View Lab Activity as it unfolds in the webpage area below: There are many steps... so read each instruction carefully and look at the accompanying photographs carefully for additional insight.
Step 1:
graphing y = arcsinx
Carefully tear out a single piece of standardnotebook paper.
Step 2:
Start with a regular 8.5 x 11 inch paper in portrait orientation as shown in the picture...
Step 3:
Next fold the bottom half up toward the top and crease it soundly to produce a very evident fold as shown in the picture...
Step 4:
Next fold the right half over toward the left and crease it soundly to produce a very evident fold as shown in the picture...
Step 5:
Next unfold the paper and reorient it in its original position. Notice the evident folds you have created should divide the paper into four equal quadrants as shown in the picture...
Step 6:
Next fold the paper over left top corner down at a diagonal. BE CAREFUL TO FOLLOW THESE DIRECTIONS CAREFULLY
Make sure your diagonal fold intersects with the (0,0) formed by the intersection of your other folds
When you are folding the diagonal, look at the creases at the bottom center (near the middle notebook binder hole) for guidance
These creases need to be perfectly aligned one atop of the other
When you do these three things you will notice the two offset edges of the paper at the bottom are PARALLEL...insure this is true BEFORE you crease the diagonal fold. When all is aligned correctly the folded paper shall look as shown in the picture...
Step 7:
Next unfold the paper and reorient it in its original position. Notice the evident folds you have created should divide the paper into four equal quadrants with the diagonal fold making a perfect 45 degrees sloping in a positive direction from the 3rd into the 1st quadrant as shown in the picture...
Step 8:
Next draw in the "axis lines" and label them with the respective letters as shown in the picture...
Step 9:
Continuing to draw in the "axis lines" and label them with the respective letters as shown in the picture...
Step 10:
Next draw in the diagonal crease with a long dashed, thin line and label it y = x as shown in the picture...
Note: Considering y = mx + b
the slope "m" is 1 up and 1 over (rise over run)... thereforem = 1
The line intersects the axis at (0,0) since "b" is the y-intercept... therefore b = 0
so we have y = 1x + 0 which simplifies to the above formula y = x as labeled on your artifact
Step 11:
Now we will mark a unit on the "y axis" (one to two line spaces on the notebook paper is a good length)...
as shown in the picture...
Step 12:
Using the unit on the "y axis" as a guideline for all other units we will be creating...
...we then record that distance on a piece of scrap paper as shown in the picture...
Step 13:
Next measure down from the horizontal axis to locate the units mark on the negative part of the "y axis"...
....using the distance we recorded on the piece of scrap paper as shown in the picture...
Step 14:
Now continue to mark the units on the "y axis" using the distance we recorded on the piece of scrap paper as shown in the picture...
Step 15:
Next mark the units on the "x axis" using the distance we recorded on the piece of scrap paper as shown in the picture...
Step 16:
Now continue to mark the units on the "x axis" using the distance we recorded on the piece of scrap paper as shown in the picture...
Step 17:
Now continue to mark the units on the "x axis" in the negative direction, using the distance we recorded on the piece of scrap paper as shown in the picture...
Step 18:
Next reorient you paper and you now have an axis with graduations ready to graph.
Prepare to graph as shown in the picture...
We will start with: y = sin x
Step 19:
Since sine function starts at (0,0) and moves upward as we go to the right toward 90 degrees (pi/2 in radians)...
...we will make a light upward extension from the first tick (pi/2) to the right on the "x axis" until it reaches the "y value" height of the amplitude of our function
(Amplitude "A" is represented by the single tick height on the "y axis")
Step 20:
Now connect (0,0) to the light upward extension you just created from the first tick (pi/2) to the right on the "x axis" at the amplitude "A" high, with a smooth curve representing the first quarter of the wave shape as shown in the picture...
Step 21:
Continuing to sketch the wave shape we shall gently turn back toward the "x axis" and intersect with it at the second tick to the right (pi) as shown in the picture...
This represents half of the full wave shape.
Step 22:
Now extend the wave smoothly downward and connect to the light down ward extension you can created from the third tick (3pi/2) to the right on the "x axis" at the amplitude "-A" high.
This smooth curve represents the three quarters of the wave shape as shown in the picture...
Step 23:
Finally continue with a smooth turn upward toward the "x axis" intersecting at the fourth tick to the right, which represents a full rotation around the "Unit Circle" (2pi). You have just created the entire wave we will be considering on the positive "x" side...
Your function should look basically the same as shown in the picture...
Step 24:
Continue your smooth wave function to the left toward the negative "x axis"
intersect with each tick as appropriate to create a representation of a full "negative" rotation around the "Unit Circle" (-2pi).
You have just created the entire wave we will be considering on the across two full periods...
Your function should look basically the same as shown in the picture...
Step 25:
Now burnish, or darken your wave line with your pencil. Get a lot of graphite onto the line (it might thicken a bit but that is OK)
When you are finished you should have a bold, dark wave line as shown in the picture...
Step 26:
Next we shall refold our paper at the diagonal fold line, crossing the upper left corner across to the lower right as shown in the picture...
Step 27:
If we examine carefully we can detect the outline of our burnished line through the paper.
I am concerned with the wave form seen here as shown in the picture...
Step 28:
Next we will burnish firmly on the back of our paper (keep it folded when you burnish) tracing the wave form as seen in its entierty as shown in the picture...
Step 29:
Do every part as you inspect the paper...
Notice the little piece of the wave up near the diagonal fold at the top right portion of the picture as shown here...
Step 30:
Next we will make sure all of the wave function on this side is complete, in preparation for flipping the paper and repeating this process on the other side...
Step 31:
Next we flip our paper right to left so the diagonal fold line is now to our right.
notice I can now see the remaining portion of the wave through the paper as shown in the picture...
Step 32:
Inspecting closer we see there is the large portion as well as a small piece near the lower right of our picture...
Step 33:
Next we will burnish firmly on the back of our paper (keep it folded when you burnish) tracing the wave form as seen in its entierty as shown in the picture...
Step 34:
The completed burnish should look as shown in the picture...
Step 35:
Now we are ready to examine the outcome of our burnishing efforts...
Step 36:
Opening the page with the back facing us we see we have replicated the function...
But interestingly it is basically "flipped" upside down. This represents the function in a new way...
y = -sin x
Step 37:
However, when we examine the from we see a ghostly impression of our effort along the "y axis" moving vertically up and down said axis as shown in the picture...
Step 38:
Examining closely might help you to see this line better...
Step 39:
Notice its position relative to the "y-axis is very similar to the position of our original function along the "x axis"...
Step 40:
What we have created along the "y axis" is the "inverse sine x graph" or sometimes referred to as "arcsin x"
So to help us distinguish we label or original as y = sin x
and our new function as y = arscin x...
Step 41:
With this process well described...the challenge is now for you to use your background knowledge of graphing all trig functions to create a sheet for each and transfere the Inverse functions of all trig functions.
Including this one, we will have six sheets in all: