## Coordinate Geometry and Passport to Advanced Math

Coordinate geometry, or more colloquially graphing, is another area which is filed under Passport to Advanced math, as long (and here’s the important part) the equation is a polynomial. Most of the times, this will mean a parabola. Sometimes, though, you’ll get a graph of some monstrous polynomial like .

The goods news is you’ll probably only have to decipher the graph to figure out how many times it crosses through the x-axis or something else relatively straightforward.

It’s really the parabola that is going to show up more often. What you need to know is that parabolas are symmetrical, meaning that each side occupies the same area on both sides of either y-axis or x-axis.

The equation of a parabola can be defined as , where a, b, and c are constants (meaning they are some fixed number). Since a will often equal 1, it helps knowing good-old fashioned FOIL, as the following question shows.

## SAT Coordinate Geometry Practice Question 1

What is the sum of x-intercepts of the equation in the xy-plane?

Explanation:
Really all we are doing here is old-fashioned FOIL so that can find out the solutions for x. Those solutions are the same as an x-intercept, since when you plug either value for x back into the equation, f(x), or the y-coordinate, will equal 0.

The sum equals 6.

Sometimes, the test might ask you to find something that requires a little more knowledge of parabolas. One useful form is , where a, h, and k are constants and (h,k) is the vertex of the parabola.

Since a polynomial usually isn’t in that form, you’ll often have to get it there by “completing the square” as we’ll see in the next problem.

## SAT Coordinate Geometry Practice Question 2

Which of the following is an equivalent form of the equation in the xy-plane, from which the coordinates of the vertex V can be identified as constants in the equation?

A)
B)
C)
D)

Explanation:
It helps to know the following equation for a parabola:

Getting the equation into the form above will help us determine the vertex. First, we need to complete the square, because does not lend itself to the format.

Notice how the two ‘1’ cancel each other, thereby leaving us with the original equation. Why did we even put the ‘1s’ in the first place? Well becomes . This is called completing the square, which I did by dividing the quantity that has the ‘x’ in it (in this case 2x) by ‘2’ and squaring it. Whatever number results (in this case positive ‘1’), I take the negative value of it and stick it at the end of the equation (in this case the ‘-1’). So now I have:

Therefore, h is equal to 1 and k is equal to -25. So the vertex (1, -25). This question did have us write the vertex out, but asked for the way we could best identify it. That’s the equation directly above.

## Recap of Passport to Advanced Math

Passport to Advanced Math only includes 16 of the 58 questions spread out over the two math sections. However, if you’re already comfortable with the other math (which many are), you should spend more time in this area. The reason is, you are likely to get very flustered by this question type on test day and this can affect your performance on easier question types.

To get a sense of all the different type of concepts that pop up in the Passport to Advanced Math section, check out the Official Study Guide. Take the practice tests to see the 16 questions per test. If you miss a question because of conceptual misunderstanding, you’ll want to go back to pages 263-276.