# `x=t^2+t , y=t^2-t` Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter.

The graph is described by the parametric equations in x, y and t:

`x(t) = t^2 + t, quad y(t) = t^2 - t `

A sketch of the graph is as pictured, with (as standard) the horizontal axis being the x-axis and the vertical axis being the y-axis.

...

## Unlock This Answer Now

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

The graph is described by the parametric equations in x, y and t:

`x(t) = t^2 + t, quad y(t) = t^2 - t `

A sketch of the graph is as pictured, with (as standard) the horizontal axis being the x-axis and the vertical axis being the y-axis.

To express the function in rectangular form, we eliminate the parameter `t ` .

Firstly, note that

`x + y = 2t^2 `   and `(x-y ) = 2t ` ` `` `

so that

`2(x+y) = (x-y)^2 `

We can then write the function in rectangular form, in terms of `x ` and `y ` only as

`(x-y)^2 - 2(x+y) = 0 `

Since `x ` and `y ` are interchangeable in this function, the graph is symmetric about the line `y = x `

Approved by eNotes Editorial Team