Question

How do you graph `y=-2(x-2)^2-4`?

Answer 1

Graph:
graph{-2(x-2)^2-4 [-6.54, 13.46, -12.2, -2.2]}

See explanation below.

Explanation:

There are more rigorous ways to draw the graph of an parabola by hand (using calculus, mostly), but for our purposes, here's what we're going to do:

Step 1: Identify the Vertex
This is just because you have your parabola in vertex form, which makes this process very easy. For a parabola in vertex form `y = a(x-h) + k`, the vertex is simply `(h, k)`. Therefore, your vertex would be `(2, -4)`.

Step 2: Identify Intercepts
This is `x` intercepts (where `y = 0`), and `y` intercepts (where `x=0`). Let's find these:

`x`- intercepts:

`0 = -2(x-2)^2-4`
`4 = -2(x-2)^2`
`-2 = (x-2)^2`

Now we can stop right there, as we can see that we're gonna end up with the square root of a negative number. Hence, we have no real `x`-intercepts.

`y`-intercepts (these are considerably easier):

`y = -2(0-2)^2-4`
`y = -8`

Step 3: Identify The Direction of the Parabola

This is pretty straightforward -- this basically depends on the sign of the `a` value at the front of your equation. In our case it is negative, so our parabola is going to be pointed downward.

Step 4: Easy Points

This just means you plug in some values of `x` for which it is pretty straightforward to find the value of `y`. `2`, for example, works really well since it cancels out everything under the square, and leaves you with `4`. `1` also works out pretty well, and gives you `-6`.

Now, you just put everything together, and see what turns out:
graph{-2(x-2)^2-4 [-6.54, 13.46, -12.2, -2.2]}

Does that have everything we just calculated?

Hope that helped :)