Parametric Equations

by Kassie Smith


Graph for some appropriate range for t. Interpret. Is there anything to vary to help understand the graph?



In order to make the discussion about my exploration easier I will rewrite the two equations as x = t + and y = t + or

I am going to explore the effects on the graph for varying t, , , , and .


Let's first explore the equations as given, , with t ranging from 0 to 1. This results in the following graph. We can see that the graph has a slope of 2 and an x- intercept at 1.5. If the graph were to continue the y-intercept would be at -3. X values on the graph range from 1 to 2 and y values on the graph range from -1 to 1. The length of the graph is .






If we now let t range from 0 to 2, we get this graph. The graph still has the same slope of 2. And the x-and y- intercepts are or would be the same. The only difference is the length of the graph. The length of this graph is .








The pattern of changing t and its effect on changing the length of the graph changes. Mathematically, we would say that the domain is changing. As the domain increases the length of the graph increases. We are able to change the interval of t to include negative numbers and this just extends the graph to the left.


Now, let's look at the effect of varying and . When and vary, the slope of our line varies. The slope will be . This makes sense if we consider slope as rise over run. The rise would be the because y is on the vertical axes and the run would be because x is measured on the horizontal axes. Because the slope changes, our x- and y- intercepts also change. Note here that depending on the signs of and , our slope can also become negative. Below is an animation that shows the effect of just varying from -5 to 5and keeping =2.


And now, we will look at varying and . First, it is worth noting that the graph passes through the point ( , ) in our original graph and will continue to do so for any and . If  remains the same and we just let vary, then this results in vertical shifts of the graph. If  remains the same and we let vary, then this results in horizontal shifts of the graph. In the following animations I restricted t to vary from 0 to 1 in order to make it easier to see what was happening.

When varies:

Note here that a negative value for results in a leftward shift in the graph and a positive value for results in a rightward shift in the graph.

When varies:

Note here that a negative value for results in a upward shift in the graph and a positive value for results in a downward shift in the graph. This is due to the negative sign in the expression.