Up Your Score (46 page)

In the circle above, only the line
p
is a tangent line.

Example 1: A circle of the equation
x
²
+
y
²
= 25 has a tangent line at the point (4,3). What is the equation of the tangent line?

First of all, we have to visualize this circle. Because there’s nothing after the
x
’s and the
y
’s, the center of the circle is on the origin (0,0). The radius is the square root of 25, and so it’s 5.

The tangent line is perpendicular to the radius, so we have to figure out what the equation for the radius is at this point. We know the center is (0,0), and the point is (4,3), so:

So now we know that the slope of the radius is ¾. We’re trying to get the tangent line, so we need the perpendicular slope, which is –
.

Now all we have to do is plug everything into an equation:

And we have our tangent line.

Note that this also could have been written as

Trigonometry

The College Board says there are problems on the SAT that can be solved using trigonometry. Unless you’re already comfortable with it, though, we suggest you just memorize the properties of the similar triangles (see
page 213
).

Basic Transformations

A transformation of a graph of a function means moving it. Here are a few basic transformations that you might need to know:

Changing the basic function
y
=
x
³

Some handy dandy example graphs:

JaJa says: If you forget some transformations, do some trial-and-error on your graphing calculator, but only if you have time.

As if the SAT didn’t already have enough ridiculous things on it, the Serpent came up with the funny symbol question. Here’s how funny symbol questions work.

The Serpent gives you some symbol, like this,

and tells you what it does. You’re supposed to apply it. Easy. Don’t freak out and say, “Oh, no! We never went over
in
my math class, I guess I can’t do this one.” The fact is that no one has ever seen that symbol in math class. The Serpent just dreamed it up. (Yes, he loses sleep at night thinking up funny symbols and how to make them exceptionally cruel.)

And he will tell you exactly what the symbol means.

The Evil Testing Serpent defines the symbol, usually using
x
and
y
(or
a
and
b
or whatever) in terms of arithmetic commands ( +, −, ×, ÷), which you know how to use. You take the numbers given to you in the question and do the arithmetic the funny symbol represents. Be careful that you do it in the
same order
as in the definition of the funny symbol.

Okay, here are some examples:

1. If
x
y
=
x
²
+ 2
y
then what is 3
4?