Math Study Guide for the SAT Exam

The Coordinate Plane

Algebra has close ties to graphing, so you need to be familiar with the coordinate plane. It provides a two-dimensional space with which to graphically represent points, lines, and specific types of equations known as functions. This plane is also known as the \(xy\)-axis.

The Axes and Quadrants

The horizontal axis is the \(x\)-axis, and the vertical axis is the \(y\)-axis. The origin is the intersection of those two axes and is located at point \((0,0)\). This intersection creates four quadrants:

These four quadrants begin with the top right as quadrant I. The quadrants increase in number in a counterclockwise fashion. The quadrant to the left of quadrant I is designated quadrant II. The quadrant below quadrant II is quadrant III, and the remaining quadrant is quadrant IV.

• Quadrant I contains positive \(x\) and positive \(y\) values.
• Quadrant II contains negative \(x\) and positive \(y\) values.
• Quadrant III contains negative \(x\) and negative \(y\) values.
• Quadrant IV contains positive \(x\) and negative \(y\) values.

Variables on the Coordinate Plane

The coordinate plane is most useful for showing the relationship between two variables, known as a relation. The \(x\) variable is commonly referred to as the independent variable, meaning the value of the variable is chosen. The \(y\) variable is commonly referred to as the dependent variable, meaning its value is determined by the function containing the independent variable.

Points, given in the form \((x,y)\), are commonly graphed on the coordinate plane. The point \((-3,2)\), for example, would be located three units to the left of the origin \((0,0)\) and two units above the \(x\)-axis.

Here are several points graphed on the coordinate plane:

Graph Concepts: Slope and Intercept

When we graph a linear function (discussed in greater detail below), it creates a straight line. The slope of that line is the ratio of the change in \(y\) values to the change in \(x\) values:

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

where \((x_1, y_1)\) and \((x_2, y_2)\) are points on the line.

It is helpful to think of the slope of a line as the rise over the run. It is also important to understand that a line is straight because the change in the \(y\) values is in proportion with the change in the \(x\) values. If this were not so, the graph would exhibit a different characteristic.

Here are lines with varying slopes:

Aside from the slope, graphs of lines will also exhibit \(x\)- or \(y\)-intercepts. These are points at which lines cross the \(x\)- or \(y\)-axis. These points are important because they show the \(x\) value of the function when \(y\) is equal to \(0\) and the \(y\) value of the function when \(x\) is equal to \(0\).

This line shown below, which can be expressed as the function \(y = \frac{5}{3}x + 5\), crosses the \(x\)-axis at \(-3\) and the \(y\)-axis at \(5\):