Introduction to :

We can describe trajectories of things with parametric equations.

Before we go forward, we should review parabolas via Super Mario Brothers. When Mario jumps, we can parameterize his trajectory d(x,y) = as a function of time t by writing d(t) = <x(t),y(t)>.

To graph this parametric equation, we graph a bunch of individual points. To find the coordinates of a point we take a t value and find the corresponding x and y values.

How do we know if Mario landed safely on the other side of the canyon? We need to know if point (x₁,y₁) is on the trajectory.

We are going to reverse the graphing process. We'll start with point (x₁,y₁) and find the t-value that produced it, if there is one.

To determine if a particular point (x,y) is on a parametric graph we need to find a single value of the parameter t that produces the correct x- and y-values at the same time.

Example 1

Determine if the point (5,-5) is on the graph of the parametric equations

x = 4t + 1

y = 2t-7.

Example 2

Determine if the point (17,0) is on the graph of the parametric equations x = 4t + 1 y = 2t-7.

Example 3

Determine if the point (4,-8) is on the graph of the parametric equations

x = t²

y = t³.

Example 4

Determine if the point (4,3) is on the graph of the parametric equations

x = t²
y = t + 2

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Exercise 1

Determine whether each point is on the graph of the parametric equations

x = t³y = t⁴

(8,16)
(8,-16)
(-27, 81)

Exercise 2

Show that the point (5,-6) is on the graph of the parametric equations

x = t² + 1y = 3t

Exercise 3

Show that the point (11,9) is not on the graph of the parametric equations

x = t² + 1y = 3t

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