Question

How do you graph `y = x^2 - 3`?

Answer 1

See below

Explanation:

I'm assuming that you are familiar with the graph of `f(x)=x^2`, since it is the "standard" parabola:

graph{x^2 [-3, 3, -4.5, 10]}

Now, the function you want to graph is not exactly this one, but they're quite similar: `f(x)=x^2-3` is a transformed version of `f(x)=x^2`.

The tranformation belongs to the following family of transformations:

`f(x) \to f(x)+k`

(in your case, `k=-3`)

This kind of tranformations translate the graph vertically, upwards if `k>0`, downwards otherwise. So, in your case, there will be a shift of three units down. Compare this graph with the first one:

graph{x^2-3 [-3, 3, -4.5, 10]}

you can see how the "shape" is identical, and the only difference is the vertical translation.