Student Exploration: Direct and Inverse Variation

Transcription

1 Name: Date: Class Period: Student Eploration: Direct and Inverse Variation Vocabular: constant of proportionalit, direct variation, inverse variation Overview:. Michelle makes $0 an hour babsitting. A. How much will she make in hours? in 4 hours? B. How does doubling the babsitting time affect the amount Michelle makes?. A car moving at a speed of 0 miles per hour (mph) will travel 0 miles in one hour. A. How long will it take to cover 0 miles at a speed of 60 mph? B. How does doubling the speed affect the time? Activit A: Direct variation Graph =. A. Substitute the -values shown in the table to the right into the equation to find several points on the line. 0.5 B. Calculate the values of in the last column of our table. What is true about the ratio of the variables in a direct variation?.5.5

2 C. How much does change when increases b? D. How does the change ou described above relate to the slope of the line =? E. In the direct variation equation = k, k is the constant of proportionalit. Look at the last column of our table and our answers to the questions above. What three things are equal to the value of k? Use a different value for k in the general equation = k. Stud the resulting graph and table to see what happens to the -value for the following changes in the -value. A. If the -value is multiplied b, what happens to? B. If the -value is multiplied b 5, what happens to?.5 C. If the -value is divided b, what happens to? D. If the -value is divided b 4, what happens to?. If varies directl as, then = k for some value k. A. Suppose varies directl as and = 0 when = 5. What is the constant of proportionalit, k, in this situation? Eplain how ou found k. B. Using the k value ou found above. Find four other (, ) pairs that occur in this direct variation function. (, ) (, ) (, ) (, )

3 4. Suppose ou re echanging mone from one form of currenc to another, where $.00 in currenc A equals $.70 in currenc B. A. If = amount of currenc A and = amount of currenc B, write an equation to model this currenc conversion. B. Graph our equation. C. How much is $8.00 in currenc A equal to in currenc B? Eplain how ou found our answer. D. How much is $7 in currenc B equal to in currenc A? Eplain how ou found our answer. 5. Write an equation to solve each problem. Then solve the problem. A. If varies directl as, and = when =, find when = 0. equation: If = then = 0. B. Joe gets paid b the hour. If he earns $54 for working 6 hours, how much will he earn when he works 5 hours? equation: If Joe works 5 hours, he will make

4 Activit B: Inverse variation. Graph =. A. Substitute the -values shown in the table to the right into the equation to find several points on the graph. B. Calculate the product in the last column of our table. What is true about this product in an inverse variation? C. In the inverse variation equation, k is the constant of proportionalit. What do ou notice about the value of k for this equation and the values in the last column of our table? D. How much does change when increases b? 8 k. Use different values for k in the general equation =. Stud the resulting graph and table to see what happens to the -value for the following changes in the -value. A. If the -value is multiplied b, what happens to? B. If the -value is multiplied b 5, what happens to? C. If the -value is divided b, what happens to? D. If the -value is divided b 4, what happens to?. If varies inversel as, then = k. Suppose varies inversel as and = 4 when = 6. A. What is the constant of proportionalit, k, in this situation? B. Eplain how ou found k.

5 4. For anthing in motion, distance traveled, d, is equal to the average speed (average rate), r, multiplied b the time traveled, t. This is often abbreviated as d = rt. A. Suppose ou need to drive 75 miles. In the boes to the right, solve d = rt for time, and use 75 for d. This should give ou an equation that shows that time varies inversel with rate. = B. Graph our equation. C. How fast would ou need to drive to make the trip in.5 hours? Eplain how ou found our answer. 5. Daniel s baseball team raised $90 to bu new baseballs. The team needs to know how man baseballs the can bu, based on how much each one costs. A. Write an equation to model this situation. (Hint: Your equation should show that the number of baseballs the team can afford varies inversel with the price of each baseball.) = B. What does each variable in our equation represent? C. How man baseballs can the team bu if each baseball costs $.50?