Light It Up!Light It Up is the newest attraction at the Springfield Fair. In this game, a laser pointer slides along a string at a height of 1.5 meters from the ground. A 10-cm platform is positioned 25 cm from a wall, and a mirror is placed on top of the platform, parallel with the ground. The object is to slide the laser pointer so that when the bean is reflected off the mirror, it will hit a target that is 1 m off the ground, as shown below.

A player is given just one attempt to hit the target. If she does, she then tries to hit a target that is 3 m off the ground. And if she hits that one, too, she tries for a target that is 10 cm off the ground. If she hits all three, she wins the Grand Prize: A DVD player and a DVD copy of Itchy and Scratchys Greatest Hits.Bart and Lisa have played the game several times, yet they havent been able to hit the target. Lisa is sure that there must be a mathematical equation that would allow them to determine the correct position of the laser pointer.There is such an equation, it is a special kind known as a reciprocal function. Before you can help Lisa and Bart win this game, you must first understand the behavior of these functions.Over the next several days in your groups you will: Understand reciprocal functions Graph reciprocal functions Collect data Play the game using your function

The Climbing WallThe Climbing Wall is a popular event at the Springfield Fair. The wall is 100 meters high. Bart can climb at a rate of 8 meters per minute, but other participants climb at different speeds.1. Complete the table to show that the time it takes to reach the top of the wall depends on the climbers speed. Graph the function on the coordinate plane (be sure to include labels and scales!).

2. Write an equation that describes how you determined the values in the table. Explain what your variables mean in the context of the problem.

3. List the domain and range of your function. Use proper notation.

4. Why doesnt it make sense to use negative numbers in your domain for the context of this problem?

5. Graph your function to include negative values in the domain on the coordinate plane above.Trip to the FairMs. Crabapple, a teacher at Springfield Elementary, is sponsoring a school trip to the fair. The total cost for transportation, parking and entrance fees is $1200. The total cost will be divided among all the students who go on the trip.1. Complete the table to determine how much each student must pay to cover the expenses. Graph the function on the coordinate plane (be sure to include labels and scales!).

2. Write an equation that describes how you determined the values in the table. Explain what your variables mean in the context of the problem.

3. List the domain and range of your function. Use proper notation.

4. Why doesnt it make sense to use negative numbers in your domain for the context of this problem?

5. Graph your function to include negative values in the domain on the coordinate plane above.

6. How are the graphs of The Climbing Wall and Trip to the Fair similar and different? You may want to give your functions different names so that your thoughts are easier to record.

SimilaritiesDifferences

A Closer Look at the Trip to the FairMs. Crabapple was just informed that the new attraction, Light It Up, will cost an additional $5 per student.1. Discuss with your team members how this new information will change the amount each student will have to pay. Complete the table below to reflect this new information and graph your values on the coordinate plane provided.

2. Write an equation that describes how you determined the values in the table. Explain what your variables mean in the context of the problem.

3. Some rational functions are a certain type, called reciprocal functions. Reciprocal functions are written in the form:

where and are constants, and . If your equation is not already written this way, rewrite your equation in this form.

4. What is the domain of the function you found in question 3? How does the domain make sense in the context of the problem?

5. What is the range of the function you found in question 3? How does the range make sense in the context of the problem?

6. Use your equation from question 3 to extend the graph to include negative values.

7. How are the values of and from your equation reflected in your graph?

An Even CLOSER Look at the Trip to the FairMs. Crabapple just received some more information. The trip must include chaperones. She decides that ten sixth-graders will serve as chaperones, and the chaperones will not be required to pay for their trip. The expenses will be covered by the other students.1. With your group members, discuss how this new information will change the amount each student has to pay. Complete the table below to reflect this new information and graph the values on the coordinate plane provided.

2. Describe your method for determining the values of the table.

3. Write an equation that describes the table.

4. What is the domain of your function? How does the domain make sense in the context of the problem?

5. What is the range of your function? How does the range make sense in the context of the problem?

6. Extend your graph into negative values.

7. Rewrite your equation in reciprocal form (if it is not already).

8. How are the values of and reflected in your graph?

Writing Reciprocal Functions

Write an equation in the form for the following rational functions. Check your equations using a graphing calculator. For now, you may leave in your equation.1.

2.

3.

4.

5. Algebraically, how can you determine the value of in the equation for the graphs of 1-4? Think about how you solve for in a linear equation.

6. Use your explanation from part 5 to solve for in each of the functions 1-4. Record your functions (with ) below. Show work!1: _____________________________

2: _____________________________

3: _____________________________

4: _____________________________

7. What do you notice about the value of for 1 and 2 that is different from 3 and 4? How does this affect your graph?

Name: _______________________________________The Light It Up GameFor this experiment you will need: Laser pointer Small, flat mirror Tape measurers or yard sticks (x2) Textbook TapeProcedure:Tape one of the tape measurers to the wall, beginning at floor level. Place the other tape measurer along the floor to measure the distance from the wall to the flashlight. Place the textbook 25 cm from the wall. Place the mirror on top of the book. Record where exactly you place the mirror, you want this to be consistent throughout the entire experiment.Stand on the side of the mirror opposite the wall. Aim the laser pointer toward the center of the mirror so that its image is reflected onto the tape measure attached to the wall. (See figure below.)

It is very important that you hold the laser pointer at the same height throughout the experiment. Let represent the distance from the wall to the laser pointer. Let represent the distance that the reflection appears up the wall. Measure both and in centimeters.

Record:The distance from the wall to the center of the mirror: ____________________The distance from the laser pointer to the floor: _________________________The distance from the floor to the mirror: ______________________________

Data:Collect at least eight data points and enter them in the table below.

Graph:Ketch a graph of your data below.

Equation:Write an equation for your function.

Analysis:If the information above is also true for Bart and Lisas situation, determine the distance that the should stand from the wall if the target isa. 1 meter from the floor: __________________________

b. 3 meters from the floor: _________________________

c. 10 meters from the floor: ________________________

Your mirror will be placed ________________ cm from the wall. How far away from the mirror will you need to stand?

At the fair, Lisa stumbled upon a game called Cool Down. In this game, there is a container with 8 cups of hot water (at a temperature of ). The object is to determine how many cups of cold water (at a temperature of ) to add to the container to reduce the temperature to .a. Find an equation expressing the temperature (in ) of the water in the container as a function of the volume (cups) of cold water added.

b. Explain how you obtained the equation in part a above.

c. Use your equation to determine the amount of cold water that must be added to reduce the temperature to .