Calculus of a Single Variable, Chapter 10, 10.2, Section 10.2, Problem 6

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