Contents: This page corresponds to § 1.1 (p. 76) of the text.
Suggested Problems from Text:
Page 58 #67, 69, 71,77
Page 84 #3, 5, 7, 11, 13, 15, 19, 21, 25, 27-33, 37, 41, 63, 65, 73
Equations Graphs of Equations Using the Java Grapher Graphs of Circles
In this course we will work with several types of equations, and we will often be interested in finding solutions of equations. Consider the following example.
Equation 1:
x2 - x - 6 = 0.
A solution for Equation 1 is a number such that when we substitute that number for x, the resulting statement is true. So x = -2 is a solution for equation 1, because (-2)2-(-2)-6 = 0 is a true statement.
Exercise 1:
Are there any other solutions for Equation 1?
Now we consider a different kind of equation, one with two variables.
Equation 2:
2x + 3y = 5.
A solution for equation 2 is a pair of numbers a and b such that if we substitute x = a and y = b, then the equation becomes a true statement. One solution for equation 2 is easy to see: x = 1, y = 1.
Exercise 2:
Find three other solutions for equation 2. You can check your answers in the box below by typing numbers in the boxes and clicking the check button.
equation:
x =
y =
Note: You can type arithmetic expressions into the x and y boxes. For example, with the equation that appeared initially, entering x = 2/3 and y = (5 - 4/3)/3 will return True. Also, you can use this program to check other equations, just enter the equation you want to work with into the equation box. Be careful when you enter the equation. Unlike some calculators, you must put in the * for multiplication. If you entered the equation as 2x + 3y = 5, then the Check button would always return False.
One way to find solutions for an equation in x and y is to solve for y. In the case of equation 2 we have
2x + 3y
=
5.
3y
5 - 2x.
y
(5-2x)/3.
Now you can find solutions by substituting any number for x and then computing the corresponding value of y. For example, if you let x = 6, then you find y = -7/3.
Equation 2 has infinitely many solutions. A convenient way to visualize the set of solutions is to associate a solution x = a , y = b with the point (a,b) in the plane. We call the point (a,b) a solution point for the equation.
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The set of all solution points for an equation is called the graph of the equation.
Exercise 3:
Plot the solution points you have found for equation 2, and note that all the points lie on a line. That line is the graph of equation 2.
The most straight forward technique for graphing equations is to plot solution points until a geometric pattern emerges
Equation 3:
y = 4x - x2 - 2.
Exercise 4:
Find five points on the graph of equation 3 by completing the following table. x 0 1 2 3 4 y = 4x - x2 -2 Now plot the five solution points and sketch the graph.
Find five points on the graph of equation 3 by completing the following table.
x
0
1
2
3
4
y = 4x - x2 -2
Now plot the five solution points and sketch the graph.
The Java Grapher can be used to graph equations. To graph an equation in the variables x and y with the Grapher, the equation must be solved for y. Since equation 3 is in the right form, no additional steps will be needed to see the graph.
Open the Java Grapher and type the formula for y into the f text box. Remember that you must use a * for multiplication and ^2 for squaring, so you would type in
4*x - x^2 - 2
and then click the Graph button. (The spaces we used in the formula above are not necessary.) If a graph had already been drawn, then the viewing rectangle may be set so that the graph is not visible. Clicking the Default button in the lower right-hand corner sets the default viewing rectangle, which is -10 < x < 10 and -10 < y < 10.
If you hold the mouse button down and move it while it is in the viewing window you change the viewing rectangle. The next exercise explores this feature.
Exercise 5:
(a) Type the formula 14 + (x3 - 64x)/25 in the f function box of the Java Grapher. Of course, on the Grapher it will be 14+(x^3-64*x)/25 . Now click the Graph button. ( Note that if you leave a formula in f and enter this formula in g, then both will be graphed when both boxes to the right are checked. If you uncheck the box to the right of a function box, then the graph of that function will not be drawn.) If the default viewing rectangle is set, then your window will appear as below. Put the mouse pointer near the top of the viewing window, press the button and drag the mouse to the bottom of the viewing window. Now your window will look similar to the following picture. Now click the Get Values button in the lower left-hand corner to see the new viewing rectangle settings. If you dragged straight down, then the the Xmin will still be close to -10 and Xmax close to 10. The Ymin will be around 5 and the Ymax around 25. The Xscl (x-scale) and Yscl values determine how far it is between tick marks on the axes, and these are not changed by dragging the graph. Videos: MS Avi File, or Real Video File (b) Enter the formula x^3 - x in the f box, graph and set the default viewing rectangle. The important features of the graph are near the origin, and we want to get a better look at that area. Click the Zoom In button, then click near the origin (0,0) of the graph. The point where you click in the graph becomes the new center of the viewing rectangle, and you zoom in on that point. Note that if you want to zoom in again, you must click the Zoom In button again. Videos: MS Avi File, or Real Video File (c) Set the Default viewing rectangle again and click the Zoom Box button. Now as you drag the mouse in the viewing window (with the button down), a rectangle appears. When you release the mouse button, the viewing rectangle is reset so that the portion of the plane that was in the rectangle fills the viewing window. Drag a rectangle around the important features of the graph and release. The Zoom Box is a fast way to set the window coordinates to display the portion of the graph that is of interest.
(a) Type the formula 14 + (x3 - 64x)/25 in the f function box of the Java Grapher. Of course, on the Grapher it will be 14+(x^3-64*x)/25 . Now click the Graph button. ( Note that if you leave a formula in f and enter this formula in g, then both will be graphed when both boxes to the right are checked. If you uncheck the box to the right of a function box, then the graph of that function will not be drawn.)
If the default viewing rectangle is set, then your window will appear as below.
Put the mouse pointer near the top of the viewing window, press the button and drag the mouse to the bottom of the viewing window. Now your window will look similar to the following picture.
Now click the Get Values button in the lower left-hand corner to see the new viewing rectangle settings. If you dragged straight down, then the the Xmin will still be close to -10 and Xmax close to 10. The Ymin will be around 5 and the Ymax around 25. The Xscl (x-scale) and Yscl values determine how far it is between tick marks on the axes, and these are not changed by dragging the graph. Videos: MS Avi File, or Real Video File
(b) Enter the formula x^3 - x in the f box, graph and set the default viewing rectangle. The important features of the graph are near the origin, and we want to get a better look at that area. Click the Zoom In button, then click near the origin (0,0) of the graph. The point where you click in the graph becomes the new center of the viewing rectangle, and you zoom in on that point. Note that if you want to zoom in again, you must click the Zoom In button again. Videos: MS Avi File, or Real Video File
(c) Set the Default viewing rectangle again and click the Zoom Box button. Now as you drag the mouse in the viewing window (with the button down), a rectangle appears. When you release the mouse button, the viewing rectangle is reset so that the portion of the plane that was in the rectangle fills the viewing window. Drag a rectangle around the important features of the graph and release. The Zoom Box is a fast way to set the window coordinates to display the portion of the graph that is of interest.
Note: With the Java Calculators and with the TI-82, the absolute value of x is denoted abs(x). You will need this for some of the exercises from the text.
The circle with center point (h,k) and radius r is the graph of the equation (x - h)2 + (y - k)2 = r2. (See p. 53 of the text) For example, the graph of (x - 3)2 + (y + 2)2 = 16 is a circle with center (3,-2) and radius 4.
The circle with center point (h,k) and radius r is the graph of the equation (x - h)2 + (y - k)2 = r2. (See p. 53 of the text)
For example, the graph of (x - 3)2 + (y + 2)2 = 16 is a circle with center (3,-2) and radius 4.
As we have pointed out, the Java Grapher can only graph equations that have the form y = some formula in x. The process of solving the above equation for y looks like this:
.
Notice that there is not a unique solution. The solution is of the form "y = something," or "y = something else." To graph the circle with Java Grapher we must graph both of the formulae.
Exercise 6:
Enter -2+sqrt(16 - (x-3)^2) in the f box, and -2-sqrt(16 - (x-3)^2) in the g box and graph. The result will be the circle with center (3,-2) and radius 4.
Enter -2+sqrt(16 - (x-3)^2) in the f box,
and -2-sqrt(16 - (x-3)^2) in the g box and graph. The result will be the circle with center (3,-2) and radius 4.