## Algebra: Patterns, Functions, and the Coordinate Plane

**Coordinate Plane**

A **coordinate plane** is composed of a horizontal number line (the x-axis) and a vertical number line (the y-axis). The point where the two axes intersect is the **origin**. A location or point on the coordinate plane is identified by an **ordered pair**, such as (3, 2), which names the coordinates of that location or point. The first number, or the x-coordinate, tells how far to the right or left the point is located in the horizontal direction. The second number, or the y-coordinate, tells how far up or down the point is located in the vertical direction. Therefore, the point (3, 2) is not the same as point (2, 3). The coordinates of the origin are (0, 0).

The coordinate axes divide the plane into four regions called **quadrants**. The quadrants are numbered I, II, III, and IV as shown. All points in Quadrant I have two positive coordinates. All points in Quadrant II have a negative x-coordinate and a positive y-coordinate. All points in Quadrant III have two negative coordinates. All points in Quadrant IV have a positive x-coordinate and a negative y-coordinate.

The coordinates of the endpoints of a line segment can be used to find the length of a line segment on the coordinate plane. At this grade level, only the length of horizontal or vertical line segments are found. The length of a horizontal line segment AB with endpoints A(x_{1}, y_{1}) and B(x_{2}, y_{1}) is given by AB = | x_{1} − x_{2}|

The length of a vertical line segment CD with endpoints C(x_{1}, y_{1}) and D(x_{1}, y_{2}) is given by CD = |y_{1} − y_{2}|

Since the absolute value is used, the order used to subtract the coordinates does not change the result.

**Functions**

A function is a rule that associates one and only one value of one variable with each value of another variable. The function y = 2x expresses y in terms of x. For each value of x, there is one and only one value of y. An equation determines y as a function of x, if, for each x, the equation can be solved to give exactly one value of y. For example, the equations 3x + 2 = 14, x − y = 7, and y = x + 3 determine y as a function of x. The equation y = x^{2} determines y as a function of x, but the equation x = y^{2} does not determine y as a function of x because for some values of x, such as 4, the two values y = ^{–}2 and y = ^{+}2 both satisfy the equation.

In general, the graph of an equation that can be written Ax + By = C, where A, B, and C are fixed numbers, is a straight line. Each point on the line is a solution to the equation. To graph a linear equation, first make a table of values for x and y. Then graph the ordered pairs. Finally, draw a line through the points.

Example: Graph the equation: y = x + 3.

x |
y |

^{–}2 |
+1 |

^{–}1 |
+2 |

0 | +3 |

+1 | +4 |

+2 | +5 |

These values can be interpreted as the x-coordinate and y-coordinates of points in the coordinate plane. Graphing these points means placing a dot at the points (

^{–}2, 1), (

^{–}1, 2), (0, 3), (1, 4), and (2, 5). To graph the equation, connect the points with a straight line.

**More About Integers**

In preparation for solving linear equations that involve integers, students need to realize that the order of operations applies to integers in the same manner as it applies to whole numbers, decimals, and fractions. The same is true for evaluating and simplifying expressions involving integers.

Solving equations involving integers is completed in the same manner as for whole numbers, decimals, and fractions. Point out to students that by writing each step involved in solving such an equation, many errors can be avoided.

**Teaching Model 13.2:** Sequences and Patterns