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

Nov 1, 2017

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 $y = a \left(x - h\right) + k$, the vertex is simply $\left(h , k\right)$. Therefore, your vertex would be $\left(2 , - 4\right)$.

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

$x$- intercepts:

$0 = - 2 {\left(x - 2\right)}^{2} - 4$
$4 = - 2 {\left(x - 2\right)}^{2}$
$- 2 = {\left(x - 2\right)}^{2}$

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):

$y = - 2 {\left(0 - 2\right)}^{2} - 4$
$y = - 8$

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 :)