NCES: MA.N.1.1; MA.A.2; MBC.A.1.2 NCSCS:

Transcription

1 Unit 1-Functions, Equations and Systems Lesson 1- Direct and Indirect Variation-Investigation 1 NCES: MA.N.1.1; MA.A.2; MBC.A.1.2 NCSCS: Algebra I 1.03; Algebra II 1.05 Name Date By the end of this investigation you should be able to answer: How do course length and steepness affect run time for a downhill race and how can the relationship between those variables be expressed in symbolic form? On a Roll Platform Height, Ramp Length, and Ride Time (p. 4) 1. The height of a platform and length of the ramp affect the time it takes to roll down the ramp. What are your predictions for:

2 a. For a fixed ramp length, time as platform height increases. b. For a fixed platform height, time as ramp length increases. c. Two ramps, one twice as long as the other.how do you adjust their platform heights so that skateboarders will reach the bottom at the same time? 2. a. The following table gives data for the time T that it takes a ball to roll down 2 ramps of different heights as the ramp length L increases from 3 to 8 feet. Ramp Length L (in feet) Roll Time T (in sec) at 0.5 ft height Roll Time T (in sec) at 0.25 ft height b. Make a plot of (L,T) relating roll time T to ramp length L.one for each platform height. 0.5 ft Platform Height 0.25 ft Platform Height Roll Time T (sec) Roll Time T (sec) Ramp Length L (ft) Ramp Length L (ft) c. For each platform height describe the relationship between roll time and ramp length.

3 3. a. The following table gives data for the time T that it takes a ball to roll down 2 ramps of different lengths as the platform height H increases from 0.25 feet to 1.5 feet. Platform Height H (in feet) Roll Time T (in sec) for 8-ft length Roll Time T (in sec) for 4-ft length b. Make a plot of (H,T) relating roll time T to platform height H.one for each ramp length. 8-ft Ramp Length 4-ft Ramp Length Roll Time T (sec) Roll Time T (sec) Platform Height H (ft) Platform Height H (ft) c. Relationship between roll time and platform height for each fixed ramp length. 4. Compare your results from Problems 2 and 3 to your predictions in Question 1. Any surprises? Why do the results make sense?

4 5. Refer to the functions in #5 on page 5 to fill in the following table.

5 Basic Variation Patterns (p. 6) Direct Variation: or y varies directly with x y is directly proportional to x The ratio of y to x is constant. Ex: for some constant k (constant of proportionality) All of these mean: Ex: 7. Explain why the perimeter P of a square is directly proportional to the length s of a side. s a. Direct variation equation: b. Constant of proportionality: s s c. As the value of s increases, P.. s How is this pattern related to the constant of proportionality? 8. There are two relationships from Problem 5 that are direct variations. Find these two and fill in the chart, one column per relationship. Direct variation relationships (brief description of function) a. How does the dependent variable change as the independent variable increases? b. Fill in the blanks: c. Relationship in an equivalent symbolic form. The variable is directly proportional to, with constant or proportionality. The variable is directly proportional to, with constant or proportionality.

6 Inverse Variation: or for some constant k (constant of proportionality) y varies inversely as x y is inversely proportional to x The product of y to x is constant. All of these mean: Ex: Ex: 9. Time t required to download a 4-megabyte music file on the internet is inversely proportional to the rate r at which the data is transferred. a. time it takes to download a 4-megabyte file if the transfer rate is 2.5 megabytes per minute. if the transfer rate is 0.8 megabytes per minute b. symbolic form for relationship between t and r c. As r increases, t.. How is this pattern related to the constant of proportionality? In our experiments with ramp height, platform length, and roll time, what were the constants of each? a. Ramp Length vs. Roll Time: b. Platform Height vs. Roll Time: 6. Using your graphs (data plots) from Problems 2 and 3 and your new knowledge of direct and inverse variation, can you find function rules that might be good models for the relationships between (make sure you label what your constant is in the context of each graph): a. roll time T and ramp length L: b. roll time T and platform height H:

7 Substitute in the corresponding table values for k in your above equations. Do your equations match the data closely? Explain. 10. There are two relationships from Problem 5 that are inverse variations. Find these two and fill in the chart, one column per relationship. Inverse variation relationships (brief description of function) How does the dependent variable change as the independent variable increases? Relationship in two different equivalent symbolic forms. Fill in the blanks: The variable is inversely proportional to, with constant or proportionality. The variable is inversely proportional to, with constant or proportionality. 11. For the tables given, check whether the relationship is direct or inverse variation and give the constant of proportionality. Table Direct? Inverse? I Constant of Proportionality II III

8 Summarize The Math (p. 9) a. Suppose y is directly proportional to x with constant of proportionality k>0. i. if x increases by 1, y ii. if x doubles, y. iii. graph of the function looks like. b. A function with rule y = mx + b where b 0 is not a model of direct variation. i. differences between graph of linear function and related direction variation y = mx. similarities between graphs.. ii. differences between tables of same two functions. similarities between tables. c. Suppose y is inversely proportional to x with constant of proportionality k>0. i. is x increases, y.. ii. if x doubles, y.. iii. graph of the function looks like.. Check Your Understanding (p. 9) Direct, Inverse, Function Neither? a. N = 100h Constant of Proportionality Sentence Describing Function b. B = 50(2 t ) c. d. C = 0.2w e. v = 64 32t