2008-01-08 - Parametrics
Today's nonsense
- Murky classroom environment
- Homework (ws) due tomorrow
- Daniel Tammet (see 2006-05-27 logs; Tesla's "The Problem of Increasing Human Energy"; hyperlexia; 28 concepts/moment instead of 5 + or - 2 things per moment in visual thinkers)
Parametric equation
Parametric equations are much like the components of vectors in physics. None of this has anything to do with choir. For each pair of parametric equations:
1) transform to a regular quadrant
2) tell from rectangular equation what the graph should be
3) sketch the graph for t > = 0 showing direction
a) x = 2 - 3t
y = 4t + 1
What is the common multiple so that when you add the two equations, the T variables go away? In this case, 4x = 8-12T, and then the bottom is going to become 3y = 12T + 3
Part 1) Add them together and you get 4x+3y = 11
Part 2) Looking at that, what is 4x+3y = 11 look like? A line. That's all you have to say.
Part 3) If t > = 0, then my x is going to have a starting point and my y is going to have a starting point, and we don't have negative t variables so the y variable will never be negative in this case.
So. The graph. Go to your calculator. T > = 0. So when t = 0, what is x and what is y? x is 2 and y is 1. And it's greater than or equal to, so we know the point exists. (This is going to be a ray, not a line.) Go to your window and make sure your T starts at 0. So, when T = 1, plot some points. What is your x and what is your y? You should include an arrow - just watch the calculator as it graphs the graph, and put an arrow on the end of the ray. It'll go much faster if you do this on the calculator. You change it to rectangular and in part 2 you say what it should look like, but then you graph the parametric equation and you see how the domain restrictions play into all of this. Include the graph. With an arrow for direction.
b) x = T^(2) + 2
y = 3T^(2) - 2
Part 1) Solve for T^2 in the x equation. You get: y = 3x - 8.
Otherwise you would have had to multiply the x equation by -3.
Part 2) What type of graph is that (from part 1)? A line.
Part 3) Graph it.