LESSON 1: INVERSE VARIATION
WARM-UP
Factor the following expressions completely.
1.
2.
3.
5.
4.
6.
THINK ABOUT THE SITUATION
Suppose you are going on a car trip with your family. You know you need to go 750 miles to get to your destination.
What other piece of information would you need to know in order to figure out how fast you had travelled for the entire
trip?
What was your average rate if you travelled there in 25 hours? In 15 hours? In 10 hours?
What is happening to your average rate as your time decreases? Could you get there in no time at all?
What would your average rate be if you took 3 days to get there? What would it be if it took you a week? (Don’t forget
to change your time into hours for your rate!)
INVESTIGATION 1
There are times when two variables have opposite effects on one another. For instance, during the winter the
temperature decreases. Usually, that means you turn on the heat. What happens to your electric bill when you turn on
the heat? It increases! Use the table below to answer the questions.
Outside temp (Fahrenheit)
35
48
42
54
22
11
8
14
41
Daily Electric bill ($)
1.
$2.40 $1.84 $2.18 $1.66 $2.65 $3.12 $3.17 $2.94 $2.20
Graph the data using the graph below.
2. Find the regression equation of the line that best fits this graph.
3. What is the y-intercept in context?
4. Does the domain value of zero give a logical range value in this case? Explain.
5. What is the slope of the regression line telling you in context?
6. How does the cost of electricity function as the temperature increases?
7. Does the linear regression line seem to fit this data well? Explain.
8. Find the x-intercept of your equation (y=0). Does this range value give you a logical domain value in context of
the problem? Explain.
9. For what domain values would your equation be a reasonable one? Explain.
Investigation 2 - The Breaking Point
Name: ________________________
Materials:
 5 pieces of dried spaghetti
 A small cup
 String/Yarn
 Beans
 Ruler
 Tape
1. Lay a piece of spaghetti on a table so that its
length is perpendicular to one side of the table and the end extends over the edge of the table.
2.
3.
4.
Measure the length of the spaghetti that extends beyond the edge of the table. Record this information in a table of
(length, number of beans) data.
Attach the string to the cup so that you can hang it from the end of the spaghetti. (You may need to use tape to hold the
string in place.)
Place beans into the cup one at a time until the spaghetti breaks. Record the maximum number of beans that the length of
spaghetti was able to support.
Step One: Work with a partner. Follow the procedure above to record at least five data points and then compile your
results with 3 other groups.
Step Two: Make a graph of your 15 data points with length as the independent variable, x, and number of beans as the
dependent variable, y. Does the relationship appear to be linear? If not, describe the appearance of the graph.
Step Three: Write an equation that is a good fit for your plotted data.
Length (cm)
#beans
Step Four: Brainstorm 2 real world situations in which the relationship between
the independent and dependent variables would be considered an inverse
relationship.
#beans
Length (cm)
Investigation adapted from Discovering Advanced Algebra. Key Curriculum Press, 2004
INVESTIGATION 3*
The Daytona 500 is a stock car race that is 500 miles long. Because speed varies inversely with time, we can write the
following function for the race:
.
1. Fill in the chart below using the inverse variation equation above to fill in the chart.
Time to finish (hrs)
4.25
4
3.75
3.5
3.25
3
Speed of car (mph)
2.75
2.5
2. Graph the data collected below.
Daytona 500 Data
450
400
350
300
250
200
150
100
50
0
-10
-5
-50 0
5
10
15
20
-100
3. Consider the equation. Are there any domain values that would not make sense in this problem? For any of
those values, use the instructions from Investigation 2 to create a vertical asymptote to reflect this on the graph.
4. As the domain values continue to increase, what seems to be the range value that the graph is approaching?
For that value, use the instructions from investigation 2 to create a horizontal asymptote (limit) on the graph.
5. Is there ever a time when someone will be able to complete the race in zero hours? How can we see this on
your graph?
6. Can someone complete the race by going 0 mph? How can we see this on your graph?
7. Can someone choose to go so slow that they could complete the race by taking a very long time? How is this
explained by the concept of a limit?
*Adapted from Core Math 1 and 2
CHECK YOUR UNDERSTANDING
The fuel efficiency of a car on a 300 mile test run can be calculated by the formula
, where E(g) is fuel
efficiency and g is the gallons of gas used.
1. Fill in the chart below to reflect fuel efficiency as a function of gallons of gas used.
Gas used (gallons)
5
8
10
13
17
Fuel efficiency (miles per gallon)
2.
18
20
Create a graph to represent the function from problem 1.
Fuel Efficiency Data
300
250
200
150
100
50
0
-15
-10
-5
0
5
10
15
20
-50
3. If there is a vertical asymptote, draw it on your graph.
4. What is the vertical asymptote telling us in context of the problem?
5. Is there a limit to the graph? If so, what is it?
6. What is the limit telling us in context of the problem?
7. Draw the horizontal asymptote on your graph if there is one.
25
30
35
23
HOMEWORK
1.
* The frequency(F) of the swing of a pendulum depends inversely on the length(L) of the pendulum arm. It can
be represented by the equation
√
.
a.
Make a table representing the function with a domain
b.
Sketch a graph representing the function.
, increasing by 10.
Pendulum Frequency Data
c. If there is a vertical asymptote, draw it on your graph.
d. What is the vertical asymptote telling us in context of the problem?
e. Is there a limit to the graph? If so, what is it?
f.
What is the limit telling us in context of the problem?
g. Draw the horizontal asymptote on your graph if there is one.
On a sheet of graph paper, for each rational equation below:
a. Make a table with a domain of
b. Graph each equation.
c. Sketch the vertical asymptote and the horizontal asymptote.
1.
3.
5.
2.
4.
6.
For each equation below:
a. List any domain values that would not be defined (Use your calculator if necessary).
b. Use your calculator to decide whether there is a range limit for the equation.
1.
3.
5.
2.
4.
6.