Cynthia Lanius

performance to stress graph
Pictured:The Yerkes-Dodson Law

Stressed Out

Slope as Rate of Change


It's the night of the big game (or play). You're in the locker room (or dressing room). The coach (make that director) is pumping the team (cast) up.
"Now, I know you people are nervous. That's okay, in fact, that's what we want. You're going to perform better on the court (stage) if you're a little nervous".
Does the graph above confirm what the coach (director) told you?

Here are some questions for you based upon the graph.

  1. Write a statement that describes performance as stress increases.

  2. Which part of the graph illustrates where stress is highest? What is performance at that point? Which part of the graph illustrates where performance is highest? Which part of the graph illustrates where performance is increasing? Which part of the graph illustrates where performance is decreasing?

    performance to stress graph with line of symmetry

  3. Notice that the graph is symmetric about a vertical line. What would you say this indicates about performance?

  4. What if this were a graph of your bank balance over a year. Describe what the graph would indicate about your saving and spending practices.

 

 

Slope as Rate of Change

There is a direct relation between the rate of change of a function and the slope of its lines.

speed to time graph Study the graph to the left. Carlos and Maria are on their way to school. Answer the following questions with values of time (t). (For example, AB includes all t between A and B.)

  1. For what values of time (t) are Carlos and Maria stopped?
  2. For what values of t do they increase their speed?
  3. Which of these increase at the same rate?
  4. Explain how this question relates to the slope of the lines?
  5. For what values of t do Carlos and Maria decrease their speed?
  6. At what time (t) do they achieve their highest speed?
  7. For what values of t do Carlos and Maria drive at a constant rate?


Consider the diagram below. We are familiar with the slope of straight lines. If a function is decreasing, the line is falling, and the slope is negative. If a function is increasing, the line is rising, and the slope is positive. If a function maintains a constant value, the line is horizontal, and the slope is 0. What about a vertical line?

Slippery Slopes

lines with positive and negative slopes

 
To answer that, let's think about this. Where would a line be on the diagram that had a slope of 10? How about 100? 1000? 1,000,000? A Billion? A trillion? A trillion trillion? Or a negative trillion trillion? As the lines get closer and closer to a vertical line, the slope is getting bigger and bigger, what mathematicians call approaching infinity. Or in the case of negatives, getting smaller and smaller, approaching negative infinity. So we say the slope of a vertical line is undefined, since it would be positive or negative infinity.

 
performance to stressLet's go back to our original performance-to-stress graph. How does slope relate to rate of change here? You may be saying, how can we talk about slope because the lines aren't straight. Well, a curved line will have more than one slope. It's not a constant like it was on the straight lines. A straight line has the same slope everywhere on the line, but a curve won't. We are going to define slope at a point on a curved line as the slope of the tangent line drawn to the curve at that point.

The slope of the curve at point A equals the slope of the tangent line at A.
performance to stress

performance to stress

Notice that the slope of a curved line can be positive, where the function is increasing, negative, where the function is decreasing, or 0, where it reaches that highest point.

vertical tangentSome curves can also have vertical tangent lines. Do you remember what the slope of this line would be?

To the right is the graph of the height of a clock's minute hand as it travels around the clock. Answer these questions about the slope of the graph.
  1. What is the range of time where the graph has a positive slope?
  2. What is the range of time where the graph has a negative slope?
  3. At what times does the graph have slope = 0?
height to time
clock
 

Assignment

Find some other phenomenon like stress to performance or speed over time and draw a graph of the function. Write at least 5 questions about the function. Make sure that some of the questions pertain to how slope and rate of change are related.  

Teachers' Notes


More Math Lessons from Cynthia Lanius

More on Slope from the Web

Slope and Line Equations
Linear Function - Slope Intercept Form Java Applet

From the Dr. Math archives

From Ask Dr. Math, the Forum's math question & answer service (High School: Mathematics After High School)
  • Solving problems the hard way (Fan)

  • Slope of 3-Dimensional Equations (Keenan)
    Is it possible to find the slope of three-dimensional equations?

  • Graphing (Rider)
    I've got questions concerning graphing.

  • Using the Slope-Intercept Formula (Plotkin)
    How can I write an equation for the line when I have the x-intercept and the slope?

  • Preparing for an Algebra Test (Nerlekar)

  • The Ratio of the Speed of a Colonel to his Soldiers (Sismondo)

  • Graphing an Equation (Garb)
    How do you graph 2x-y = 10?

  • Equations for Lines (Milburn)
    Write an equation of the line containing the point P(1,-3)...


    More Math Lessons from Cynthia Lanius

    ©1998-2008 by Cynthia Lanius