SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (29 page)

The SAT will never ask you to graph a linear function. It will only ask you to use graphs to figure out other information, or to identify an answer choice that correctly graphs a function.

A quadratic function is a function where the
x
variable has an exponent of 2.

Example:

y
=
x
2
is a quadratic function.

Quadratic functions are NEVER linear.

The SAT never asks you to draw the graph of a quadratic function. It will only ask you to use given graphs to answer questions, or to identify which answer choice correctly graphs a given function.

Quadratic functions always extend infinitely in some direction (up or down).

Example:

The graph of
y
=
x
2
extends “up” infinitely, and looks like this:

The graph of
y
= - (
x
2
) extends “down” infinitely, and looks like this:

Note that the “direction” of the graph of a quadratic equation is really just a question of its range. When the range extends to negative infinity, the graph “opens down.”  When the range extends to positive infinity, the graph “opens up.”

When a quadratic function
“opens down,” its highest point is the (
x, y
) pair that has the greatest
y
value.

When a quadratic function
“opens up,” its lowest point is the (
x, y
) pair that has the lowest
y
value.

Sometimes you’ll be asked to find the “zeros” of a quadratic function. The zeros are the points where the graph of the function
touches the
x
-axis. To find the zeros, just set
f
(
x
) equal to zero, and then solve the resulting equation by factoring, just like we did above.

Example:

To find the zeros of
f
(
x
) = (
x
2
)/3 – 3, we set
f
(
x
) equal to zero and then solve for
x
by factoring:

0 = (
x
2
)/3 – 3

3 = (
x
2
)/3

9 =
x
2

x
= 3  
or  x
= -3

So the zeros of
f
(
x
) = (
x
2
)/3 – 3 are 3 and -3.

A unique line can be drawn to connect any
two points.

Between any two p
oints on a line, there is a midpoint that is halfway between the two points.

Any three
or more points may or may not fall on the same line. If they do, we say the points are collinear.

Degrees are the units that we use to measure how “wide” or “big” an angle is.

Example:

This is a 45-degree angle:

This is a 90-degree angle, also called a “right angle:”

This is a 180-degree angle, which is the same thing as a straight line:

Sometimes angles have special relationships. The two types of special relationships that the SAT cares about the most are vertical angles and supplementary angles

Vertical angles are the pairs of angles that lie across from each other when two lines intersect. In a pair of vertical angles, the two angles have the same degree measurements as each other.

Example:

Angles
ABC and
DBE are a pair of vertical angles, so they have the same degree measurement. Angles
ABD and 
CBE are also a pair of vertical angles, so they have the same measurements as each other as well.

Supplementary angles are pairs of angles whose measurements add up to 180 degrees. When supplementary angles are next to each other, they form a straight line.

Example:

ABC
and
ABD are a pair of supplementary angles, because their measurements together add up to 180 degrees—together, they form the straight line CD.

The SAT loves to ask about
triangles.

The sum of the measures of the angles in any triangle is 180 degrees, the same as it is in a straight line.

In any triangle, the longest side is always opposite the biggest angle, and the shortest side is always opposite the smallest angle.

In an “equilateral” triangle, all the sides are the same length.

In an equilateral triangle, all the angles measure 60 degrees each.