Graph the following parabola.

We need a few points to graph this dude.

The vertex and y- and x-intercepts are all relatively easy to find, so let's go with them.

The vertex is at (3, 1).

The y-intercept is at x = 0, so plug that in.

So, the y-intercept is (0, 7).

To find the x-intercepts, if they exist, we need to multiply everything out.

Just like Bigfoot or Australia, the x-intercepts may not actually exist. Check the discriminant to see if they exist or not.

The discriminant is negative, so Australia isn't real. And neither are the x-intercepts for this function.

We do need a few more points, though. We're going to show x = 1 and x = 4, but whatever you choose should work out the same.

Collecting all of the points we've found into a table gives us the following.

x	y
3	1
0	7
1
4
6	7
5
2

The last three points come from the function's symmetry, centered on the vertex. For instance, x = 6 is 3 away from the vertex, and so it shares a y-value with (0, 7).

Graph the following parabola.

y = -3x² + 1

Try to find the easy points first, to minimize how many calculations you have to do.

Notice that our function is equivalent to y = -3(x – 0)² + 1. This means the vertex is at (0, 1).

We don't need to solve for the y-intercept, because it happens to be our vertex.

Looking at the discriminant, we see that 0² – 4(-3)(1) = 12. We have two real number roots, which means the function crosses the x-axis twice.

0 = -3x² + 1

3x² = 1

 and 

Our x-intercepts are  and . This are equal to about (0.577, 0) and (-0.577, 0).

We now have enough points to graph the function.

Graph the following function.

f(x) = 2(x + 2)² – 2

Start off by finding the vertex and the y-intercept.

The vertex is (-2, -2).

f(0) = 2(0 + 2)² – 2 = 2(4) – 2 = 6

The y-intercept is (0, 6).

Expanding the function, we find:

0 = 2(x² + 4x + 4) – 2

0 = x² + 4x + 4 – 1

0 = x² + 4x + 3

0 = (x + 3)(x + 1)

We would check the discriminant, but the function is so easily factored that we don't need to bother. That's pretty handy. The x-intercepts are (-3, 0) and (-1, 0).

And now, we graph.