Plot3D doesn't generate the ellipitic paraboloid it's supposed to

Question

I need a contour plot of the function $f(x,y) = (x-1)^4 + (2x+y)^2$. Using Mathematica, I get

I thought this looks strange. The contour plot is supposed to show ellipses. After all, this is what the graph of $f$ looks like (elliptical paraboloid) using GeoGebra

I thought that I was doing something wrong with ContourPlot. But then I used Plot3D on that same function $f$, and I got this

So I don't know what's going on here. I don't understand why Plot3D doesn't generate the elliptical paraboloid that it is supposed to. But what I really need is the contour plot showing the ellipses.

Thanks in advance for any help, and sorry if the images in the body of the questions are too big.

Answer 1

Clear["Global`*"]

The default PlotRange is obscuring the detail that you want.

Plot3D[(x - 1)^4 + (2 x + y)^2, {x, -8, 8}, {y, -8, 8},
 PlotRange -> {0, 6},
 ClippingStyle -> None]

ContourPlot[(x - 1)^4 + (2 x + y)^2,
 {x, -8, 8}, {y, -8, 8},
 PlotRange -> {Full, Full, {0, 20}}]

Answer 2

To understand why the plot you made (which is correct although distorted) looks like it does and also see the region that really interests you without distortion, make use of the options AspectRatio and Contours. Like so:

ContourPlot[(x - 1)^4 + (2 x + y)^2, {x, -2, 4}, {y, -14, 9},
  AspectRatio -> Automatic,
  Contours -> 2^Range[-2, 6],
  ImageSize -> Large]

Observe that the countours in the range around y = 0 become more and more vertical as z increases. That explains your result.