6a - Lab #6 Symmetry of functions and function modifications.sagews

##Lab 6 Symmetry of functions and their modifications

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First we'll look at the three types of symmetry--**x-axis, y-axis and origin**.  Recall that a function that is symmetric with respect to the y-axis is classified as an **even** function and one that is symmetric w.r.t. the orgin is classified as an **odd** function.  A relation that is symmetric w.r.t. the x-axis is NOT a function as it obviously fails the vertical line test!!  You will also see a new *plot* command called **implicit_plot

First we'll look at the three types of symmetry--x-axis, y-axis and origin.

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y-axis symmetry means that all points on the function on one side of the x-axis have mirror images on the other side.  Numerically, this means that the opposite of the x-value, *-x*, produces the same function- or y-value.  Examine and execute the command block below to see if the $f(x)=x^4-3x^2$ is symmetric w.r.t. the y-axis and thus even.
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x,y=var('x,y')
f(x)=x^4-3*x^2
f(x)==f(-x)
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Since the sides of the equation are identical, we can assume that the function has x-axis symmetry and thus is even.  You create a command block to now see if the function, $f(x)=5x^3-7x$ is symmetric w.r.t. the origin by setting f(-x)=-f(x)$.  In other words, replacing x with its opposite produces the opposite y-value.  Graphically, a function point that appears anywhere has a mirror image after rotating around the origin 180 degrees.
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Now let's look at implicitly defined functions or relations like $x^2+3y^2=18$.  Study the command block below to see how the new graphing command works.  I first run the numerical tests for x-axis and origin symmetry.
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x,y=var('x,y')
f(x,y)=x^2+3*y^2-18#this is a function of 2 variables created by setting one side equal to zero!
show(f(x,y))
show(f(x,y)==f(-x,y))#testing for x-axis symmetry
show(f(x,y))==f(-x,-y)#testing for symmetry w.r.t. origin
#here comes the new implicit plot command:
implicit_plot(f(x,y),(x,-5,5),(y,-5,5),axes='true')# you can other plot options to spruce up the plot as well.  The three required arguments for this command are in this order:  (1)a function of 2 variables, (2)horizontal variable with its range, (3)vertical variable with its range.
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The relation we looked at above was not a function.  But it does have both x-axis and y-axis symmetry.  It does not have symmetry w.r.t. the origin.  Practice your skills as you test these for symmetry and create plots.  
1.  $f(x)=5x^4-3x^2+4$
2.  $g(x)=x^(1/3)-x/4$
3.  $x^2-3xy+y^2=0$  Use the implicit_plot command here.

The relation we looked at above was not a function. But it does have both x-axis and y-axis symmetry. It does not have symmetry w.r.t. the origin. Practice your skills as you test these for symmetry and create plots.

1. $f(x)=5x^4-3x^2+4$

2. $g(x)=x^(1/3)-x/4$

3. $x^2-3xy+y^2=0$ Use the implicit_plot command here.

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Let's investigate modifications to the graphs of functions.  One general rule to keep in mind is this: **When you want to alter a graph vertically to adjust the function's output.  Horizontal modifications are achieved by adjusting the input(x-variable)**.
It's going to take a little extra code to get the job done.  You'll also be introduced to the idea of a *for* loop that does the same thing with different values of a list repeatedly.  The blocks of code below will eventually result in single graphs of multiple functions graphed together.  I am using two separate blocks since I have to use the answers from the first block to create the second block of commands.  There is a more direct way but we don't want to go there just yet!  The first two blocks will demonstrate how to shift a function **vertically** and the second two will shift the same function horizontally.  The final two blocks will shift a function both vertically and horizontally simultaneously.
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vert_adjustment=[-2,-1,1,2]#list of values for shifting
f(x,v)=x^2+v#This is just the basic x-squared function just using two variables
#the loop sequence that generates for different function expressions:
for v in vert_adjustment:
    f(x,v)

x^2 - 2 x^2 - 1 x^2 + 1 x^2 + 2

#I use copy and paste with solutions above to create single plot of the list of 4 functions above.  You execute to see plot
plot([x^2,x^2-2,x^2-1,x^2+1,x^2+2],(x,-2,2))#notice that the brackets indicate a list!
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#Now to shift horizontally. . .
hor_adjustment=[-1,-.5,.5,1]
f(x,h)=(x-h)^2
for h in hor_adjustment:
    f(x,h)

(x + 1)^2 (1.00000000000000*x + 0.500000000000000)^2 (1.00000000000000*x - 0.500000000000000)^2 (x - 1)^2

plot([x^2,(x + 1)^2,(1.00000000000000*x + 0.500000000000000)^2,(1.00000000000000*x - 0.500000000000000)^2,(x - 1)^2],(x,-2,2))#you execute to see plot
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Your turn!  You modify the simple cubic, $x^3-x$, in three ways: (1)stretch/squeeze it vertically by the two different factors 3 and 1/3.  The final plot should consist of the original function and the two modified versions;  (2)stretch/squeeze the same cubic horizontally by the same two factors;  (3)Reflect or flip the cubic over the x-axis by negating the function.  This last item won't involve the use of the *for* loop.

Your turn! You modify the simple cubic, $x^3-x$, in three ways: (1)stretch/squeeze it vertically by the two different factors 3 and 1/3. The final plot should consist of the original function and the two modified versions; (2)stretch/squeeze the same cubic horizontally by the same two factors; (3)Reflect or flip the cubic over the x-axis by negating the function. This last item won't involve the use of the for loop.

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