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Lucas Fink

Quadratic Functions Shortcuts on the SAT

Most function questions on the SAT don’t give you a graph. In fact, a lot of SAT functions are basically just equation questions. And even if they do include a figure, the functions aren’t always quadratic, making parabolas (horse-shoe shapes).

So if you hate graphing parabolas, or you haven’t really studies it, then it’s not the end of the world. It won’t come up very often. But there’ll probably be one question on your SAT that asks you to match a graph to an equation (or vice versa).

To answer it, all you need is a couple of shortcuts.


What do quadratic functions look like?

The basic form of a quadratic function is this:


Any time you see an equation that looks like that, you can graph it as a parabola.

y=5x^2+2x+3 would be quadratic, for example. a=5, b=2, and c=3.

How do you graph it? Since the SAT doesn’t ask you to actually draw any graphs, we won’t focus on that. You may be able to find the answer to the question without even needing to know how.


Shortcut#1: a shows the direction of the parabola

The value of a will show you whether the parabola it represents is positive or negative. So if  a (the number multiplying x^2) is negative, the curve will open downward.


And the other way around is also true; if a is positive, the curve will open upward.




Shortcut #2: c shows the y-intercept

Let’s look at that quadratic function form again.


If x=0, then (fx)=c, since we can take out both of those x terms. That’s the y-intercept, where the function crosses the y-axis. In the two graphs above, the y-intercept is the same: 1. That means any equation for either of those graphs would have to include a +1 at the end of it.

On the other hand, if the graph looked like this…


…then c would have to be -1. That’s where the curve crosses the y-axis.


Shortcut #3: Use a graphing calculator!

Based on those two numbers, a and c, you can usually eliminate a couple of equations or graphs from the answer choices pretty quickly.  Then, just put in whatever info you can into your calculator.

If the question is asking about some general trend and doesn’t include specific values for a,b, and/or c, then check if you can plug some in to see the pattern.

If you don’t have a graphing calculator, then just take it one point at a time. If x is 1, what is y? If x is 2? And so on.


To put it short

Always check if the values of a and c narrow down your answer choices. That’s the quickest way to make progress when matching parabolas to their equations.


About Lucas Fink

Lucas is the teacher behind Magoosh TOEFL. He’s been teaching TOEFL preparation and more general English since 2009, and the SAT since 2008. Between his time at Bard College and teaching abroad, he has studied Japanese, Czech, and Korean. None of them come in handy, nowadays.

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