Graph Skills: Unit Eight
Determining Whether the Slope of a Curve is
Positive,
Negative, or Zero
In the unit on Slopes in Book I, 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 0.




Now, how do these generalizations work with curves.
Both graphs at the below 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.


In the graphs below, 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.


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.


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 itŐs downward sloping; therefore, the curve has a negative slope at points A, B, and C. All tangents to this curve have negative slopes.


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 ycoordinate and a slope of zero.
 A minimum point is the point on the curve with the lowest ycoordinate 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 ycoordinate 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 ycoordinate 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 ycoordinate.
 Point C is the minimum. At this point, the curve has a slope of zero with the smallest ycoordinate.
 Point A clearly has the lowest ycoordinate of the points on the curve. Point D has the highest ycoordinate. 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.