The Gentle Wave”

The Equation: $y = 10 {(0.9)}^{x} \cdot \sin (x)$

a.

Sketch of Actual Graph:

b.

What features I got right and what I missed:

c.

What I noticed:

The function affects .
The function alters .
The result of the combining operation is .

10.

Proposal #5: “The Spinning High Dive”

The Equation: $y = 100 - x^{2} + 5 \sin (4 x)$

a.

Sketch of Actual Graph:

b.

What features I got right and what I missed:

c.

What I noticed:

The function affects .
The function alters .
The result of the combining operation is .

11.

You and your friends decide to propose a different ride to the engineers. Name your proposal and write its equation. Explain why you think the features of this graph would make a fun ride.

My Proposed Ride:

The equation for my ride:
My explanation of my proposal:

Ready for More?

Find an image online of a portion of a track from an amusement park ride. Create a function to model the path of the track.

Takeaways

When two functions are combined by addition, .

When two functions are combined by multiplication, .

The output of some functions .

The absolute value of a function .

Vocabulary

composition of functions
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we observed what happens when we combine two different function types by adding or multiplying them together. We also noticed what happens to the portion of a graph of a function that lies below the $x$ -axis when we take the absolute value of the function.

Retrieval

1.

The graph shows $f (x) = \cos x$ .

a.

Write the equation of the dotted line labeled $a (x)$ .

b.

Write the equation of the dotted line labeled $b (x)$ .

c.

List everything you notice about the graphs of $f (x)$ , $a (x)$ , and $b (x)$ .

2.

List at least $5$ features of $y = x^{3}$ or $a x^{3} + b x^{2} + c x + d$ . (Think about intercepts, roots, end behavior, symmetry, degree, graph, rate of change, equation(s), etc.)