The Task

You will be required to submit three art pieces that are created using different family of functions, each with accompanying write-ups.

In order to do so, you will need to create your own function and find its derivative. Using the sequence function in GeoGebra, you will be able to create a “family of functions,” which will transform your function multiple times, creating a mesmerizing effect. It will require lots of exploration and trial and error with the equation of your function, as well as customization with the appearance.

The Details

You will be creating three different arrangements of mathematical artwork and a write-up to accompany each piece.

 

  • Each arrangement consists of: o A family of functions

o  A corresponding family of derivatives

o  A write-up responding to the different questions posed on the next page

  • For each of your functions, you must have at least two function transformations. o For example, y = kx is unacceptable because it has only one transformation
  • However, y = kx + k or y = k sin( 1k x) + k 2 are acceptable because they have two or more transformations on the function
  • One of your arrangements must include a rational function

OVERALL CHECKLIST

You should have 3 arrangements, which means there are 6 graphs in total (3 functions and 3 derivatives)

Each arrangement should be accompanied by its own write up (3 write ups in total)

In your artwork, the all gridlines and axes should be hidden, so that the only thing visible is your family of curves

The functions used in your graphs must have at least 2 transformations One of your graphs must use a rational function

Include a title page for your final product

For each arrangement, you will be creating a write-up to accompany the artwork. In each write-up, you will ensure a professional and clear appearance, while being creative. The exact details of what the write-ups look like are up to you.

 

WRITE-UP CHECKLIST (for each arrangement)

 

A clever title for every graph

The equations used to generate the family of curves and the family of derivatives

The GeoGebra commands used for the family of curves and the family of derivatives

 

The scales you used to generate the final images An explanation of what you like about your piece

  • Why did you choose it among all the pieces you generated?
  • What about it appeals to you?

Observations about your family of curves and your family of derivatives

Questions you have about your family of curves and your family of derivatives

  • These could include things that you do not understand about your graphs A comparison of your family of curves to your family of derivatives
  • Comment on their visual similarities and differences
  • Possible explanations for these differences/similarities

Explain the parent function used to generate the family of curves and give a general outline of the transformations involved to create the artwork

The write-ups should encapsulate both the artistic and the mathematical sides of what you have created.

Graphing Sequences on GeoGebra:

In order to create a function that is graphed repeatedly with various transformations, we first must learn how to use and write sequences. We can use a function in GeoGebra to generate a sequence of numbers for us. You just need to input a rule for the sequence and a few more pieces of information.

Example 1: Let’s say we want GeoGebra to generate a sequence of numbers using the rule, , where and increases by 1 each time.

 

What is the actual sequence of numbers that should be created from the information given above?

Answer: _________________________________________________

In order to direct GeoGebra to create a sequence, we need to type the following:

Sequence[pattern,variable,start value,end value,increment number]

*Pay particular attention to the fact that Sequence has a capital S and there are no spaces between entries in your command. You should have 5 entries inside the brackets.*

What do you need to type to create the sequence given above?

Answer: ________________________________________________

Once you press enter, GeoGebra will generate the sequence and it should look like this:

If you double click on the sequence that was just generated, you can alter the original input command. Alter the previous command so that now the increment changes are by 0.5. Your output should look like this:

Check your understanding:

Without actually using GeoGebra, what will the output be for this sequence?

Answer: _________________________________________________

(Now check on GeoGebra – were you correct?)

Example 2: Without using GeoGebra, what do you think will be displayed if you were to type in the following command?

Answer: _________________________________________________

(Now check on GeoGebra – were you correct?)

Thinking about function transformations, how does the k affect the base function in the example above

In mathematics, we call the collection of all functions y = kx for all values of k , a family of functions and we call k a parameter. Of course it’s not possible to graph for absolutely all values of k (since there are infinitely many) so we tend to view families of functions by plotting some of the functions.

 

Now double click on your output list (in the input panel there should be something that says ‘list 1’) to edit your original sequence.

Change the last number, the 0.25 to 0.1 and see what happens.

Change the -5 and 5 to -10 and 10 and see what happens. Play around!

Extension

If we have the family of functions f ( x) = kx 2 + k , let’s think about what the various values of k would make the function look like.

k = a super large positive number (try 10)…

The function is y = 10 x2 +10

What does the coefficient in front of the x2  do to the parabola?

What does the +10 at the end do to the parabola?

k = a small positive number (try 0.5)…

The function is y = 0.5 x2 + 0.5

What does the coefficient in front of the x2  do to the parabola?

 

What does the +0.5 at the end do to the parabola?

  • = 0

The function is y = 0

What does the function look like?

k = a small negative number (try -0.5)…

The function is y = -0.5 x2 -0.5

What does the coefficient in front of the x2  do to the parabola?

What does the -0.5 at the end do to the parabola?

k = a super large negative number (try -10)…

The function is y = -10 x2 -10

What does the coefficient in front of the x2  do to the parabola?

What does the -10 at the end do to the parabola?

Now let’s see what this family of curves looks like!

Did that match what you thought? Try fiddling around with the scales and the colours to get a more interesting piece of work.

You can also play around with the sequence by changing what transformations are occurring.

Edit your existing sequence so it graphs these family of curves (try each one separately).

 

  • f ( x) = kx2 + k2
  • g ( x ) = kx + k
  • h( x) = kx + k 2

 

Notice for each of these family of curves, you have two transformations happening simultaneously. For example, in h( x) = kx + k 2 , you can say this family of curves is the set of lines where the steeper the slope (the more positive or more negative the k is), the higher up the lines are moved.

Try other family of curves:

  • Sequence[1/k*sin(k*x),k,-10,10,0.2]
  • Sequence[k*sec(x)+(1/k)*x,k,-10,10,0.25]

 

Try fiddling with the visual appearance of your graphs and see if you can get accustomed to changing different settings.

 

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