Points, Vectors, and Functions
Topics
Introduction to Points, Vectors, And Functions - At A Glance:
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) =
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 (x_{1},y_{1}) is on the trajectory.
We are going to reverse the graphing process. We'll start with point (x_{1},y_{1}) 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 = t^{2} y = t^{3}. |
Example 4
Determine if the point (4,3) is on the graph of the parametric equations x = t^{2} |
Exercise 1
- Determine whether each point is on the graph of the parametric equations
x = t^{3}y = t^{4}
- (8,16)
- (8,-16)
- (-27, 81)
Exercise 2
- Show that the point (5,-6) is on the graph of the parametric equations
x = t^{2} + 1y = 3t
Exercise 3
- Show that the point (11,9) is not on the graph of the parametric equations
x = t^{2} + 1y = 3t