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This was a pretty interesting exercise that we were asked to do in class, I must say, but I think I may be able to help you a little bit

Think back to the first problem we had which was to sketch the x-z and y-z axis interpretations of the function

$$ f(x,y) = x^2 - y^3 $$

The x-z axis will be easier to start with because we know what a parabola looks like. We take $y$ to be a constant and we will set this constant to be $y = 0$. Now we'll see that

$$ f(x,y) = z = x^2 - y^3 $$

$$ f(x,y) = z = x^2 $$

This function will produce a graph in the x-z plane in the shape of a parabola. As $y$, the constant, changes, the position of the parabola is shifted up and down the z axis.

(insert a graph here later)

Now the second piece we have to look at is what happens when we hold $x$ constant and change only $y$. By setting $x = 0$ we now have the equation:

$$ f(x,y) = z = -y^3 $$

The graph of this is a reversed hyperbolic curve in the y-z plane because we have $-y^3$ instead of just $y^3$. This curve starts in the top left corner and makes its way down to the bottom right in a hyperbolic manner

(insert another graph here later).

Now we have an idea of what the cross sections of this function look like as we hold either $x$ or $y$ constant, but this only gives us half of the picture. Two halves to be exact.

To put these two pieces together to get a rough idea of what the 3D figure should look like.

Recapping:

• in the x-z plane, we have parabolas

• in the y-z plane, we have hyperbolas starting in the negative y moving towards the positive y

The figure we get after putting this together (at least in small scale) will look something akin to a water slide.

[note if someone knows how to work matlab and can help me get a picture of these figures either in matlab or in any other language, please let me now or post an answer below]

This was a pretty interesting exercise that we were asked to do in class, I must say, but I think I may be able to help you a little bit

Think back to the first problem we had which was to sketch the x-z and y-z axis interpretations of the function

$$ f(x,y) = x^2 - y^3 $$

The x-z axis will be easier to start with because we know what a parabola looks like. We take $y$ to be a constant and we will set this constant to be $y = 0$. Now we'll see that

$$ f(x,y) = z = x^2 - y^3 $$

$$ f(x,y) = z = x^2 $$

This function will produce a graph in the x-z plane in the shape of a parabola. As $y$, the constant, changes, the position of the parabola is shifted up and down the z axis.

(insert a graph here later)

Now the second piece we have to look at is what happens when we hold $x$ constant and change only $y$. By setting $x = 0$ we now have the equation:

$$ f(x,y) = z = -y^3 $$

The graph of this is a reversed hyperbolic curve in the y-z plane because we have $-y^3$ instead of just $y^3$. This curve starts in the top left corner and makes its way down to the bottom right in a hyperbolic manner

(insert another graph here later).

Now we have an idea of what the cross sections of this function look like as we hold either $x$ or $y$ constant, but this only gives us half of the picture. Two halves to be exact.

To put these two pieces together to get a rough idea of what the 3D figure should look like.

Recapping:

• in the x-z plane, we have parabolas

• in the y-z plane, we have hyperbolas starting in the negative y moving towards the positive y

The figure we get after putting this together (at least in small scale) will look something akin to a water slide.

[note if someone knows how to work matlab and can help me get a picture of these figures either in matlab or in any other language, please let me now or post an answer below]