## 2.3A Key Concept: The Slope of a Curve

The slope of a line or curve is a measure of how steep it is. The slope of a line is measured by “rise over run”—the change in the y-variable between two points on the line divided by the change in the x-variable between those same two points.

The slope of a curve is a measure of how steep it is and indicates how sensitive the y-variable is to a change in the x-variable. In our example of outside temperature and the number of cans of soda a vendor can expect to sell, the slope of the curve would indicate how many more cans of soda the vendor could expect to sell with each 1 degree increase in temperature. Interpreted this way, the slope gives meaningful information. Even without numbers for x and y, it is possible to arrive at important conclusions about the relationship between the two variables by examining the slope of a curve at various points.

## The Slope of a Linear Curve

Along a linear curve the slope, or steepness, is measured by dividing the “rise” between two points on the curve by the “run” between those same two points. The rise is the amount that y changes, and the run is the amount that x changes. Here is the formula:

In the formula, the symbol Δ (the Greek uppercase delta) stands for “change in.” When a variable increases, the change in that variable is positive; when a variable decreases, the change in that variable is negative.

The slope of a curve is positive when the rise (the change in the y-variable) has the same sign as the run (the change in the x-variable). That’s because when two numbers have the same sign, the ratio of those two numbers is positive. The curve in panel (a) of Figure 2A-2 has a positive slope: along the curve, both the y-variable and the x-variable increase.

The slope of a curve is negative when the rise and the run have different signs. That’s because when two numbers have different signs, the ratio of those two numbers is negative. The curve in panel (b) of Figure 2A-2 has a negative slope: along the curve, an increase in the x-variable is associated with a decrease in the y-variable.

Figure 2A-3 illustrates how to calculate the slope of a linear curve. Let’s focus first on panel (a). From point A to point B the value of the y-variable changes from 25 to 20 and the value of the x-variable changes from 10 to 20. So the slope of the line between these two points is:

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Because a straight line is equally steep at all points, the slope of a straight line is the same at all points. In other words, a straight line has a constant slope. You can check this by calculating the slope of the linear curve between points A and B and between points C and D in panel (b) of Figure 2A-3.

Between A and B:

Between C and D:

Figure 2.3: FIGURE 2A-3 Calculating the Scope
Figure 2.3: Panels (a) and (b) show two linear curves. Between points A and B on the curve in panel (a), the change in y (the rise) is −5 and the change in x (the run) is 10. So the slope from A to B is , where the negative sign indicates that the curve is downward sloping. In panel (b), the curve has a slope from A to B of . The slope from C to D is . The slope is positive, indicating that the curve is upward sloping. Furthermore, the slope between A and B is the same as the slope between C and D, making this a linear curve. The slope of a linear curve is constant: it is the same regardless of where it is measured along the curve.

## Horizontal and Vertical Curves and Their Slopes

When a curve is horizontal, the value of the y-variable along that curve never changes—it is constant. Everywhere along the curve, the change in y is zero. Now, zero divided by any number is zero. So, regardless of the value of the change in x, the slope of a horizontal curve is always zero.

If a curve is vertical, the value of the x-variable along the curve never changes—it is constant. Everywhere along the curve, the change in x is zero. This means that the slope of a vertical curve is a ratio with zero in the denominator. A ratio with zero in the denominator is equal to infinity—that is, an infinitely large number. So the slope of a vertical curve is equal to infinity.

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A vertical or a horizontal curve has a special implication: it means that the x-variable and the y-variable are unrelated. Two variables are unrelated when a change in one variable (the independent variable) has no effect on the other variable (the dependent variable). Or to put it a slightly different way, two variables are unrelated when the dependent variable is constant regardless of the value of the independent variable. If, as is usual, the y-variable is the dependent variable, the curve is horizontal. If the dependent variable is the x-variable, the curve is vertical.

## The Slope of a Nonlinear Curve

A nonlinear curve is one in which the slope is not the same between every pair of points.

A nonlinear curve is one in which the slope changes as you move along it. Panels (a), (b), (c), and (d) of Figure 2A-4 show various nonlinear curves. Panels (a) and (b) show nonlinear curves whose slopes change as you move along them, but the slopes always remain positive. Although both curves tilt upward, the curve in panel (a) gets steeper as you move from left to right in contrast to the curve in panel (b), which gets flatter.

Figure 2.4: FIGURE 2A-4 Nonlinear Curves
Figure 2.4: In panel (a) the slope of the curve from A to B is , and from C to D it is . The slope is positive and increasing; the curve gets steeper as you move to the right. In panel (b) the slope of the curve from A to B is , and from C to D it is . The slope is positive and decreasing; the curve gets flatter as you move to the right. In panel (c) the slope from A to B is , and from C to D it is = . The slope is negative and increasing; the curve gets steeper as you move to the right. And in panel (d) the slope from A to B is , and from C to D it is . The slope is negative and decreasing; the curve gets flatter as you move to the right. The slope in each case has been calculated by using the arc method—that is, by drawing a straight line connecting two points along a curve. The average slope between those two points is equal to the slope of the straight line between those two points.

A curve that is upward sloping and gets steeper, as in panel (a), is said to have positive increasing slope. A curve that is upward sloping but gets flatter, as in panel (b), is said to have positive decreasing slope.

When we calculate the slope along these nonlinear curves, we obtain different values for the slope at different points. How the slope changes along the curve determines the curve’s shape. For example, in panel (a) of Figure 2A-4, the slope of the curve is a positive number that steadily increases as you move from left to right, whereas in panel (b), the slope is a positive number that steadily decreases.

The absolute value of a negative number is the value of the negative number without the minus sign.

The slopes of the curves in panels (c) and (d) are negative numbers. Economists often prefer to express a negative number as its absolute value, which is the value of the negative number without the minus sign. In general, we denote the absolute value of a number by two parallel bars around the number; for example, the absolute value of −4 is written as |−4| = 4.

In panel (c), the absolute value of the slope steadily increases as you move from left to right. The curve therefore has negative increasing slope. And in panel (d), the absolute value of the slope of the curve steadily decreases along the curve. This curve therefore has negative decreasing slope.

## Calculating the Slope Along a Nonlinear Curve

We’ve just seen that along a nonlinear curve, the value of the slope depends on where you are on that curve. So how do you calculate the slope of a nonlinear curve? We will focus on two methods: the arc method and the point method.

The Arc Method of Calculating the Slope An arc of a curve is some piece or segment of that curve. For example, panel (a) of Figure 2A-4 shows an arc consisting of the segment of the curve between points A and B. To calculate the slope along a nonlinear curve using the arc method, you draw a straight line between the two end-points of the arc. The slope of that straight line is a measure of the average slope of the curve between those two end-points.

You can see from panel (a) of Figure 2A-4 that the straight line drawn between points A and B increases along the x-axis from 6 to 10 (so that Δx = 4) as it increases along the y-axis from 10 to 20 (so that Δy = 10). Therefore the slope of the straight line connecting points A and B is:

This means that the average slope of the curve between points A and B is 2.5.

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Now consider the arc on the same curve between points C and D. A straight line drawn through these two points increases along the x-axis from 11 to 12 (Δx = 1) as it increases along the y-axis from 25 to 40 (Δy = 15). So the average slope between points C and D is:

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Therefore the average slope between points C and D is larger than the average slope between points A and B. These calculations verify what we have already observed—that this upward-tilted curve gets steeper as you move from left to right and therefore has positive increasing slope.

A tangent line is a straight line that just touches, or is tangent to, a nonlinear curve at a particular point. The slope of the tangent line is equal to the slope of the nonlinear curve at that point.

The Point Method of Calculating the Slope The point method calculates the slope of a nonlinear curve at a specific point on that curve. Figure 2A-5 illustrates how to calculate the slope at point B on the curve. First, we draw a straight line that just touches the curve at point B. Such a line is called a tangent line: the fact that it just touches the curve at point B and does not touch the curve at any other point on the curve means that the straight line is tangent to the curve at point B. The slope of this tangent line is equal to the slope of the nonlinear curve at point B.

Figure 2.5: FIGURE 2A-5 Calculating the Slope Using the Point Method
Figure 2.5: Here a tangent line has been drawn, a line that just touches the curve at point B. The slope of this line is equal to the slope of the curve at point B. The slope of the tangent line, measuring from A to C, is .

You can see from Figure 2A-5 how the slope of the tangent line is calculated: from point A to point C, the change in y is 15 and the change in x is 5, generating a slope of:

By the point method, the slope of the curve at point B is equal to 3.

A natural question to ask at this point is how to determine which method to use—the arc method or the point method—in calculating the slope of a nonlinear curve. The answer depends on the curve itself and the data used to construct it.

You use the arc method when you don’t have enough information to be able to draw a smooth curve. For example, suppose that in panel (a) of Figure 2A-4 you have only the data represented by points A, C, and D and don’t have the data represented by point B or any of the rest of the curve. Clearly, then, you can’t use the point method to calculate the slope at point B; you would have to use the arc method to approximate the slope of the curve in this area by drawing a straight line between points A and C.

But if you have sufficient data to draw the smooth curve shown in panel (a) of Figure 2A-4, then you could use the point method to calculate the slope at point B—and at every other point along the curve as well.

## Maximum and Minimum Points

The slope of a nonlinear curve can change from positive to negative or vice versa. When the slope of a curve changes from positive to negative, it creates what is called a maximum point of the curve. When the slope of a curve changes from negative to positive, it creates a minimum point.

A nonlinear curve may have a maximum point, the highest point along the curve. At the maximum, the slope of the curve changes from positive to negative.

Panel (a) of Figure 2A-6 illustrates a curve in which the slope changes from positive to negative as you move from left to right. When x is between 0 and 50, the slope of the curve is positive. At x equal to 50, the curve attains its highest point—the largest value of y along the curve. This point is called the maximum of the curve. When x exceeds 50, the slope becomes negative as the curve turns downward. Many important curves in economics, such as the curve that represents how the profit of a firm changes as it produces more output, are hill-shaped like this.

Figure 2.6: FIGURE 2A-6 Maximum and Minimum Points
Figure 2.6: Panel (a) shows a curve with a maximum point, the point at which the slope changes from positive to negative. Panel (b) shows a curve with a minimum point, the point at which the slope changes from negative to positive.

A nonlinear curve may have a minimum point, the lowest point along the curve. At the minimum, the slope of the curve changes from negative to positive.

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In contrast, the curve shown in panel (b) of Figure 2A-6 is U-shaped: it has a slope that changes from negative to positive. At x equal to 50, the curve reaches its lowest point—the smallest value of y along the curve. This point is called the minimum of the curve. Various important curves in economics, such as the curve that represents how per-unit the costs of some firms change as output increases, are U-shaped like this.