Graphing Exercise: Curves and Slopes, Part 1—Straight Lines

Economics is the study of relationships—between wants and resources, between price and quantity, between consumption and income, just to name a few.  As you will discover throughout the text, many of these relationships can be presented graphically.  Graphs can be a handy tool to summarize quickly the relationships between two variables. 

If the two variables of interest increase or decrease together, the relationship is said to be direct and the graph of the relationship will be an upward sloping line.  If the two variables move in opposite directions, the relationship is said to be inverse and the graph of the relationship will be a downward sloping line. 

In many cases, these relationships can reasonably be represented by a straight line.  The slope of the line is the ratio of the vertical change to the horizontal change between any two points on the line.  Generally, the slope tells us the amount by which one variable changes when the other variable increases by one unit.  The general equation of a straight line is y = a + bx, where:
y is the value of the variable on the vertical axis (the dependent variable),
x is the value of the variable on the horizontal axis (the independent variable),
a is the vertical intercept, the value at which the line crosses the vertical axis, and
b is the slope
(The exception is for a vertical line, for which the slope is infinite and there is no vertical intercept.)

Exploration: How are lines, slopes, and intercepts related?

The position of any straight line is determined by two points.  The graphing applet will allow you to graph straight lines by clicking anywhere in the graph to establish an initial point and clicking again to establish a second point.  (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 given by the value of y corresponding to a zero value of x.  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, then 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 such that 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 Exercise: Curves and Slopes, Part 2—Curved Lines

Not all economic relationships are best represented by straight lines.  For example, as production of an item increases, its per-unit cost usually falls at first, then increases.  As another example, income taxes paid do not usually increase in proportion to income.  Instead, income taxes tend to rise faster as income increases.

In these and many other cases, we are still interested in the slope of the line representing the relationship—at any particular point, if one variable changes, how much will the other variable change?  For example, consider any particular level of income.  We might like to know how much taxes will rise if income increases starting from that particular point.  Or, we might like to know by how much average cost changes as output changes from its current level.  Unlike a straight line whose slope is always the same at any point on the line, the slope of a curved line will vary as we move along the curve.  The slope at any point is best represented by the slope of a straight line that is tangent to the curve.

Exploration: How do we find the slope of a nonlinear curve?

The graphing applet will calculate the slope of a nonlinear curve by drawing in a straight line tangent to the graph.  To use the graph, click and drag the blue diamond to the left or right—the slope of the line 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 curved line when X is 1?
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2. What happens to the slope of the line as X increases from a value of 1 to 3?
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3. What is the slope of the curved line when X is 5?  How do you interpret the minus sign?
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4. What happens to the slope of the line as X increases from a value of 5 to a value of 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 whether a graph opens upward or downward.
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