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WinPlot and 2-Space
WinPlot and 2-Space
Using WinPlot Graphics to Enhance Teaching and Learning
of Mathematical Concepts and Ideas in 2-Space
Students, very early in their study of mathematics, are introduced to the importance of graphs and they, in turn, develop the skill of graphing. As students progress through math courses in the various grade levels and into post-secondary studies of mathematics, graphing remains as a skill to be mastered to facilitate the teaching and learning of mathematics.
It can be challenging and somewhat difficult, however, to construct good, accurate graphs and coordinatized drawings. Effective use of technology can assist in filling this void. WinPlot is a graphic utility designed to enable students and educators to effectively and accurately perform such tasks as graphing points, segments, functions, relations, and constructing coordinatized diagrams. Features of WinPlot not only improve the presentation and accuracy of graphs and drawings, but can also facilitate the learning of certain mathematical concepts and ideas, especially with the use of sliders.
The following project has been designed for students and educators alike and will provide evidence of the power of WinPlot in the teaching and learning of various mathematical concepts found in the senior high mathematics curriculum for the Atlantic provinces.
The first worksheet involves several problems relating to linear inequalities and linear programming. These topics are studied in mathematics 1204. Several problems have been provided as well as a solution sheet.
The second worksheet includes problems with sinusoidal functions and their associated graphs. Solutions have been provided for these questions.
The activity involves an investigation of quadratic functions given in transformational form. The intent of the activity is to use sliders in WinPlot to illustrate to students the effects of changing the variables a, h, and k, and showing how the new quadratic functions compare with the basic quadratic function . In addition, students will be required to change quadratic equations, given in transformational form, to standard and/or general form in an attempt to use WinPlot options to determine the vertex and x-intercepts of a given quadratic function. A similar activity could be created for other types of functions such as linear functions, exponential functions, and sinusoidal functions(examining properties including vertical stretch, horizontal stretch, vertical translation and horizontal translation). This activity, or certain sections of it (depending on the course of study), could be used in math 1204, math 2204/05, and math 3204/05. The nature of the activity assumes that students have had some prior experience in using WinPlot.
This assignment merely ‘touches the surface’ in exploring the possibilities of using WinPlot in the teaching and learning of mathematics. Its potential goes far beyond the concepts and ideas explored here and will, no doubt, prove to be an invaluable resource in my teaching of mathematics in the future.
Worksheet 1Linear Inequalities and Linear Programming
1.Which of the following best describes the feasible region shown in the given graph?
(A).
(B).
(C).
(D).
2.Which of the following points is a feasible solution for the feasible region shown in the given graph?
(A).(1,5)
(B).(3,1)
(C).(4,2)
(D).(8,4)
3.Write inequalities to represent each of the following graphs.
(A).
(B).
(C).
4.Terra Technical Services assembles both laptop and personal computers. They make $100 profit for each laptop sold and $70 profit for each personal computer sold. Find the maximum profit under the following conditions:
it takes 4 hours to assemble a laptop computer
it takes 2 hours to assemble a personal computer
they assemble computers for a maximum of 200 hours per month
the maximum number of computers that can be made is 60 per month
5.A local painter, Ted Stuckless, classifies his paintings as either scenic paintings which he sells for $300, or portrait paintings which he sells for $200. Find the maximum profit given the following conditions:
he limits the number of paintings he does to 8 per week
it takes him an average of 6 hours to do a scenic painting
it takes him an average of 3 hours to do a portrait painting
he works a maximum of 30 hours per week.
Worksheet 1{Solutions}Linear Inequalities and Linear Programming
1.The answer is (B). There is a dotted horizontal line through , so we do not include this line in the inequality describing the graph. We do, however, include the line , since there is a solid vertical line passing through this point, hence the need for the sign.
2.A feasible solution for this problem would be (B), the point (3,1). Feasible solutions must be found in the shaded area, and (3,1) is the only point found in this region, as indicated in the graph below.
3.(A).(B).(C).
4.let x = # of laptop computers sold
let y = # of personal computers sold
let P = maximum profit
Objective Function:
Inequalities:
Graph:
Hence, we discover that the maximum profit is made when 40 laptop computers and 20 personal computers are sold, and the maximum profit is
5.let x = # of scenic paintings sold
let y = # of portrait paintings sold
let P = maximum profit
Objective Function:
Inequalities:
Graph:
Thus, Mr. Stuckless will maximize his profit if he sells 2 scenic paintings and 6 portrait paintings. The maximum profit will be
Worksheet 2Working With Sinusoidal Functions and Their Graphs
1.Maria, a member of TricentiaAcademy’s Acrobat team, is swinging back and forth on a trapeze. Her distance from a support beam with respect to time is modeled in the graph below. What is a possible sinusoidal function that describes Maria’s distance from the beam in relation to time?
2.Determine the equation of a sinusoidal function, in transformational form, for each of the given graphs.
(A).
(B).
Worksheet 2{Solutions}Working With Sinusoidal Functions and
Their Graphs
1.Two possible sinusoidal functions (given in transformational form) that can be used to represent Maria’s distance from the beam in relation to time are as follows:
(i).
OR (ii).
2.(A).Two possible sinusoidal functions include:
(i).
OR (ii).
(B).Two possible sinusoidal functions include:
(i).
OR (ii).
ActivityInvestigating the Properties of Quadratic Functions In Transformational Form
The following slider activity, using WinPlot, will enable you to investigate the various properties of quadratic functions given in transformational form. Remember that the transformational form of a quadratic function is given as follows:
Remember that the graph of a quadratic function is a parabola .
We want to discover the effect of changing the values of a, h, and k. We will initially graph the basic quadratic function, . Subsequent graphs created from changing the values of a, h, and k (using the slider feature with WinPlot) will then be compared to this graph.
Important: Please follow all instructions carefully to ensure successful completion of this activity!
Step 1
Click on WinPlot on your screen and click on Window and 2-dim, as illustrated below:
Step 2
Next, use your previously acquired skills using WinPlot to change your view(Ctrl-V) and grid(Ctrl-G) settings to the following:
If you now press View and Zoom square (or Ctrl-Q), you should have a grid that looks like the following: {Remember: To insert a zero, “0”, in your grid, use the Btns and Text feature}
Step 3
Now we are ready to begin the investigation of quadratic equations in transformational form.
First, we need to graph the basic quadratic function . To do this click Equa and y=f(x)…, or press F1. You will see the following:
When you click ok, you should obtain a graph like the following:
We can use the grid to see that for , we go over 1, up 1…over 2, up 4…over 3, up 9, and so on.
Step 4
To examine the effects of the values of a, h, and k in transformational form, we need to enter the transformational form of a quadratic equation. To do this, click on Equa, then
0 = f(x,y), or press F4 to obtain the following:
After clicking ok, we do not see any change to the graph shown on the screen. We still only see the parabola for (the slider values initialize at 0). In the inventory (Ctrl – I), we see the following:
We will now manipulate the variables a, h, and k, to determine the effect that each of these variables has on the graphs of the subsequent quadratic functions.
Step 5…What happens when we change the value for “a”?
We will now determine the effect on parabolas when we change the value for a in the transformational form of the equation. To do this we click Anim, and A, as this is the variable we are manipulating:
You should now see a window that has the following:
Now use the slider, as indicated above, to observe the changes that occur in the parabola as the values for a change.
Questions
1.What happens to the parabola, in comparison with y=x2, when the value for “a” becomes increasingly greater than 0?
2.What happens to the parabola, in comparison with y=x2, when the value for “a” becomes less than 0?…and increasingly less than 0?
3.What does the “a” value tell you about the direction of opening of the parabola?
4.How does the “a” value relate to the vertical stretch of a parabola? What happens if a=1? How does it compare to y=x2? Why?
NOTE:Make sure your value for “a” is set at 1 before
continuing!!!!
Step 6…What happens when we change the value for “h”?
We will now determine the effect on parabolas when we change the value for h in the transformational form of the equation. To do this we click Anim, and H, as this is the variable we are manipulating. If we assign a value of 5 for h we obtain a graph as shown below:
Assign other values for the variable h, and/or use the slider to answer the following questions.
Questions
1.What happens to the parabola, in comparison with y=x2, as the value for “h” increases?
2.What happens to the parabola, in comparison with y=x2, as the value for “h” decreases?
3.What can you conclude about the effect of the value of “h” on a quadratic equation in transformational form?
NOTE:Make sure your value for “h” is set at 0 before
continuing!!!!
Step 7…What happens when we change the value for “k”?
We will now determine the effect on parabolas when we change the value for k in the transformational form of the equation. To do this we click Anim, and K, as this is the variable we are manipulating. If we assign a value of -4 for k we obtain a graph as shown below:
Assign other values for the variable k, and/or use the slider to answer the following questions.
Questions
1.What happens to the parabola, in comparison with y=x2, as the value for “k” increases?
2.What happens to the parabola, in comparison with y=x2, as the value for “k” decreases?
3.What can you conclude about the effect of the value of “k” on a quadratic equation in transformational form?
Step 8
As a final step in this activity, assign values to all three variables simultaneously to discover the resulting parabolas, and examine how they differ from .
For example, if a = -2, h = 3, and k = -1, we obtain the following parabola(outlined in blue):
Now, consider the following:
If we change our quadratic function from transformational form to standard form, , or general form, , we can enter the equation in Equa, and y=f(x) (or press F1), and then determine the maximum or minimum value of the graph (the vertex) and the x-intercepts (or zeros) of the function. For example, if we write the equationin standard form and then general form, we obtain the following:
We enter the standard (or general) form of the equation by pressing F1, and change the colour of the curve to distinguish it from the transformational form of the function when graphed, as shown in the following frame:
When we click ok, we notice that our new form of the function gives the same graph as the transformational form (AND IT SHOULD…we still have the same equation, it is just in a different form!).
We are now ready to determine the vertex and the values for the x-intercepts (if there are any). To determine the vertex, we click on One and Extremes as shown below:
After clicking on Extremes, we obtain the following screen that indicates the maximum value for this parabola at the point (3, -1). This point represents the vertex of our parabola.
To determine the x-intercepts, we click on One, and then Zeros, as shown below:
After clicking on Zeros, we obtain the following screen, indicating that there are no zeros (x-intercepts) for our function. This should make perfect sense because the graph of our function does not cross the x-axis!
Follow-up…
Use your sliders for each of the variables to obtain a graph of each of the following quadratic functions given in transformational form. Please indicate the vertical stretch, vertical translation, and horizontal translation, which are the values of a, k, and h, respectively. Then, change each of the equations to standard or general form and, using the necessary options in WinPlot, determine the vertex and x-intercepts(zeros, roots) of the functions from (A) – (D).
(A).(C).
(B).(D).
PLEASE NOTE:
(you may include your response here as a journal entry)
In an attempt to improve this activity for other students, can you please indicate any problems/difficulties you experienced when proceeding through the various steps in the procedure. Were the instructions easy to follow? Are there certain parts of the procedure that need to be clarified?
Indicate, as well, how this activity has increased (hopefully ) your understanding of quadratic equations, their associated graphs, and some of their properties.