Graphing Skills: Linear Equations
Explore: Explore how to illustrate linear relationships with a graph.

The material presented in this problem pertains to Appendix A. This graphing exercise illustrates how to graph a straight line given the slope and the y-axis (or vertical axis) intercept.

1. What is the equation that expresses the relationship between minutes called and per minute if your long distance telephone plan charges $5.00 per month plus 10 cents per minute for calls?
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2. What is the total cost of 10 minutes of calls?
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3. The scale on the axes matter a great deal in graphing. Graph the equation for phone charges. To do this press the Plot Equation button with 5 in the vertical intercept and 0.1 in the slope boxes. Notice how the line looks -- nearly flat and difficult to read. Without changing the numbering on the axes, how might you change the slope so that the graph is more readable?
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4. Let's make this a little bit more difficult and see if you remember some of your basic algebra. Suppose you don't know the slope, but you do know the vertical intercept and one other point. If the fixed cost is $10 and we know that 4 minutes cost $26, what is the equation for the line?
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5. Without using the interactive graph, how might you have found the slope of the equation?
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6. In the examples we have used thus far the slope has been a positive number and the graph has risen from left to right. This positive slope reflects a direct relationship. However, suppose the two variables are inversely related. For example, suppose that at a price of zero people want 90 widgets per day, but as price rises they want fewer and fewer, at a rate of 7 fewer per dollar increase in price. Using the interactive graph, establish and plot the equation representing this relationship. Is the slope positive or negative?
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7. Based on part (f), what is the quantity of widgets demanded if the cost is 8 dollars each?
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Graphing Skills: Linear Equations
Explore: Find out how a linear curve can represent the relationship between two variables.

The material presented in this problem pertains to Appendix A. This graphing exercise illustrates how to graph a straight line given two points, or coordinates. You can graph a straight line in this exercise by clicking anywhere in the graph to establish an initial point, dragging the mouse to a second point, and clicking again. (X and y values are displayed above the graph to help in locating points.) Click the Plot Equation button to draw a line connecting the points. The accompanying table will list several points along the line; the vertical intercept is always the value of y when x is zero. Finally, the equation of the line is displayed above the graph.

1. Click on the point x = 0, y = 50. (The location of points can be abbreviated (x, y) where x is the value of x and y is the value of y.) Next, click on the point (10, 0) and plot the resulting line. What is the equation for this line? Does it represent a direct or an inverse relationship? What is the vertical intercept? What is the slope?
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2. Click Reset. Click on the points (3, 30) and (6, 60). Plot the line. What is the equation for this line? Does it represent a direct or an inverse relationship? What is the vertical intercept? What is the slope?
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3. Graph a horizontal line with a vertical intercept of 40. What is the equation for this line? What is its slope?
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4. Click Reset. Click on the point (2, 20). Move your mouse to any point at which x increases by 2 and y increases by 10; click on this point. What is the equation for the line? What is the vertical intercept and slope? Verify that the slope measures the ratio of the vertical change to the horizontal change of your two points.
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5. Use the values in the table to verify that the slope is the same between any two points along this line.
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Graphing Skills: Nonlinear Curves
Explore: How does slope change along a nonlinear curve?

The material presented in this problem pertains to Appendix A. This graphing exercise will calculate the slope of a nonlinear curve by drawing in a straight line tangent to the curve. Click and drag the blue diamond to the left or right--the slope of the curve is calculated and displayed in the box. Click the Open Up/Down button to change the shape of the curve.

1. Click on the Open Up/Down button until the curve is open at the bottom. What is the slope of the curve when x is 1?
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2. What happens to the slope of the curve as x increases from a value of 1 to 3?
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3. What is the slope of the curve when x is 5? How do you interpret the minus sign?
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4. What happens to the slope of the curve as X increases from a value of 5 to 7?
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5. At what x value does the line have a zero slope?
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6. Experiment on your own: If the curve is getting "flatter," is the slope rising or falling? What can you say about positive and negative slopes?
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7. Click on the Open Up/Down button until the curve is open at the top. Repeat the previous questions for this graph.
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8. Comparing the "open down" and "open up" curves, speculate as to how knowledge of the slope--and how it is changing--might indicate which graph is which.
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