A two-dimensional plane formed by the intersection of a vertical line which is called the y-axis and a horizontal line called the x-axis is known as a coordinate plane. These are a set of perpendicular lines that intersect at zero, which is known as the origin. The axes divide the coordinate plane into four sections, each of which is referred to as a quadrant.
Ordered pairs are used to graph and locate points on a coordinate plane. The ordered pairs are always in the (x, y) form where the first number is the x-coordinate and the second number is the y-coordinate. The first coordinate tells you how far to move along the x-axis, and the second coordinate tells you how far to move up or down the y-axis.
For example, to graph (4, 1), find 4 on the x-axis. Then move up 1 space. To graph (-3, 2), find -3 on the x-axis, then move up 2 spaces. To graph (3.5, -3), find 3.5 on the x-axis, then move down 3 spaces:
Ordered pairs are used to graph and locate points on a coordinate plane. The ordered pairs are always in the (x, y) form where the first number is the x-coordinate and the second number is the y-coordinate. The first coordinate tells you how far to move along the x-axis, and the second coordinate tells you how far to move up or down the y-axis.
For example, to graph (4, 1), find 4 on the x-axis. Then move up 1 space. To graph (-3, 2), find -3 on the x-axis, then move up 2 spaces. To graph (3.5, -3), find 3.5 on the x-axis, then move down 3 spaces:
All ratio tables follow a pattern. For a ratio of two quantities, equivalent ratios can be used to create ordered pairs of the form (first quantity, second quantity).
These ordered pairs can be plotted in a coordinate plane and a line drawn from point A to point B where A is the origin (0, 0).
Look for the x-coordinate on the x-axis as mentioned in the ordered pair. Then, if the y-coordinates are positive, move up the number of spaces equal to the y-coordinate (or move down if the y-coordinate is negative).
Example 1: Use the graph to represent the given relationship.
(a)
Table | 2 | 4 | 6 |
|---|---|---|---|
Chair | 3 | 6 | 9 |
(b)
Class | 1 | 2 | 3 |
|---|---|---|---|
Students | 10 | 20 | 30 |
Solution :
(a)
The ordered pairs (Tables, Chairs) are (2, 3), (4, 6), and (6, 9).
Make a graph of the ordered pairs. Draw a line through the points starting at (0, 0).
(b)
The ordered pairs (Class, Students) are (1, 10), (2, 20) and (3, 30).
Make a graph of the ordered pairs. Draw a line through the points starting at (0, 0).
Example 2: A man buys apples for $10 per pound.
(a) Represent the statement in a ratio relationship using a graph.
(b) How much will 3.5 pounds of apples cost?
Solution:
(a)
Create a ratio table with apples (pound) and cost (dollar).
Apple (Pound) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
Cost (Dollar) | 10 | 20 | 30 | 40 |
The ordered pairs (Apples, Cost) are (1, 10), (2, 20), (3, 30), and (4, 40). Make a graph of the ordered pairs. Draw a line through the points starting at (0, 0).
(b)
You can see from the graph that the price of 3.5 pounds is halfway between $30 and $40.
Hence, 3.5 pounds of apples cost $35.
Alternative method:
We can find the cost of apples by using a double number line.
The first number line will represent the pounds, and the other will represent the cost per pound.
Example 3: Every two seconds, a man walks 4 meters. Every three seconds, a woman walks 5 meters. In the same coordinate plane, graph each ratio relationship. Who walks faster?
Solution:
Make a ratio table for the man and the woman. Then, using the graph, determine who walks faster by plotting the ordered pairs (Time, Distance) from the table.
Man
Time (Seconds) | Distance (Meters) |
|---|---|
2 | 4 |
4 | 8 |
6 | 12 |
The ordered pairs for the man are (2, 4), (4, 8) and (6, 12)
women
The ordered pairs for the woman are (3, 5), (6, 10) and (9, 15).
Make a graph of the ordered pairs. Draw a line through the points starting at (0, 0) and label the points.
Both graphs start at the same point (0, 0). Since the man’s graph is steeper than the woman’s, the man walks faster.
Example 4: Every two seconds, a production machine produces 3 products. Every three seconds, the packaging machine packs 4 products. In the same coordinate plane, graph each ratio relationship. Which machine works faster?
Make a ratio for both the production and the packaging machine. Then, using a graph, determine which works faster by plotting the ordered pairs (time, distance) from the table.
Production Machinery
The ordered pairs for the production machinery are (2, 3), (4, 6) and (6, 9)
Packaging Machinery
Time (seconds) | Production (per unit) |
|---|---|
3 | 4 |
6 | 8 |
9 | 12 |
The ordered pairs for the packaging machinery are (3, 4), (6, 8) and (9, 12).
Make a graph of the ordered pairs. Draw a line through the points starting at (0, 0) and label the points.
Both graphs start at the same point (0, 0). Since the production graph is steeper than the packaging, the production machinery works faster.
A ratio table is a logically organized list of equivalent (equal value) ratios that aid in understanding the relationship between ratios and numbers. Rates are a type of ratio in which the two numbers being compared have different units.
Ratios assist us in comparing and determining the relationship between quantities. A ratio is the result of dividing one quantity by the other to obtain a comparison of two similar quantities.