Math Background

Functions and Graphing: Overview

Graphing on the coordinate plane is a way to visualize relationships. A coordinate plane has 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. (See Graphing Integers.)

coordinate plane

All x-values to the right of the origin are positive and all x-values to the left of the origin are negative. The x-value of all points on the y-axis is zero. Similarly, y-values above the origin are positive, and y-values below the origin are negative. The y-value of all points on the x-axis is zero. The axes split the plane into four quadrants, which are numbered starting in the upper right-hand quadrant and going counterclockwise as I, II, III, and IV.

Every point on the plane has a unique ordered pair that corresponds to it. An ordered pair is enclosed in a pair of parentheses. The first number represents the x-coordinate and the second number represents the y-coordinate. The origin is the ordered pair (0, 0). To locate a point on the coordinate plane, do the following.

  • Start at the origin.
  • The first number is the x-coordinate. Move to the right the appropriate number of units if the number is positive; move to the left the appropriate number of units if it is negative.
  • The second number is the y-coordinate. Move up the appropriate number of units if the number is positive and down the appropriate number of units if it is negative.

To locate point A, whose ordered pair is (3, -2) on the coordinate plane, we move three units to the right and down two units. On the coordinate plane below are the points B (-3, 5), C (-1, -4), D (0, -3), E (2, 4), and F (4, 0).

coordinate plane

In order to find the area of common geometric figures on a coordinate graph, you will need to find the lengths of line segments—that is, the distance between points. To find the length of a segment parallel to the x-axis, subtract the x-coordinates. For example, to find the length of the line segment between the points (4, 7) and (-2, 7), subtract: 4 − (-2) = 6. If you wanted to find the length of a line segment parallel to the y-axis, you would subtract the two y-coordinates. For example, the distance between (-3, 5) and (-3, 9) is 9 − 5 = 4.

To find the area of the rectangle below, students need to find the coordinates of the points and apply the formula for finding the area of a rectangle (A = lw). In the diagram below, the points are located at (-2, 1), (3, 1), (3, 4), and (-2, 4). The length of the rectangle is 3 − (-2), or 5, and the width is 4 − 1, or 3. Thus, the area of the rectangle is 5 x 3, or 15, square units. Encourage students to check their answers by counting the number of squares in the figure.

coordinate plane

In a similar manner, students can find the area of the triangle by finding the base and the height and using the formula for the area of a triangle (A = one-halfbh). Points are located at (2, -4), (6, -4), and (6, 2). The base is 6 − 2, or 4, and the height is 2 − (-4), or 6. The area of triangle PQR is one-half (4)(6), or 12 square units.

There are many ways to record and communicate information: in words, in a table, in a formula, or in a graph. A graph is a visual representation of information and can be used to get quick overviews and estimates. For example, a graph can be used to show the relationship between measurements in different measurement systems. The relationship between centimeters and inches is graphed below. If you want to find out approximately how many inches are equivalent to 20 centimeters, find 20 centimeters on the x-axis. Draw a vertical line from this point to the graph. From this point, draw a horizontal line across to the y-axis. The line crosses the y-axis at about 8 inches. Similarly, if you wanted to find out approximately how many centimeters are in 15 inches, find 15 inches on the y-axis. Read across horizontally to the graph, and then read down to the x-axis. There are about 38 centimeters in 15 inches.

graph

A function is a rule that associates one and only one value of one variable with each value of another variable; each element in the first set corresponds to one and only one element in the second set. In the equation y = 2x + 3, y is expressed in terms of x; y is a function of x. For each value of x, there is one and only one value of y. In order to graph this function, you can set up a function table like the one below. By substituting values for x in the equation, you can find the values for y. For example, if x = 4, then y = 2(4) + 3 = 11.

function table

Since y is a function of x, we graph this relationship with the x-values along the horizontal axis and the y-values along the vertical axis. By plotting the points above and connecting them with a straight line, we get the graph below.

graph

Many mathematical expressions involve more than one operation. We need agreement about the order of operations and the use of grouping symbols when simplifying and evaluating expressions. (See Writing and Solving One-Step Linear Equations in One Variable.) For example, to simplify the expression 3(-5 + 2) − (8 + 6) ÷ 2 + (-2 + 7)², we follow rules in this order:

  1. Simplify the expressions in the parentheses. 3 x (-3) − 14 ÷ 2 + (-5)²
  2. Evaluate powers.
    3 x (-3) − 14 ÷ 2 + 25
  3. Multiply and divide in order from left to right.-9 − 7 + 25
  4. Add and subtract in order from left to right.-16 + 25 = 9

Houghton Mifflin Math Grade 6