Parabolas Examples

What are the vertex, focus, directrix, and line of symmetry for 8(y – 2) = (x + 1)²?

This equation is in the conic form of a parabola. If you've already forgotten, here's what that looks like:

4p(y – k) = (x – h)²

Let's go through and tackle these, one by one. The vertex is at (h, k). Notice that both h and k are subtracted in the general equation, so we take the opposite of what we see in our equation. The vertex is (-1, 2).

Before we go any further, we're going to need p; it tells us the distance from the vertex to both the focus and the directix. The equation tells us that 4p = 8. Then it whispered "Burma shave." We're just going to ignore that.

So, p = 2. The positive sign tells us that the parabola points either up or right, depending. The value is the distance we travel to get to the focus and directrix from the vertex.

So, does it point up or right? We need to know to find everything else about the parabola. Well, our equation follows the general form of (some y stuff) = (some x stuff)². We recognize that as an old-school parabola that opens up or down. So, up it is.

The focus is going to be even farther up, then, up in the chest of the parabola. Two up, to be precise, from the vertex. That's (-1, 2 + 2) = (-1, 4). The x value doesn't change, just y.

The directrix will be below the parabola's vertex. Moving 2 down from the vertex puts us at y = 0.

The line of symmetry is perpendicular to the directrix, and it passes through the focus and vertex. You can find the slope between those two and plug them into the equation for a line, or notice that they both have -1 as their x term. The line of symmetry is of the form x = (something), where (something) = -1.

Graph the parabola (y + 2)² = -12(x – 4).

Here we are, in the conics section (aisle 5), so we may as well use the conics formula to graph this parabola.

The vertex is sitting at (4, -2). The squared term is y, so we have an equation of the type x = y². That's backwards compared to what we're used to (just like the equation itself), so we're dealing with a sideways parabola. We don't know p yet, but we can see that it's negative, so the graph opens to the left.

Now's a good time to find p, though. We already checked behind the garage and under the car, so it must be hiding elsewhere. Let's check inside -12. If we divide by 4, a p of -3 pops out.

Tag, you're it, p. Now hunt down the focus and directrix for us. The focus is going to be to the left of the vertex, because that's inside the parabola's bendy arms. Moving 3 left from the vertex gets us to (4 – 3, -2) = (1, -2). The directrix was hiding on the other side of the vertex, at x = 7. It's "x =" because the graph of the parabola would eventually touch any "y =" line.

The last thing to grab is the line of symmetry. It passes through the vertex and focus, so it must be y = -2. Now we can put everything up on the big board.

Plugging in one or two points, to be sure that they fit on your plot, wouldn't be a bad idea.

Convert y² – 2x = 3y + 7 into conic form.

To convert an equation of a parabola into conic form, we need to first get the x's and y's on separate sides. Segregation is the name of the game here, so that we can complete the square. y is being squared, so we'll put those on the right side.

-2x = -y² + 3y + 7

The -2 on x is fine for now, but the -1 on y is unacceptable. We need a = 1 to complete the square. Purge the minus sign by multiplying everything by -1.

2x = y² – 3y – 7

The next step is to add and subtract  from the right-hand side of the equation. This will lead to ???, which will lead to profit.

Ah, it led to the completed square, which leads us to profit. At least, as long as we've learned to profit from right answers. Hopefully not by selling them off, though.

The last thing to do is rearrange the constants around x.

We have to factor out the 2 from x. Be careful with factoring with the fraction—once you think you have the right answer, multiply the new fraction by 2 to see if you get  back.

Yep, factoring a fraction like this makes the denominator smaller. At least now we have the completed conic function.