Parametric

Let's look at the parametric equations of the form:

eq1

What shapes will the graph represent? What do the values of a and b determine for the graph? We'll start by looking at a few examples to just understand what happens as we change a and b.

Notice how the graphs change in relation to a and b. When a is larger than b, we see how the our circle is stretched sideways. If b is larger than a than the circle is stretched up and down. This stretched circle is called an ellipse.

We can also take note of how as the distance between a and b becomes larger, the more stretched the circle becomes. In other words, as the as either a or b becomes significantly larger than the other the ellipse becomes more elongated.

The form of the parametric equations give us a very clearly represented picture of what our ellipse should look like. Where a gives us the semimajor axis and b gives us the semiminor axis. Therefore, 2a = major axis of our ellipse and 2b = minor axis of the same ellipse.

Look below at the two graphs.

The graphs look identical, while the equations below show an obvious difference. Even though b is now -b, the negative sign does not seem to effect the graph's overall shape and look. Let's see if we can locate where the negative sign does interact with our graph.

Since we have recognized before that to get only half of the graph to show we need to only take a partial length of a circle. So we will have 0 < t < pi.

Now we can see where our negative sign effects our graph. The negative simply allows for the graph to trace the opposite direction than from when a and b are both positive. From our observations of how a and b interact prior, what can we know about if a is negative and b is positive? What if both are negative?

Continue working with your graphing calculator or graphing program until you feel confident in your predictions.

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Assignment 11: Every Rose has its Thorns