Equations and Graphs

Overview: In this module we review the essentials of interpreting and preparing graphical data and revisit the graphical addition of vectors.

Skills:


In the sciences, many times it is necessary to be able to interpret graphs as well as be able to graph certain equations. Often data is available in a graphical format and you must be able to extract the necessary information. Other times, it may be helpful to plot an equation in order to fully understand a problem. However, graphing can be difficult for some students. The format in this section is a little different. The first part will be simply showing what some special equations look like in graphical form and the second part will be a series of questions to help you understand graphs better.

This graph shows a straight line and the corresponding equation is y = mx + b. Y is the y value (distance along the y-axis, which is the vertical axis) and x is the x value (the distance along the x-axis, which is the horizontal axis). The slope of the line is m, which is also the rise/run or the Dx/Dy. The y-intercept of the line is b. This is the y value where the line crosses the y-axis (when the x = 0). In the graph here, the slope is 1 and b is +2. Notice that a line crosses each axis only once (at most).

This graph represents a quadratic function, which is y = ax2 + bx + c. The parameters a, b and c are constants. In this graph the actual equation is y = 2x2 + 3x - 2. Notice that in this graph the function crosses the y-axis once and the x-axis twice. This has to do with the fact that x is squared in the equation and y is not. The function crosses the axis (up to) the same number of times as the power of that value. In this graph x is squared so the line crosses the x-axis up to two times.

This graph shows a cubic equation, which is expressed mathematically as y = ax3 + bx2 + cx + d. For this particular function, y = x3 + 2 x2 - 2 x - 3. Because x is raised to the third power, the function crosses the x-axis 3 times. It crosses the y-axis only once however.

This is a log plot. Notice that the x-axis is quite different than in the other graphs. In this graph, if we look at the y-axis we see that the distance from 1 to 10 is the same as that from 10 to 100. But we know that the span from 10 to 100 represents a much larger range of x-values than that from 1 to 10. This is a property of a log plot. Be sure to look at the axis on a log plot (as well as all graphs) in order to understand exactly what the graph is trying to show.

In order to answer the following questions, you need to have a good understanding of the preceding subjects in the tutorial.



1.

A particle's movement is plotted on the right. Use the information to answer the following questions.

  • a. How far has the particle moved at 4 s? Answer
  • b. How long does it take for the particle to move 6 m? Answer
  • c. What is the particle's velocity? Answer
2.

Another particle moves on a trajectory described in the plot on the right.

  • a. How long does it take for the particle to move 16 m from its starting position? Answer
  • b. How far does the particle travel in the first 5 s? Answer
  • c. Is the particle accelerating? Answer
3.

A ball is dropped from the top of a building and its fall is plotted in the graph to the right.

  • a. Is the ball accelerating? Answer
  • b. What is the velocity at 1 s? Answer
  • c. How long does it take for the ball to reach a velocity of 49 m/s? Answer
  • d. What is happening to the ball's potential energy as it falls? Answer
4.
  • a. How many times would the following function cross the y-axis?
    5y6 + 7y3 + 5x4 – 2x2 = 0? Answer
  • b. For the same equation, how many times will it cross the x-axis? Answer
 

Summary

In this module we have reviewed the essential skills necessary for interpreting data presented in a graphical format. Interpretation of graphical data is a routine skill used by the practicing scientists, physicians, and engineers. You will develop these skills further in your specific discipline at Washington Univeristy.


Return to the Chemistry Subject Index