Unit 3: Determining Whether the Slope of a Curve is Positive, Negative, or Zero

In the unit on slope in the previous tutorial, we made some generalizations concerning the slopes of straight lines. The pattern for slope was:

 If the line is sloping up to the right, the slope is positive (+). If the line is sloping down to the right, the slope is negative (-). Horizontal lines have a slope of zero (0).

Now, how do these generalizations work with curves.

Curves with a Positive Slope

 Both graphs at the right show curves sloping upward from left to right. As with upward sloping straight lines, we can say that generally the slope of the curve is positive. While the slope will differ at each point on the curve, it will always be positive

To check this, take any point on either curve and draw the tangent to the curve at that point.

 What is the slope of the tangent? Positive. For example, A, B, and C are three points on the curve. The tangent line at each of these points is different. Each tangent has a positive slope; therefore, the curve has a positive slope at points A, B, and C. In fact, any tangent drawn to the curve will have a positive slope.

Curves with a Negative Slope

 In the graphs at the right, both of the curves are downward sloping. Straight lines that are downward sloping have negative slopes; curves that are downward sloping also have negative slopes. We know, of course, that the slope changes from point to point on a curve, but all of the slopes along these two curves will be negative.

In general, to determine if the slope of the curve at any point is positive, negative, or zero you draw in the line of tangency at that point.

 A, B, and C are three points on the curve. The tangent line at each of these points is different. Each tangent has a negative slope since it's downward sloping; therefore, the curve has a negative slope at points A, B, and C. All tangents to this curve have negative slopes.

Example

 In this example, our curve has a: positive slope at points A, B, and F, a negative slope at D, and at points C and E the slope of the curve is zero. (Remember, the slope of a horizontal line is zero.) Make sure you understand the logic here before you move along.

Maximum and Minimum Points of Curves

In economics, we can draw interesting conclusions from points on graphs where the highest or lowest values are observed. We refer to these points as maximum and minimum points.

• Maximum and minimum points on a graph are found at points where the slope of the curve is zero.
• A maximum point is the point on the curve with the highest y-coordinate and a slope of zero.
• A minimum point is the point on the curve with the lowest y-coordinate and a slope of zero.

 Maximum Point Point A is at the maximum point for this curve. Point A is at the highest point on this curve. It has a greater y-coordinate value than any other point on the curve and has a slope of zero. Minimum Point Point A is at the minimum point for this curve. Point A is at the lowest point on this curve. It has a lower y-coordinate value than any other point on the curve and has a slope of zero.

Example

 Identify any maximum and minimum points on the curve. The curve has a slope of zero at only two points, B and C. Point B is the maximum. At this point, the curve has a slope of zero with the largest y-coordinate. Point C is the minimum. At this point, the curve has a slope of zero with the smallest y-coordinate. Point A clearly has the lowest y-coordinate of the points on the curve. Point D has the highest y-coordinate. However, at neither one of these points is a slope of the curve zero.

As you may have already guessed, by using this definition of maximum and minimum we can have curves that have no maximum and minimum points.

 On this curve, there is no point where the slope is equal to zero. This means, using the definition given above, the curve has no maximum or minimum points on it.

You are now ready to try a practice problem. If you have already completed the first practice problem for this unit you may wish to try the additional practice.