Leonardo da Vinci totally <3’d parabolas.

The parabola is actually a hugely important mathematical concept with tons of forms, properties, and even its own history. It can open up, down, left, right, or any other direction. It can be used to graph the trajectory of my last AT&T cell phone that I threw in a lake when it dropped one too many calls. But if you’re interested in that stuff, you should go to the Wikipedia parabola article, and tattoo the word NERD on your forehead while you’re at it.

On the SAT, there’s actually not much you need to know about parabolas. So let’s keep it simple, huh?

Parabolas Are Symmetrical

This is the most important thing to remember about parabolas, because this is the key that unlocks most of the SAT’s most difficult parabola questions. The awesome thing is that you probably already knew this. The not awesome thing is that the SAT still finds ways to make you miss these questions. Let’s look at an example:

  1. The graph above represents the parabolic function f(x). If the function’s minimum is at f(–3), and f(0) = 0, which of the following is also equal to 0?
    (A) f(3)
    (B) f(–1)
    (C) f(–4)
    (D) f(–5)
    (E) f(–6)

Right. So let’s translate this into English first (if you’re having trouble with the function notation, have a look at this post for some help). What they’re saying here is that the line of symmetry for this parabola is at x = -3, and that the function goes right through the origin: f(0) = 0 means that this graph contains the point (0,0). They’re basically asking us to find the other x-intercept.

Here’s where they whole symmetry thing really comes in. If we know the line of symmetry, and we know one of the x-intercepts, it’s CAKE to find the other. Put very simply, they have to be the exact same distance from the line of symmetry as each other. Since (0,0) is a distance of 3 from the line of symmetry at x = -3, our other x-intercept has to be a distance of 3 away as well!

So we’re looking for the point (-6,0). Choice (E) is the one that does that for us: f(-6) = 0.

Can they find ways to make symmetry questions difficult? You bet. Will you be ready for them? Yarp.

The equation of a parabola:

y = ax2 + bx + c

Again, let me say that there are many more equations of parabolas. If you go on to do advanced math in college, you’ll need to learn some of them. If you’re doing three-dimensional math now maybe you know some of them. But you won’t need any of them for the SAT. Know this one (and what the coefficients signify), and you’re good to go.

In this equation:

  • a tells you whether the parabola opens up or down. If a is positive, it’s a smiley face. If a is negative, it’s a frowny face. Easy to remember, no?
  • b is pretty useless for you, as far as the SAT is concerned.
  • c is your y-intercept. If there is no c, that means your parabola has a y-intercept of 0 (which is to say, it goes through the origin).

Ready for some practice problems?

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For a little bit more practice, see this previously posted parabola problem (with explanation).