Jumping for Joy
You have explored several situations where variables change over time or as a result of changes in other variables. These changes that occur are called ratios, unit rates, constants of proportionality, and/or slope. For example, in the Unit Price activity, you can buy a 2 oz candy bar for $1.50, which means that every ounce of the candy bar costs $0.75. This is said to be a direct variation because for every ounce of candy, the price increases by $0.75.
When this situation is represented using symbols, an equation of the form is made, where y is the price of the candy bar, x is the number of ounces, and k is the unit rate of $0.75. The value of y depends on the value of x, which is why x is considered the independent variable and y is the dependent variable. This is a direct variation equation, where the amount of change from one point to another is the same.
Direct variation equations are linear equations because they have graphs that are straight-lines and data patterns that show a constant rate of change.
1. Can you identify any other characteristic about the graphs of these types of equations?
2. Why do you think direct variation equations pass through the origin?
Let us extend the ideas of direct variation to focus on linear models that have constant patterns of change in tables, graphs, and equations that do not always pass through the origin. An example of one such model is the way that forces stretch things such as rubber bands for bungee jumping or compress things such as a spring diving board.
1
I. The graph below shows the relationship between the length of a rubber band and weight for a bungee jump.
1. What are the variables represented in this problem?
2. What is the relationship between weight and the length of a rubber band as weight is added?
3. How is this relationship shown in the graph?
4. Which variable is the independent variable?
5. Which variable is the dependent variable?
1
Bungee Jumping
There is a lake in the same aquatic park as the pool, where people bungee jump from a tall bridge that crosses over the lake. The bungee jump apparatus uses industrial strength rubber bands for people of different shapes and sizes to bungee jump from a bridge.
6. For bungee jumping off the bridge, the length, L, of the rubber band is related to the amount of weight, w, attached to the rubber band by the equation: .
a. Use the equation to complete this table of sample (w, L) coordinates for the weight of a person and the length of the rubber band after jumping.
Weight (w) in lbs. / 0 / 10 / 20 / 30 / 40 / 50 / 100 / 200 / 500Length (L)in ft.
b. How does the pattern of change shown in your table compare with that shown in the graph on the previous page?
c. What is the length of the bungee cord when there is no weight on it? ______
How can this information be seen in the equation: ?
How can this information be seen in the table of values above?
How can this information be seen in the graph on the previous page?
d. How many additional feet does the bungee cord stretch for every 10 pounds is added to the cord? ______
How can this information be seen in the equation: ?
How can this information be seen in the table of values above?
How can this information be seen in the graph on the previous page?
7. The next table shows a different bungee jumping company, Jumping for Joy, and the length of their bungee chord with various weights.
Weight in lbs. (w) / 10 / 20 / 30 / 40 / 50Length in ft. (L) / 60 / 80 / 100 / 120 / 140
a. Does the length of the bungee cord appear to be a linear function of the weight added to the cord? Explain.
b. Assume that the length of the bungee cord is a linear function of the weight added to the cord. Calculate the missing entries in the table below.
c. What is the length of the rubber band without any weight attached? ______
Weight in lbs. (w) / 0 / 10 / 15 / 20 / 25 / 30 / 40 / 50 / 100 / 101Length in ft. (L) / 60 / 80 / 100 / 120 / 140
Rates of Change
A key feature to any function is the way the value of the dependent variable changes as the value of the independent variable changes. Notice that as the weight increases from 30 lbs. to 40 lbs., the length of the bungee cord stretches from 100 ft. to 120 ft. This is an increase of 20 ft. in length for an increase of 10 lbs. of weight on the cord, or an average of 2ft. per pound. The length of the bungee cord increases at a rate of 2ft. per 1lb. of weight added.
8. Using your table from part b, complete the entries in the table below by studying the rate of change in the Jumping for Joy Company as the weight increases.
Change in Weight (in lbs.) / Change in Length (in ft.) / Rate of Change (in ft. per lb.)10 to 20
20 to 25
25 to 40
50 to 100
a. What do you notice about the rate of change in the length of the bungee chord as the weight of the person jumping increases?
9. Analyze the following equations:
a. Which of the equations shows how to calculate the length of the bungee chord after jumping at the Jumping for Joy Company? How do you know?
b. What do the numbers in the equation(s) you selected in part a, tell you about any jump at the Jumping for Joy bungee jumping company?
The diagram below presents graphs of the length of the bungee chord using three different rubber bands when people of various weights have jumped.
10. The equations that correspond with the graphs are:
a. Match each equation with its graph. Explain how you can make the matches without any calculations or graphing tool.
Series 1: ______
Series 2: ______
Series 3: ______
b. What do the numbers in the equation for each situation tell you about the relationship between the length of a stretched bungee chord and the weight of the person who jumped?
Series 1: ______
______
Series 2: ______
______
Series 3: ______
______
Springboard Diving
In some places where springs are used, the springs are made shorter by pressure from force or weight. For example, the springs in a diving board are compressed when a person jumps on them.
The diagram below shows graphs of three different linear relations. These graphs model data from an experiment that tested springboards for diving using spring length in centimeters and weight on the board in kilograms.
1. Using the chart above, make three tables of (weight, length) coordinates. Make one table for each linear model; include weights starting at 0 kilograms.
Series 1
Weight (x) / 0 / 20 / 40 / 60 / 80 / 100 / 120Length (y)
Series 2
Weight (x) / 0 / 20 / 40Length (y)
Series 3
Weight (x) / 0 / 20 / 40 / 60 / 80 / 100Length (y)
2. How long are the springs with no weight applied? ______
3. What are the rates of change in spring length as weight increases?
Series 1 ______
Series 2 ______
Series 3 ______
4. How can you calculate these rates using points on the line?
5. How can you calculate these rates using pairs of values in the tables?
6. For each starting point and each rate of change, write an equation expressing L (length in centimeters) as a function of W (weight in kilograms).
Series 1 ______
Series 2 ______
Series 3 ______
7. Given each linear equation below that models a spring that is stretching or compressing, identify the following.
a. b.
Initial length of spring: ______Initial length of spring: ______
Rate of change: ______Rate of change: ______
Stretch or compression?: ______Stretch or compression?: ______
8. Linear models relating any two variables (x) and (y) can be represented using tables, graphs, or equations. Important characteristics of a linear model can be seen in each representation.
a. How can the rate of change between two variables be seen:
• In a table of values: ______
• In a linear graph: ______
• In an equation: ______
b. How can the y-intercept be seen:
• In a table of values: ______
• In a linear graph: ______
• In an equation: ______
Jumping for Joy Answer Key
You have explored several situations where variables change over time or as a result of changes in other variables. These changes that occur are called ratios, unit rates, constants of proportionality, and/or slope. For example, in the Unit Price activity, you can buy a 2 oz candy bar for $1.50, which means that every ounce of the candy bar costs $0.75. This is said to be a direct variation because for every ounce of candy, the price increases by $0.75.
When this situation is represented using symbols, an equation of the form is made, where y is the price of the candy bar, x is the number of ounces, and k is the unit rate of $0.75. The value of y depends on the value of x, which is why x is considered the independent variable and y is the dependent variable. This is a direct variation equation, where the amount of change from one point to another is the same.
Direct variation equations are linear equations because they have graphs that are straight-lines and data patterns that show a constant rate of change.
1. Can you identify any other characteristic about the graphs of these types of equations?
All graphs pass through the origin.
2. Why do you think direct variation equations pass through the origin?
When the value of x is 0, the value of y is 0.
Let us extend the ideas of direct variation to focus on linear models that have constant patterns of change in tables, graphs, and equations that do not always pass through the origin. An example of one such model is the way that forces stretch things such as rubber bands for bungee jumping or compress things such as a spring diving board.
1
I. The graph below shows the relationship between the length of a rubber band and weight for a bungee jump.
1. What are the variables represented in this problem?
The variables represented are weight and rubber band length.
2. What is the relationship between weight and the length of a rubber band as weight is added?
As weight increases, the length of the rubber band increases.
3. How is this relationship shown in the graph?
The line increases (goes up) as the graph goes left to right.
4. Which variable is the independent variable?
The independent variable is weight.
5. Which variable is the dependent variable?
The dependent variable is the length of the rubber band.
1
Bungee Jumping
There is a lake in the same aquatic park as the pool, where people bungee jump from a tall bridge that crosses over the lake. The bungee jump apparatus uses industrial strength rubber bands for people of different shapes and sizes to bungee jump from a bridge.
6. For bungee jumping off the bridge, the length, L, of the rubber band is related to the amount of weight, w, attached to the rubber band by the equation: .
a. Use the equation to complete this table of sample (w, L) coordinates for the weight of a person and the length of the rubber band after jumping.
Weight (w) in lbs. / 0 / 10 / 20 / 30 / 40 / 50 / 100 / 200 / 500Length (L)in ft. / 20 / 25 / 30 / 35 / 40 / 45 / 70 / 120 / 270
b. How does the pattern of change shown in your table compare with that shown in the graph on the previous page?
As the weight increases by 10, the length increases 5.
c. What is the length of the bungee cord when there is no weight on it? 20
How can this information be seen in the equation: ?
The length with no weight is represented by the constant.
How can this information be seen in the table of values above?
The length with no weight is found in the table when the weight is 0.
How can this information be seen in the graph on the previous page?
The length with no weight is represented by the y-intercept.
d. How many additional feet does the bungee cord stretch for every 10 pounds is added to the cord? 5 feet
How can this information be seen in the equation: ?
Substitute 10 for w; .5 times 10 is 5.
How can this information be seen in the table of values above?
As the weight increase 10, the length increases 5.
How can this information be seen in the graph on the previous page?
Every time the weight increases 20, the length increases 20.
7. The next table shows a different bungee jumping company, Jumping for Joy, and the length of their bungee chord with various weights.
Weight in lbs. (w) / 10 / 20 / 30 / 40 / 50Length in ft. (L) / 60 / 80 / 100 / 120 / 140
a. Does the length of the bungee cord appear to be a linear function of the weight added to the cord? Explain.