The Cartesian Plane |
Contents: This page corresponds to § P.5 (p. 49) of the text.
Suggested Problems From Text:
Page 55 #1, 3, 5, 7, 11, 25, 37, 38, 39, 41, 47, 51, 53, 59.
Coordinates in the Plane
Distance Formula
Midpoint Formula
Scatter Plots
The Cartesian plane, named after the mathematician Rene Descartes (1596 - 1650), is a plane with a rectangular coordinate system that associates each point in the plane with a pair of numbers. The basic definitions and terminology are covered in section P.5 ( p.49) of the text.
The location of a point P is determined by an ordered pair of numbers (a,b).
The Java applet (program) below shows a coordinate plane and the point (-2, 1). As you drag the point the coordinates are reported in the text box below the graph. (When we say "drag the point" we mean click the mouse button on the point, then move the mouse with the button held down.) If you click and drag at a spot away from the indicated point, the visible portion of the graph changes. You may also edit the coordinates in the text box, then press enter to show the position of the point. Note that the coordinates need to be entered as number,number. (Some browsers return an error if there is a space before or after the comma.)
(a) Draw a set of coordinate axes and plot the points (-2,3), (4,5), (3,-4), and (-1,-3). Use the toolbook program Plotter to check your work.
(b) If a point is 3 units left of the y - axis and 4 units above the x - axis, then what are its coordinates?
The distance from point (x1 , y1) to point (x2 , y2) is given by
If A and B are points, then d(A,B) stands for the distance from A to B.
Note on technology:
The distance formula above is actually an image file (picture) that is being displayed in this page. Currently, that is the most practical way to put mathematical formulas into Web pages, and it is rather cumbersome. If there are many formulas to display, then your browser must download many image files, and this adds significantly to the time you have to wait to see the results. We will insert formulas with image files when necessary, but we will also use text names for some of the mathematical symbols.
For example, sqrt(4) stands for the square root of 4, and x^2 stands for x squared. Using these conventions, the distance formula above would become
d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
These are the same symbols that are used in the Java Calculator that can be called from these course pages. There will be a link to the Calculator at the top of all content pages for this course.
Clicking the Java Calculator link opens a new browser window that is just large enough to hold the calculator, so the current page is still visible. If you click the current page while the calculator is open, then the calculator window goes behind the current window, but it can be recovered by clicking the Calculator button on the task bar in a Windows environment. More help on the calculator is available by clicking the Calculator Help link in the box with the calculator links. Try using the calculator for the next exercise.
(a) Verify that the distance from (-3,5) to (2,1) is 6.403 (rounded to three decimal places). The calculator formula is
sqrt((2 - (-3))^2 + (1 - 5)^2).
Calculations such as this one are done in the box at the top of the calculator. After typing the formula press the enter key. (In most cases, after you press enter, the result is displayed and the cursor moves to the next line. Then you are ready to do another calculation. With some versions of Internet Explorer the cursor stays where it was when you pressed enter. In this case you have to move the cursor down to the new line after the result in order to do the next calculation. You can use the down arrow key for this.)
The calculator keeps a record of the last result in a variable called "a". Here is an example of the use of the "a" variable.
Type 6 + 7 and then press enter. The result 13 is displayed. If you now type a^2, the result 169 will be displayed, because after the first calculation, a was set equal to 13. The value of a can be checked by typing a and enter.
(b) Verify that the distance from (4,3) to (-1,-2) is (approximately) 7.071.
The midpoint of the line segment connecting the point (x1 , y1) to point (x2 , y2) is
Midpoint = ((x1+x2)/2 , (y1+y2)/2).
For example, the midpoint of the segment connecting (-1,2) and (3,9) is ((-1+3)/2 , (2+9)/2) = (1, 5.5).
(a) M = (1, -2) is the midpoint of the segment from A = (-2, -3) to B = (4, -1). As a result, d(A, M) = d(M, B). Verify that the two distances d(A,M) and d(M,B) are equal.
(b) Find the midpoint of the segment from (-5,3) to (2,-1).
(c) Let A = (3,2) and M = (0,4). If M is the midpoint of the segment from A to a point B, then what are coordinates of B? Hint: Plotting the points will probably help.
The most important reason for plotting points is to study relationships between variables. Consider, for example, the fund raising history of a small public radio station. During its first year of operation listener contributions amounted to $70,000. The table below gives contribution amounts for several years.
|
Year |
1 |
3 |
5 |
8 |
11 |
14 |
|
Contributions |
$70,000 |
$145,000 |
$190,000 |
$230,000 |
$270,000 |
$290,000 |
In order to have a useful plot of this data we will need to have different scales on the two axes. The first coordinate of each of the six points will be the year, and we will measure contributions in $10,000 increments, so the first point will be (1,7), the second (3,14.5), etc.
Scatter Plot of Contribution Data
From the scatter plot one sees easily that while contributions are rising, they increased more rapidly in the early years. This visual representation of the data makes trends and patterns more obvious than they are from the data table.
Make a scatter plot of the data in the following table.
3
7
9
11
8
4
12
2
6
1
10
14
12
7
13
12
4
7
14
3
There is a Java tool called Scatter Plot which may be used to make scatter plots. To enter a point you type its coordinates into the text box labeled Coordinates and press the enter key. The format for entering coordinates is number,number. For example, to enter the first point from the table above you would type 3,10 and then press enter. If your entry is in the correct form, then the text box clears and is ready to accept the next point. The pull down list at the top of the editor shows which point is being edited. After entering a point, you can change its coordinates by selecting that point from the pull down list. Once you have entered all the points, click the Plot button and the scatter plot will appear in the graph window. Once you have the scatter plot has been displayed, if you change a point or add a new point, then you must click Plot again to update the graph.
