`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.

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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 `

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