Graphing a Parametric Equation

Question

I need to graph and show the work for this problem. The graph needs to include arrows on the curve to show the direction of motion and I need to label the t-values graphed.

$$c(t)=(2+4t, 3+2t)$$

So to start off do I need to take the parametric equation and put it in a y=f(x) form? I've never had to graph one of these before nor have I seen one graphed so I'm a bit lost. Thanks!

Answer 1

Probably the easiest way to do this is to choose a bunch of equally-spaced values of $t$ and compute, for each $t$, the coordinates $c(t)$. Plot those points and connect them with a smooth curve or straight line, using arrows to show the forward motion.

For example, with $t=0$ you have $c(0)=(2,3)$ and with $t=1$ you have $c(1)=(6,5)$. Plot those, and also $c(-2), c(-1), c(2), c(3)$, etc.

Answer 2

c(t) = x(t), y(t)

x(t)= 2+4t => t=(x-2)/4

y(t)= 3+2t => t=(y-3)/2

t = t

(x-2)/4 = (y-3)/2

2(x-2) = 4(y-3)

2x - 4 = 4y -12

x - 2 = 4y -6

x + 4 = 4y

y = (x+4)/4

Set values for your "new" function, and you will have the graph.