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.
Quadratic functions
A quadratic function is a function where the x variable has an exponent of 2.
Example:
y = x2 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 = x2 extends “up” infinitely, and looks like this:
The graph of y = - (x2) 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) = (x2)/3 – 3, we set f(x) equal to zero and then solve for x by factoring:
0 = (x2)/3 – 3
3 = (x2)/3
9 = x2
x = 3 or x = -3
So the zeros of f(x) = (x2)/3 – 3 are 3 and -3.
Points and lines
A unique line can be drawn to connect any two points.
Between any two points 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.
Angles in the plane
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.
Triangles
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.
Example:
In the equilateral triangle EQI below, all the sides are of equal length, and all the angles are 60 degrees.
In an “isosceles” triangle, two of the three sides are the same length as each other, and two of the three angles are the same size as each other.
Example:
In the isosceles triangle ISO below, side IS is the same length as side SO. Also, SIO and SOI have the same degree measurement as each other.
A “right” triangle is a triangle that includes a ninety-degree angle as one of its three angles.
A special relationship exists between the measurements of the sides of a right triangle: If you take the lengths of the two shorter sides and square them, and then add those two squares together, the resulting amount is the square of the length of the longest side.
Example:
In the right triangle below, a2+ b2 = c2
The expression of this relationship, a2 + b2 = c2, is called the “Pythagorean Theorem.”
A “Pythagorean triple” is a set of three numbers that can all be the lengths of the sides of the same right triangle. Memorizing four of these sets will make your life easier on the SAT.