We all know that locations such as mile markers on the highway or places on maps or street addresses may be specified by numbers. We locate points on a number line by assigning a number to each point on the line. One point is arbitrarily labeled with a 0 and another point with a 1. All other points are labeled accordingly as shown on the line below, using equal spacing between every pair of consecutive whole numbers:
Generally, on a horizontal number line, the positive numbers are to the right of zero and the negative numbers are to the left of zero.
On a vertical number line, the positive numbers are above zero and the negative numbers are below zero as shown above.
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Positions in two dimensions, such as points on a computer screen or locations on a map require two intersecting number lines, one to show the horizontal location (left and right) and the other to show the vertical location (up and down) as shown below:
Often on a map one of the number lines will be labeled with letters rather than numbers and the other number line will be labeled with positive numbers. Some examples of point location on such a map are shown in the next four examples:
Example 1. Write the letter A at location E3 on the coordinate system which follows:
Answer:
See the coordinate system below.
Example 2. Write the letter B at location C9 on the following coordinate system:
Answer:
See the coordinate system below.
Example 3. What is the location of the triangle?
Answer:
L6
Example 4. What is the location of the circle?
Answer:
C11
The two intersecting number lines are called the coordinate axes, and the two numbers used to locate a point are called the coordinates of the point. The first coordinate gives the horizontal location of the point, while the second number gives the vertical location of the point. The two axes cross at the point (0,0), which is called the origin. Such a system for labeling is called a Cartesian coordinate system. Some examples of this system follow:
Example 5. Place the letter A on the point (-4,2).
Answer:
See picture below.
Example 6. Place the letter B on the point (0,3).
Answer:
See picture below.
Example 7. Place the letter C on the point (-1,-2).
Answer:
See picture below.
Example 8. Place the letter D on the point (3,4).
Answer:
See picture below.
Example 9. Place the letter E on the point (2,-1).
Answer:
See picture below.
Example 10. Place the letter F on the point (-2,0).
Answer:
See picture below.
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Quadrants are sometimes used to show the general location of points. As shown in the figure below, the upper right hand region is referred to as Quadrant I or the first quadrant, the upper left hand region is referred to as Quadrant II or the second quadrant, and so forth.
Example 11. Using the figure above, determine in what quadrant the point (-5,3) is located?
Answer:
II (2nd quadrant)
Example 12. In what quadrant is the point (5,-4) located?
Answer:
IV (fourth quadrant)
Example 13. In what quadrant is the point (0,3) located?
Answer:
None. Points on the axes are not in any quadrant. They are between quadrants.
Example 14. In what quadrant is the point (-7,0) located?
Answer:
None. See the previous example.
Each of the coordinate axes may be named by a letter. The most commonly used letter for the horizontal axis is the letter X, and the most commonly used letter for the vertical axis is the letter Y. The axes are then drawn as shown on the following coordinate system:
We now call the horizontal axis the x-axis and the vertical axis the y-axis. However, other letters are used in various applications. Once the axes are labeled as shown, the first coordinate of a point is called the x-coordinate and the second coordinate is called the y-coordinate. Using this system for the point (7,2), the x-coordinate is 7 and the y-coordinate is 2.
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Using a coordinate system with the horizontal axis labeled with the letter x and the vertical axis labeled with the letter y allows graphing of equations involving the letters x and y. We sometimes call the x-coordinate the input for the equation and the y-coordinate the output of the equation. To graph an equation means to graph all points (x,y) for which the equation is true.
Example 15. For the equation y = 5 - 2x, find y when x is 2.
Answer:
Replacing x with the number 2 gives the result y = 5 - 2(2) or y = 1. In other words, the point (2,1) is on the graph of the equation y = 5 - 2x.
Example 16. For the equation 5x - 2y = 40, find y when x = 10.
Answer:
Replacing x with 10 gives 5(10) - 2y = 40. Solving for y gives 50 - 40 = 2y or y = 5. In other words, the point (10,5) is on the graph of the equation 5x - 2y = 40.
By replacing the x value and the y value of an ordered pair in the equation, we can determine if the point satisfies the equation. To satisfy an equation is to find a solution of the equation. If the x and y values of the point are a solution to the equation, then the point is on the graph of the equation.
Example 17. For 3x + 2y = 12, determine if the point (2,3) satisfies the equation.
Answer:
Replacing x with 2 and y with 3 gives 3(2) + 2(3) = 12. Simplifying gives 6 + 6 =12. Since 12 = 12 is true, the point (2,3) satisfies the equation 3x + 2y = 12, and it is on the graph of the equation.
Example 18. For 3x + 2y = 12, determine if the point (1,-3) satisfies the equation.
Answer:
Replacing x with 1 and y with -3 gives 3(1) + 2(-3) = 12. Simplifying gives 3 - 6 = 12. Since -3 ≠ 12, the point (1,-3) does not satisfy the equation. Therefore, the point (1,-3) is not on the graph of the equation.
Example 19. For 
, determine if the point (5,10) satisfies the equation.
Answer:
Replacing x with 5 and y with 10 gives 
. Simplifying gives 
, 
, 
. Since 5 = 5 is true, the point (5,10) satisfies the equation.
The way to graph an equation is
a) select some x values
b) use these x values and the equation to find the y values
c) plot the points you have found
d) draw a line through the plotted points
It may be more convenient in some cases to select some y values first.
For this course, all of the equations we graph will be straight lines. However, not all graphs are straight lines. For instance, the graph of y = x2 is U shaped and is called a parabola.
Two convenient points to use for graphing a line are the intercepts of the line. The y-intercept is found by replacing x with 0 and finding y. The x-intercept is found by replacing y with 0 and finding x. The y-intercept is the point on the y-axis where the line crosses, and the x-intercept is the point on the x-axis where the line crosses.
Example 20. Find the intercepts of the line 3x + 8y = 48.
Answer:
Replacing x with 0 yields 3(0) + 8y = 48. So y = 6 and the y-intercept is the point (0,6). Replacing y with 0 yields 3x + 8(0) = 48. So x = 16 and the x-intercept is the point (16,0).
Example 21. Graph the equation y = 3x - 4 on the axes below.
Answer:
Replacing x with 0 in the equation gives y = -4. Replacing x with 2 gives y = 2. Replacing x with 3 gives y = 5. Thus, we must graph the corresponding points (0,-4), (2,2), and (3,5) and then draw a line through them. The x values must be selected intelligently so that the resulting points are not located out of the region covered by your coordinate axes. The three points and the line through them are shown below:
Example 22. Graph the equation y = x.
Answer:
This means to graph every point whose y-coordinate has the same value as its x-coordinate. Such a graph would consist of points such as (-3,-3) , (0,0), (1,1), (4,4) and all other points where the y-coordinate is equal to the x-coordinate. The graph is shown below:
Example 23. Graph the equation y = 2x - 1.
Answer:
This means to graph those points whose y-coordinate is 1 less than twice the x-coordinate. A few points on this graph would be (0,-1), (3,5), (100,199), and so on. The graph is shown below:
Example 24. Graph the equation 
.
.
Answer:
This means to graph those points whose y-coordinate is 1 more than two-thirds the x-coordinate. A good idea would be to plot the point where x = 0 and y = 1 and then points whose x values are multiples of three. For example, when x = 3, y = 3; when x = -3, y = -1; when x = 6, y = 5. Plot the points (0,1), (3,3), (-3,-1), and (6,5). The graph is shown below:
Example 25. Graph the equation 
.
Answer:
When x = 0, y = 1. Therefore, the y-intercept is the point (0,1). When y = 0, x = 4. Therefore, the x-intercept is (4,0), The graph is shown below:
Example 26. Graph the equation 
. This equation means that y can be any value, but x is always 4. Therefore, a few points on this line are (4,0), (4,-2), (4,2), and (4,5).
Example 27. Graph the equation 
. This equation means that x can be any value, but y is always 4. Therefore, a few points on this line are (0,4), (-2,4), (2,4), and (5,4).
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The coordinates of a point may represent something other than a location in the two dimensions. For instance, the first coordinate might represent the year and the second coordinate might represent the number of units sold by a business during that year. So, the point (1992,300,000) would tell you that the business sold 300,000 units during the year 1992. Or in another case the first coordinate might be your average speed on a trip and the second coordinate might be your gasoline mileage. Here, the point (68,29) would convey the information that when you averaged 68 mph on a trip, your gasoline mileage was 29 mpg.
Sometimes in cases like this, one or both of the axes are drawn with only positive numbers on them since none of the quantities considered could be negative. In other words, the graph will often lie entirely in the first quadrant.
Example 28. Suppose your gasoline mileage is represented by y and your average speed is represented by x. If y = 35 - 0.1x gives the relationship between your gasoline mileage and your average speed for speeds of 30 miles per hour or more, find your gasoline mileage when you average 60 miles per hour.
Answer:
Replacing x with 60 yields y = 35 - 0.1(60) or y = 29 mpg. Your gasoline mileage will be 29 miles per gallon when you average 60 miles per hour.
Example 29. Graph the relationship in the previous example.
Answer:
Since we know that the equation is only true for speeds of 30 mph and above, we find the value of y when x is 30. We get y = 35 - 0.1(30) = 35 - 3 = 32. This corresponds to the point (30,32). From the previous example we have the point (60,29). Then graph the two points as shown below and connect them with a line. The points are circled on the graph.
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