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Graphing f '(x):

Increasing, Decreasing, Concavity
Activity Description Activity Guide Resources


Teacher Note:
This activity is designed to follow the activities From Secant Lines to Tangent Lines and Slopes and Derivatives.

To do this activity, you will need a Sketchpad Custom Tool we refer to as the Tangent Line Tool. (For more information about tools, see below). You can download or create it yourself. To download it, click on the link to the sketch tangentline.gsp which contains the Tangent Line Tool as well as illustrates the construction of a tangent line to a curve. Save this sketch to the Tools Folder in the Sketchpad Folder on the C drive. As a result, every time Sketchpad is opened on your computer, the Tangent Line Tool will be easily accessible by holding the left mouse button down on the Custom Tools icon (double arrow icon) in the tool bar. If you want to construct the Tangent Line Tool yourself, detailed instructions can be found under the Resources link above.

What is a tool? Custom Tools are generalized recordings of sketches. It is sometimes convenient to record the steps of particular sketches (e.g., tangent lines to curves) that you have made for subsequent playback. For example, when a future sketch requires a constructed tangent line, you can just use your custom tool rather than reconstructing one. Sketchpad can generate a custom tool of a construction while you are sketching it. You can use custom tools repeatedly to generate figures, or portions of figures, while sketching. Individual custom tools can be used as foundations for larger sketches, leading to potentially more complex constructions. (This explanation was adapted from: http://www.teacherlink.org/content/math/activities/skpv4-construcisitro/guide.html )


Part 1: Increasing, Decreasing, Maximum, Minimum

  1. Plot the function f(x) = -x3 + 3x2 - 1 in Sketchpad. Determine the intervals along the x-axis over which f(x) increases and decreases; i.e., the graph rises and falls. At what x-values does f(x) attain its absolute maximum value, if at all? Its absolute minimum value, if at all? Explain your reasoning.

  2. According to the instructions below, plot a draggable point A on the x-axis and plot its image point A' with coordintes (xA, f(xA)). Use these points to numerically verify your answers to Task 1.

    How to Plot A and A':
    Point A and its image point A'(xA, f(xA)) can be automatically plotted using the Tangent Line Tool. To use the Tangent Line Tool, follow the instructions below. Hide the tangent line by selecting the tangent line then under Display, selecting Hide Line. Points A and A' will remain in the sketch.

    How to Use the Tangent Line Tool:

    • Hold the mouse button down on the double arrow tool in the tool bar until a menu appears. Select the Tangent Line Tool.
    • Click on the algebraic form of the function through which the tangent will pass. As you move the cursor, a point will appear next to the cursor. Position this point on the x-axis. (Note: When the point lies on the x-axis, the x-axis will change color.) Click the mouse button to put the point in place.

  3. At what x-values does the function f(x) = -x3 + 3x2 - 1 attain local maximum values? Local minimum values? Explain your reasoning.

  4. In the graph of g(x) below, identify the x-values, if any, at which g(x) has absolute maxima, absolute minima, local maxima and local minima. Can absolute extreme values always be classified as a local extreme values as well? Explain.

               
Teacher Note: To download this sketch, click on the link maxminsketch.gsp.

Part 2: The Graph of f(x) and the Sign of f '(x)
  1. Construct a tangent line to the curve f(x) = -x3 + 3x2 - 1 using the Tangent Line Tool (or display the hidden tangent line). Display the tangent line's slope. Describe the behavior of the tangent line as you transverse point A along the x-axis.

    How to Display a Hidden Object:

    • Under Display, select Show Hidden Objects.

    • Over what intervals along the x-axis is the slope of the tangent line positive? Describe the behavior of the function on the interval(s) where the slope of the tangent line is positive.

    • Over what intervals along the x-axis is the slope of the tangent line negative? Describe the behavior of the function on the interval(s) where the slope of the tangent line is negative.

    • At what x-values is the slope of the tangent line zero? Describe the behavior of the function on the interval(s) where the slope of the tangent line is zero.

  2. Recall the relationship between the slope of the tangent line at a point and value of f '(x) at that point. Create a textbox next to the value of the tangent line's slope which labels the slope as f '(xA), assuming A is the point through which the tangent line passes. Based on your analysis of f(x) and the slope of the tangent line to the graph of f(x), make conjectures about the following which might apply to continuous functions in general:

    • The relationship between the sign of f '(x) and the intervals along the x-axis over which f(x) increases and decreases.

    • The value of f '(x) at a local extreme value.

      Teacher Note: These conjectures will be refined during the extension portion of the lesson.

Part 3: A Begininning Sketch of f '(x)

  1. For the function f(x) = -x3 + 3x2 - 1 how would the intervals over which the slope of the tangent line is positive, negative, and zero help you to draw a graph of f '(x)? Using only this information, draw a possible graph of f '(x) on your own paper.

  2. Compare your graph to those of a few friends. You graphs will vary, but the intervals over which your graphs fall below the x-axis, cross the x-axis, and rise above the x-axis should be the same. Reconcile any differences concerning these intervals.

Part 4: Concave Up, Concave Down, and f '(x)

  1. Hide the slope of the tangent line and focus on the graph of the tangent line to f(x) = -x3 + 3x2 - 1 as you transverse A along the x-axis. Over what intervals along the x-axis is the slope of the tangent line to f(x) = -x3 + 3x2 - 1 increasing? Decreasing? Verify these intervals numerically by showing the tangent line's hidden slope.

  2. A student, Anna, is confused because the slope of the tangent line is getting less and less steep for x <= 0 but the value of f '(xA) is increasing for x <= 0. What can you tell Anna to help her clear up her confusion?

  3. Over what intervals is f(x) = -x3 + 3x2 - 1 concave up? Concave down? What are the inflection points of f(x) = -x3 + 3x2 - 1?

  4. Discuss some ways to quickly identify if a portion of a graph is concave up or concave down. Test your ideas on the graph of f(x) = x3. (See the tip about editing functions in the gray box below.)

    Teacher Note: The students might say that a portion of a graph is concave up if is it is shaped like a cup, if it could hold water, or if the tangent line is always situated below that portion of the graph. They might say a graph is concave down if could not hold water or if the tangent line is always situated above that portion of the graph.



    How to edit a function:

    • Double click on the algebraic form of the function shown in the sketch.
    • Edit the function in the window that appears. Click <OK>. All dependent items in the sketch will change simultaneously.

Part 5: Refining the Sketch of f '(x)

  1. For the function f(x) = -x3 + 3x2 - 1 use the intervals over which f '(x) is increasing and decreasing to refine your pencil-and-paper sketch of f '(x).

  2. In Sketchpad, for the function f(x) = -x3 + 3x2 - 1, plot point B with coordinates (xA, f '(xA)) and select it to be traced. As point A is transversed along the x-axis, the graph of what function will be traced by B?

    How to plot the point B (xA, f '(xA)):

    • First, click on the point's abcissa shown in the sketch; e.g. xA.
    • Second, click on the point's ordinate shown in the sketch; e.g., the slope of the tangent line.
    • Under Graph select Plot as (x,y).

    How to label a point or change its label:

    • Select the text tool from the tool bar on the left side of the sketchpad window.
    • Double click on the point to be labeled.
    • Enter the new label in the window that appears.  Also, select Show Label if it is not already selected. Click <OK>.
    • Drag the label to a suitable location next to the point.  When you are finished, select the arrow tool again.

    How to select a point to be traced:

    • Select the point. Be sure only that point is selected.
    • Under Display, select Trace Point.
    • As the point is moved, its path will be traced.

    How to erase the traces:

    • Under Display, select Erase Traces.

  3. As you transverse point A along the x-axis to trace the graph of f '(x), compare the graph of f(x) to the traced graph of f '(x). Compare the traced graph of f '(x) to your sketch of f '(x). Reconcile any differences concerning the sign of f '(x) and the intervals where f '(x) increases and decreases.

  4. Display the algebraic form of f '(x) using Sketchpad. Relate the algebraic form of f '(x) to the graph of f '(x).

    How to display f '(x):

    • Select f(x).
    • Under Graph select Derivative.
Extension:
  1. Clean up your sketch according to the instructions below. Edit the function f(x) to read f(x) = 3x2/3. This graph contains a cusp at x = 0. Determine the intervals along the x-axis over which f(x) increases and decreases and the x-values at which f(x) has local extreme values. Determine the intervals along the x-axis over which f(x) is concave up and concave down.

    To Clean Up Your Sketch:

    • Delete point B and erase the traces.
    • Delete the algebraic form of f '(x).
    • Hide the tangent line and its slope.


    Note: Sketchpad is not able to adequately plot the graph of 3x2/3 close to the origin. In reality, the graph has a relative minimum at (0, 0), not at (0, 0.25) as it appears. You can see this by zooming in and watching where point A' travels as point A is dragged slowly across the origin.


  2. Using what you have learned in this lesson, relate the intervals and x-values you determined in Task 15 to the slope of the tangent line to f(x). Predict the slope of the tangent line at x = 0 and explain your reasoning.

  3. Display the hidden tangent line and its slope. Use the tangent line and its slope to both graphically and numerically verify your answers to Task 16. Was your prediction correct? Why or why not?

    Note: To better understand what happens to the slope of the tangent line near the origin, create a movement button to move point A slowly to the origin. Also, evaluate the derivative of f '(x) at xA and compare this value to the value of the tangent line's slope at the origin.
    What happens to the tangent line at the origin? Why?
    As the tangent line approaches zero from the left, what value does the slope of the tangent line approach?
    As the tangent line approaches zero from the right, what value does the slope of the tangent line approach?

  4. In Task 6, for a continuous function f(x), you conjectured about the value of f '(x) at a local extreme value of f(x). Verify your conjecture for f(x) = 3x2/3 and make revisions as necessary.

  5. Sketch a possible graph of f '(x). Be sure to also consider where the value of f '(x) approaches zero.

  6. In Sketchpad plot the point B with coordinates (xA, f '(xA)). As you trace the path of point B to draw the graph of f '(x), compare the graph of f(x) to the graph of f '(x). Next, compare the traced graph of f '(x) to your sketch of f '(x) . Reconcile any differences.

  7. Display the algebraic form of f '(x) in your skethc. Relate the algebraic form of f '(x) to the graph of f '(x).

Glossary of Terms:

Absolute maximum value/Absolute minimum value: The highest and lowest values attained by a function are called the absolute maximum value and absolute minimum value, of f(x), respectively, or simply, the absolute extreme values of f(x).

Concave down: The graph of a function is concave down on an interval if f '(x) is decreasing over that interval.

Concave up: The graph of a function is concave up on an interval if f '(x) is increasing over that interval.

Local maximum values/Local minimum values: A function may not possess an absolute maximum or absolute minimum value, but it may have local maximum or local minimum values, sometimes referred to simply as local extreme values. They are the highest or lowest values of f(x) compared to other values "nearby". In other words, they are the highest and lowest values "locally."

Inflection points: The points where the concavity a function changes are called inflection points.


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Last modified on April 3, 2004.