In this activity you will investigate relationships between points, describe how one point depends on another, and formulate your own definition of function.

(List items in bold below contain questions for you to answer, as you work to analyze and understand your results.)

Vary the Variables

Begin by varying independent variables and observing how several functions behave.

  1. Look at the Learning Goal and then pressStart Activity.
  2. Try to drag each point on the page that appears 1. (You can drag some points but not others.)
    The points you can drag are independent. The ones that move only when you drag another point are dependent.
  3. Drag points to determine which ones are related. Then list the independent and dependent points, and describe the relationship.
  4. What’s the relative speed and direction of the variables? At what locations do they come together with each other? (These locations are called fixed points.)
    Some independent points may control more than one dependent point, while some may not control any dependent point at all.
Independent
Variable
 Dependent
Variable
 Relationship
→




RC,θ

DC,s

  1. When you start, only the independent variable x is showing. Drag x around to see that you can move it anywhere.
  2. Press Show Reflection Function to show the mirror m and the dependent variable Fm(x).
    In this activity we use F to stand for a reflection function. (If we used R, reflection might be confused with rotation.) You can think of Fm(x) as “the flip across m of independent point x.
  3. Drag x around and observe the behavior of Fm(x).
  4. What do you notice about the relative rate of change (both speed and direction) of the two variables? Record your observations in the first line of a table like the one below.
    5. Transformation Speed of dependent variable
    (slower, the same, faster)
    		 Direction of dependent variable
    (always same, always different, or varies)
    Fm

    Tv

    RC,θ

    DC,s

  5. Hide the reflection function and show one of the other functions. As before, drag x around and pay attention to the relative rate of change of the variables.
    Use tracing to help you make accurate observations.
  6. Investigate the remaining functions. For each function record in your table the relative rate of change (both speed and direction) of its variables.

Restrict the Domain to One Dimension

Now observe the effects of restricting the domain to a one-dimensional object.

  1. Click the pointer to go to page 2 of the sketch.
  2. On this page, the independent variable x is restricted to line segment d (for domain). Drag point x to observe its behavior.
  3. Press Show Reflection and Range and then drag x back and forth on the domain to see how Fm(x) moves along the range.
  4. Drag an endpoint of the domain to see how the range depends on the domain.
  5. Compare the relative length and direction of the domain and range, and describe your observations.
  6. Show each of the three remaining functions, one at a time. For each function, drag x and then modify the domain.
  7. Use a table like the one below to summarize your observations for each function.
    8. Transformation Length of range relative to domain
    (shorter, the same, longer)     		Angle of range relative to domain
    (always parallel, always different, or varies)
    Fm

    Tv

    RC,θ

    DC,s

  8. How do the results of this table compare to the table from #4 above? Explain why this makes sense.

Make the Range Collinear with the Domain

You probably noticed that two of the transformations produce a range that’s always parallel to the domain. (If not, recheck each function and your conclusions.) Now you’ll experiment with these transformations to figure out how to put the domain and range on the same line.

  1. Show one of the transformations that has a range parallel to the domain.
  2. Adjust the defining elements of the transformation (center point, vector, mirror, or parameters) to move the range so it’s lined up with the domain.
  3. Describe what you did to the first transformation to line up the range with the domain.
  4. Show the other transformation that has its domain and range parallel. Adjust this function’s defining elements (center point, vector, mirror, or parameters) to line up its range with the domain.
  5. Describe what you did to the second transformation to line up the range with the domain.

Explore More

Experiment with the other two transformations (the ones whose range was not originally parallel to the domain) to see if there’s any way to adjust the transformation to make the range and domain parallel. Describe your efforts and results.

You can expand or contract the directions below the sketch, depending on your screen size.