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Direct Variation

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides word problems which develop concepts related to direct variation.

Exercise 1

Suppose I make $6/hour. Let tt represent the number of hours I work, and mm represent the money I make.

  • a. Make a table showing different tt values and their corresponding mm values. (mm is not how much money I make in that particular hour—it’s how much total money I have made, after working that many hours.)
    Table 1
    time (tt) money (mm)
       
       
       
       
  • b. Which is the dependent variable, and which is the independent variable?
  • c. Write the function.
  • d. Sketch a quick graph of what the function looks like.
  • e. In general: if I double the number of hours, what happens to the amount of money?

Exercise 2

I am stacking bricks to make a wall. Each brick is 4" high. Let bb represent the number of bricks, and hh represent the height of the wall.

  • a. Make a table showing different bb values and their corresponding hh values.
    Table 2
    bricks (bb) height (hh)
       
       
       
       
  • b. Which is the dependent variable, and which is the independent variable?
  • c. Write the function.
  • d. Sketch a quick graph of what the function looks like.
  • e. In general: if I triple the number of bricks, what happens to the height?

Exercise 3

The above two scenarios are examples of direct variation. If a variable yy “varies directly” with xx, then it can be written as a function y=kxy=kx, where kk is called the constant of variation. (We also sometimes say that “yy is proportional to xx,” where kk is called the constant of proportionality. Why do we say it two different ways? Because, as you’ve always suspected, we enjoy making your life difficult. Not “students in general” but just you personally.) So, if yy varies directly with xx

  • a. What happens to yy if xx doubles? (Hint: You can find and prove the answer from the equation y=kxy=kx.)
  • b. What happens to yy if xx is cut in half?
  • c. What does the graph y(x)y(x) look like? What does kk represent in this graph?

Exercise 4

Make up a word problem like Exercises 1 and 2 above, on the subject of fast food. Your problem should not involve getting paid or stacking bricks. It should involve two variables that vary directly with each other. Make up the scenario, define the variables, and then do parts (a) – (e) exactly like my two problems.

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