Translation, rescaling, and reflection

If you are given the graph of a function z = f (x, y), how can you find the graph of the function z = a f (x - x₀, y - y₀) + c for some numbers a, c, x₀, and y₀?

This is exactly analogous to what you've probably already done in one-variable calculus. Changing these numbers (we'll call them parameters) results in a translation, rescaling, and/or reflection of the graph.

Below is a CVT that illustrates these manipulations using the function f (x, y) = x² + y². Before you play with the sliders on this CVT, can you tell what will happen as you change the parameters? Once you start to move them, it is obvious. Make sure you understand why the parameters behave as they do. In particular, why do we need to put a minus sign in front of x₀ and y₀ but not in front of c? After all, they each introduce translations.

Here's the graph of the function z = a$\left(\vphantom{(x-x_0)^2 + (y-y_0)^2}\right.$(x - x₀)² + (y - y₀)²$\left.\vphantom{(x-x_0)^2 + (y-y_0)^2}\right)$ + c.

Changing x₀, y₀, or c translates the graph. If we change a but keep its sign constant (for example, keep a positive), then we rescale the figure. But if we change the sign of a, then we reflect the graph across the plane z = c. For example, the graph of z = - f (x, y) is the reflection of the graph of z = f (x, y) across the xy-plane (the plane z = 0).

Implicit surfaces

The same principles hold for surfaces defined implicitly, i.e., surfaces defined by an equation F(x, y, z) = 0. For example, the equation for a sphere of radius 1 centered at the origin is x² + y² + z² = 1. If F(x, y, z) = x² + y² + z² - 1, then F(x - a, x - b, x - c) = 0 is the equation for a sphere of radius 1 centered at the point (a, b, c). Can you see that the equation

$$\left(\vphantom{\frac{x-a}{r},\frac{y-b}{r},\frac{z-c}{r}}\right. {\frac{{x-a}}{{r}}} {\frac{{y-b}}{{r}}} {\frac{{z-c}}{{r}}} \left.\vphantom{\frac{x-a}{r},\frac{y-b}{r},\frac{z-c}{r}}\right)$$

is the equation for the sphere of radius r centered at the point (a, b, c)? I don't have a CVT prepared for this. Let me know if you think you need one. You'll go one step beyond this in the computer lab (though you won't cover translations).