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Investigating by Graphing
(Who's In Charge?)
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By Shirley Treadway
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Initialization Cells
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"The use of technology in instruction should alter both the teaching and learning of mathematics." - NCTM Standards
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Introduction
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This project will develop three lessons which will investigate and compare the graphs of different functions. The objective of the first lesson is to help the students become familiar with the properties of odd and even functions and also to see what happens when a function is negative. The second lesson takes polynomial functions of the same degree and compares their graphs as x gets very large. The third lesson looks at the growth of the square root function, the sin function, the natural log function, a polynomial function, and last of all the exponential function. Students will be given an "Investigation Sheet" on which to record the discoveries they make on the computer.
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Exploring Polynomial Functions
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Lesson 1
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We will begin our study of polynomials by investigating
the shape and direction of certain curves using the
graphing capabilities of the software, Mathematica..
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First we must learn some "simple" Mathematica
commands for graphing. Once we learn these it
will be easy to just copy and revise them for
each curve that we wish to investigate. Let us look at the
example below. It should graph the curve y = xÛ.
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2:2,24,16,CalcMath,0,12,0,0,0;1,24,16,CalcMath,2,12,0,0,0;
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l. First we must define the function so that Mathematica understands what we are wishing to graph. To do that use the "f[x_]=" after we use the Clear[f,x] command that makes sure that we haven't left some other value for f or x in the machine. Let's try it and see what happens.
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5:0,0;45,1;56,0;122,2;123,0;280,-1;
3:3,24,16,CalcMath,0,12,0,0,0;1,24,16,CalcMath,2,12,0,0,0;1,13,10,Courier,1,12,0,0,0;
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Clear[f,x]
f[x_]= x^2
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Notice that the machine now knows that the curve we want is y=xÛ. So we are ready to begin our plotting of this curve. we will name it "aplot". The first letter needs to be lower case because if it is a capital letter the machine will think it might be a Mathematica programming command. The {x,-100,100} tell us what our domain is and the others add a little flourish to our graph. Let's give it a try and take a look!
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2:2,24,16,CalcMath,0,12,0,0,0;1,24,16,CalcMath,2,12,0,0,0;
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aplot=Plot[f[x],{x,-100,100},
PlotStyle->{{Thickness[0.01],Maroon}},
AxesLabel->{"x","f(x)"}];
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*On your Investigation sheet, please make a sketch of this curve.
2. Now we would like to sketch y = xÜ and make a sketch of it on our Investigation Sheet. To begin with we will need to clear and redefine--that's easy but if you have forgot, just scroll back up to find out how!
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Clear[f,x]
f[x_] = x^3
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Okay!! It looks like our machine knows that we are wanting a new function and that is good. Now we would like to graph it for
the same domain and with the same pretty Maroon color. We could retype that entire mess from above or we could use this super technology and just copy and revise. Here is how you do that---First scroll back up to our commands for graphing "aplot" and highlight all of that cell. Now pull down the edit menu and highlight "copy". Then move to where we want our new graph and click the mouse. Now go back to the edit menu and click on paste. You should have the very same command that we used to plot our first graph!!
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But we really don't want the same graph--but remember that we redefined f(x) so maybe we will be okay--let's take a look and see if we have the same curve.
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aplot=Plot[f[x],{x,-100,100},
PlotStyle->{{Thickness[0.01],Maroon}},
AxesLabel->{"x","f(x)"}];
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Would you look at that "hugger"!!! This machine may make us look smart yet!! When we renamed f(x) the machine understood and it graphed the new function. Before we get too excited we had better return to our investigation sheet and sketch a graph.
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We are about to proceed--we want to continue our investigation by looking at y= xÝ. Remember to first redefine your function and then copy.
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Remember to record on your Investigaton Sheet.
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This is getting interesting, right? This one looks a little like another of our graphs doesn't it? Let us continue. We may become technological giants, yet.
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3. Repeat this procedure for y = xÞ through y = xÚâ and be sure to record each sketch on your investigation sheet.
4. Now let's see what conclusions we can draw from our sketches.
Record the answer to the following questions on your investigations sheet.
A. Are the sketches of some of these similar?
B. Do some seem to fit in the same group?
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C. Why might we call some of these "even" functions and
some "odd"?
D. How do they differ?
E. Describe each group in your own words.
5. Now let's get busy and check out what happens if these functions need an attitude adjustment--that is that they are negative. Let's graph y = -xÛ through y = -xÚâ and record these. Remember to use our Mighty Mac and Mathematica to make this happen quick and easy. Record these sketches and compare these to their positive cousins from our work above.
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3:0,0;372,1;383,0;521,-1;
2:2,24,16,CalcMath,0,12,0,0,0;1,24,16,CalcMath,2,12,0,0,0;
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Bonus time: Just to see if you really are becoming a technological giant let's try something. As you know one of these days we may consolidate and it will not be so important to be Maroon--in fact we might find orange and blue to be as attractive at RHS as it is at my old alma mater, U of I. So try to regraph your favorite curve making it orange instead of maroon.
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Exploring Polynomials of the Same Power And Dominance
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Lesson 2
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The task of this lesson is to investigate what happens to certain functions as x gets very, very large. Hopefully this will bring to mind the famous mathematical term "limit". We also will see why it is so helpful to move to the computer to make these discoveries.
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l. We will graph three functions together on the same
axis and then change our domain "until x gets very large" to see if we might make a good conclusion about what part of the
expression for each curve is most important. The commands that tell the mighty Mac what to do are a little bit more complicated but I'll bet we can handle it! Just as before we will need to tell the machine that we want to start with a clean slate--so let's begin by clearing f and x and g and h and j because we will need to see at least three functions at the same time. Then we will need to to define our functions. Let's begin with three functions that all are quadratic, but some have more baggage than others.
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Clear[f,x,g,h,j]
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f[x_]= x^2
g[x_]= x^2 - 5 x
h[x_]= x^ 2 + 8x -25
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Now our machine knows what three functions we are wanting to see--poor thing it turned these around--it obviously missed the Algebra I lecture about descending order-- or maybe it has decided that this very important matter really isn't so important after all. Whatever--let's get to the graph!!
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dominantplot=Plot[{f[x],g[x],h[x]},{x,-10,10},
PlotStyle->{Red, Blue, Green,},
AxesLabel->{"x","f(x)"}];
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2. These folks look a lot alike but let's see what happens as we look at them from a much "wider" persepective--in math jargon---let's change the domain and see what happens. Remember how we just copied before --well this time it is even easier. As soon as you make a sketch on your Investigation Sheet, take the cursor and go back to the commands before the graph. We learned in lesson 1 that {x,-10,10} determines the domain of our graph. Remembering that we are headed to infinity(to find the limit we want to go big time!) so let's increase this to {x,-100,100} by just going up there and changing it.
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3. Son of a gun as Dr. Uhl says these huggers are "sharing ink". Just for fun let's try {x,-1000,1000}.
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And this time they really "share ink"--that is if you did your graph correctly. But the big question is --who wins??? That is which one of the original curves do the others seem to mimick?If you agree that it is y = xÛ then we can say that xÛ is the dominant term--that is that in g(x) and h(x) the xÛ term dominates the other terms. So as x gets really large all three of these graphs look like the graph of what?
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4. Now let's repeat this for three new functions, y = xÜ , y = xÜ - 3xÛ + 5x - 4 y = xÜ - 6x . Don't forget to clear x,f,g, and h. Then define the three functions.
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Clear[x,f,g,h]
f[x_]= x^3
g[x_] = x^3 - 3x^2 + 5x - 4
h[x_] = x^3 - 6x
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5. Let's continue by copying our graphing commands from before and repeat as we did before---That is begin with a domain of -10 to 10 , then 100 to -100, and last -1000 to 1000.
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6. As a result of the exercise above, please record on your Investigation Sheet what term seems to be dominant in each of these functions.
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Note: In exercise 5 would we have had to go as far as the last domain to learn what we wanted to know? In Calculus and Mathematica we learn a term, "global scale" that stems from this situation. What do you think the meaning of "global scale" might be?
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7. Continue this with the following sets of functions (our Mighty Mac will not be afraid of these even though all would agree they are a bit messy):
A. y = xÝ, y = xÝ - xÛ + 8 , y = xÝ - 7xÜ + 2 xÛ -5 + 7
B. y = xà, y = xà-5xÝ + 13x Ü -26xÛ + 34 x - 728 , y = xà-5xß + 3xÞ + 8xÝ -9xÜ+11xÛ+27x - 95
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8. Now let's try some conclusions.
A. In each of the functions in exercise 7, what term seems
to be dominant?
B. If y = x Þ- 4xÝ + 6xÜ -4xÛ + 8x - 3 , what is the
dominant term?
C. Go to your Investigation Sheet and circle the dominant
term of each function listed there.
D. In your own words please tell me what this would mean
in terms of the limit of these functions as x gets very
large.
Bonus: Just for fun, you may go back to any one of the sets of curves above and add another(j(x)) to the set and begin the graph---feel free to be creative with your new function.--Is the end result still the same?Make a sketch on your Investigation Sheet.
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Exploring the Growth of Different Functions
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Lesson 3
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Here we go again with another mathematical investigation. This time we are going to become familiar with other functions that are not necessarily polynomial. Some of these are much more "dominant" than others. Lesson 4 will be an investigation of the comparative "dominance" of these fellows by speculation and investigation with our Mighty Mac, but for now we just would like to know each when we see it.
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l. Let's begin by looking at a function that is the square root of x.
Remember how we use Mathematica to graph. Because square root is involved be sure to think carefully about the domain of x.
Also just for fun let's call this plot oblongplot.
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Clear[f,x]
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f[x_]= x^.5
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Now you write the command to plot oblongplot.
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2. Let's look to see if this is what happens if we alter it a little. How about y = 3Žx ?
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3. Not much change is there? Because our Mac is so easy to use, we might just make that coefficient 100. Now we can feel pretty
comfortable about the nature of this fella!!! In your own words, describe the "growth" of this curve on your Investigation Sheet. Include in your answer why you think this seems like a logical picture for this plot. (As we make comparisons with other functions, we'll see why we need to be able to put this in words--also we can guess why we call it oblongplot.)
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4. Next out in the forest we have our friend "log". Let's graph y = log[x]. Our first plot was oblongplot, now let's make this one palestineplot and call it g(x) instead of f(x), so these will be different. If you don't remember how to put this in computer, let me know. For our friendly software package, Mathematica Log means the natural log or what we usually refer to as "ln". Also you will need to watch the domain---think a minute about what we know about the log function and what it can have as a domain. If you happen to get "red lined" the problem is probably with the domain. Describe it's growth on your Investigations Sheet.
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5. Now let's plot log, and square root on the same axis---just to be able to see the difference in growth(remember that has to do with what happens as x gets very big or maybe the word limit).
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Good for us! We have the functions in. Now let's try to graph them. This time instead of rewriting the plot command and doing both, we can use a new command, Show. If we tell our machine to Show{oblongplot, palestineplot}, this will give us both graphs and we'll be in good shape. You will need to go back and activate each of the two plots and then use Show---a little quick trick from the world of Mathematica.. On your Investigations Sheet, please record a sketch of each of these and also record which one appears to be dominant at this point.
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6. We might try hutsonvilleplot which happens to be a the sin function from the world of trigonometry. It now becomes h(x), Let's plot these three together. Since their colors are orange and block, let's try orange for h(x). Record your observations about the growth of each of these three.
7. Just a minute and let's think where we are---we've looked at logs, square roots, and now, let's look at some power growth and we'll call it robinsonplot. Let's plot a very friendly polynomial of your choosing, which now becomes j(x).
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Whew, we knew that robinson was good, but that growth is the best yet. Record a sketch of it on your Investigation Sheet and indicate how its growth compares with the rest of the county.
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8. Remember exponential growth--our old friend from the national debt. This time your task is to plot some real number raised to the x power(7ô would be fine--and let's call it crawfordplot). Give it the color of your choice--just not one that we've already used. Sketch the graph of 7ô and see how it grows. Describe its growth on your Investigation Sheet.
9. For our grand finale--we will plot each town and the whole county on our map(graph). The domain again is very important, because of some of the functions we need to be careful--experiment but be sure to look at x from 1 to 10. Remember we need to look at x as time goes on to get a true picture. Write a short paragraph describing and contrasting the graphs of all. Be sure to include how the picture changes as x gets larger. For a bonus point or two, explain the significance of the naming of these graphs.
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Investigation Sheets
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Investigation Sheet
Lesson 1
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l. Record a sketch of y = xÛ.
2. Record a sketch of y = xÜ and y = xÝ.
3. Now record each of the functions for y = xÞ through xÚâ.
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2:0,0;27,1;142,-1;
2:1,24,16,CalcMath,0,12,0,0,0;1,24,16,CalcMath,1,12,21845,21845,21845;
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4. A. Are the sketches of some similar?
B. Do some seem to fit in the same group?
C. Why might we call one group "odd" and the other "even"?
D. How do they differ?
E. Describe each group in your own words.
5. Record the sketches of the "negative functions" and compare to
the positives functions.
BONUS:
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Investigation Sheet
Lesson 2
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1. Computer activity.
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2. Make a sketch for each of three domains.
3. A. Which one is mimicked by the others?
B. So as x gets really big all three looks like the graph of which one?
4. Computer Activity
5. Repeat exercise two for the functions in exercise 4.
6. Repeat exercise three for the function in exercise 4.
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7A. Repeat exercises 2 and 3 for these functions.
B. Repeat exercises 2 and 3 for these functions.
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8. A. In each of the functions in exercise 7, what term seems to be dominant?
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B. Dominant term is :
C. Circle the dominant term.
l. y = xÝ - 6xÜ + 5xÛ + 3x - 7
2. y = 15x Û - 8 x + 756
3. y = 95 x¡ + 36 x à - 8x + 456
4. y = 25 + 5x -7xÛ + 8xÜ + 9xÝ - 13 xÞ
5. y = 17x + 25 - 8xÝ + 3 xÜ + 5xÛ
D. The limit as x gets very large is
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Bonus: Sketch
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Investigation Sheet
Lesson 3
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l. Sketch of y =Ž x .
2. Sketch.
3. Description and explanation.
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4. Growth description.
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5. Sketch and circle the dominant one.
6. Sketch and observations about the growth of all three.
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7. Sketch and observations.
8. Sketch and description of growth.
9. Paragraph of explanation.
BONUS:
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