1/10/13 and 1/11/13 (Wasita)

1/10/13 - Thursday

On Thursday we went over homework (p. 136: 70-74 and another worksheet), wherein we were given formulas and had to plot and sketch transformations that matched up with the given transformed functions.

The problems consisted of the following transformations in several combinations:
*Horizontal shift
*Vertical shift
*Reflection over the x-axis
*Reflection over the y-axis
*Vertical stretch
*Vertical shrink
*Horizontal stretch
*Horizontal shrink
Here I will give an example of each transformation by itself:
Horizontal Shift:
f(x-2)
*Since whatever is in the parentheses is the opposite of what it actually is, similar to the vertices of conic sections, the ‘x-2’ means that it is a horizontal shift by 2 units to the right/positive.
*To plot this, find all of the coordinates from your given baseline formula, and then plug in all of your x-values.
*Say you had the point (1,2). Plug 1 into the f(x-2) formula. You will get 3. Since the y is not adjusted nor transformed in the new formula, the y-value, 2, stays the same. As a result, your (1,2) from your baseline formula turns into (3,2) with the transformed formula.
*All of your x-value points undergo this change. Here is a visual example of a horizontal shift by 2 units to the right: In this example, the blue line is the original function y=x^2 and the red line is the transformed function y=(x-2)^2. Vertical Shift:
f(x)-2
*Since whatever is outside of the parentheses contributes to what the actual function looks like, we do not do the opposite operation of what is given, unlike for x-value/horizontal changes.
*That means that the -2 really is a vertical shift by 2 units down/towards negative.
*If you had the point (1,2), the 2, when plugged into the formula [f(x)-2] comes out as a 0. The x-value (1) stays the same, giving you the new point (1,0).
*All of the y-values from your baseline/given formula undergo this change. Here is a visual example of a vertical shift by 2 units down: In this example, the blue line is the original function of y=x^2 and the red line is the transformed function y-2=x^2.
Reflection Over the x-Axis:
-f(x)
*Because there is a negative in front of the f, that means the formula is a reflection over the x-axis.
*This means that you take all of your y-values in your coordinates and multiply them by -1, or make them their opposite integers.
*If you had the point (1,2), after multiplying the y-value, 2, by -1, you get -2. Your new point would then be (1,-2). Here is a visual example of a reflection over the x-axis: In this example, the blue line is the original function: y=|x| and the red line is the transformed function: -y=|x|.
Reflection Over the y-Axis:
f(-x)
*Because there is a negative in front of the x, that means that the x/all of the x-values get flipped. You can also think of it as multiplying all of the x-values by -1.
*Plugging the point (1,2) into the formula would give you (-1,2).
*All of your x-values undergo this change. Here is a visual example of a reflection over the y-axis:
In this example, the blue line is the original function y=.25x^3+.5x^2 and the red line is the transformed function y=.25(-x)^3+.5(-x)^2.

***Another example of a reflection of the y-Axis from our worksheet (one that I had trouble grasping at first, actually) is:
-g(-(x+3))+4
*You do not distribute the negative (the second one/the one in front of the parenthesis that is in front of the x) throughout the formula. What you do do is understand that the negative sign in front of the x means that the x is flipped.
*The +3 after the x, as per usual, means that the x is shifted horizontally by 3 units to the left/towards negative.
*Granted, all of the other transformations act like usual (the -g means that the y-values are flipped and the +4 means that you add 4 to every y-value).
*Plugging in the point (1,2) into the equation would give you (2,2)
Here is a step-by-step explanation of the plugging in of this point:
*Your x-value, 1, is first multiplied by -1 → -1
*+3 is then added to the -1 → 2, your resulting x-value.
*2 is flipped/multiplied by -1 → -2
*+4 is then added to the -2 → 2, your resulting y-value. Vertical Stretch:
2f(x)
*The 2 in front of the f means that all of your y-values get multiplied by the coefficient 2.
*Plugging in the point (1,2) would give you (1,4).
*All of your y-values undergo this change and your x-values remain the same.  Here is a visual example of a vertical stretch: In this example, the blue line is the original function y=(sin x) and the red line is the transformed function y=4(sin x).
Vertical Shrink:
½f(x)
*The ½ in front of the f means that all of your y-values get halved. You can also think of this as multiplying all of your y-values by ½ or .5.
*Plugging in the point (1,2) would give you (1,1) Here is a visual example of a vertical shrink: In this example, the blue line is the original function y=5(sin x) and the red line is the transformed function y=2(sin x).
Horizontal Stretch:
f(1/2x)
*The ½ in front of the x means that all of your x-values get multiplied by 1/2’s reciprocal: 2.
*In other words, all of your x-values get multiplied by 2/get doubled.
*Plugging in the point (1,2) would give you (2,2). Here is a visual example of a horizontal stretch: In this example, the blue line is the original function y=x^3 and the red line is the transformed function 50y=x^3. Horizontal Shrink:
f(2x)
*The 2 in front of the x means that all of your x-values get divided by 2. *Plugging in the point (1,2) would give you (1/2,2). Here is a visual example of a horizontal shrink: In this example, the blue function is the original function 8y=x^3 and the red function is the transformed function y=x^3. We then worked on a worksheet in class, which was then assigned for homework. The worksheet gave us transformations (where more than 1 transformation occurred) and we had to graph/sketch the transformations ourselves. That assignment basically pulled together everything that we've been working on. Homework from the book, which contained similar exercises, was also assigned: p. 137: 86-89, 107, 109.


1/11/13 - Friday On Friday we went over the homework (p. 137: 86-89, 107, 109) assigned from last class. In the homework we were given the function f(x)=x^3-3x^2, which was our baseline function. In each problem, we were then given a graph which contained a function, which was the "transformed function" that derived from the baseline function. We had to identify the transformations and come up with the formulas for each graph. Now, I will do an example of such transformations, that being problem number 86 from the homework: You are given the formula f(x) = x^3 - 3x^2 (pictured left) as your baseline function. *You can see that the points from 86's graph show a vertical shrink by 1/2. (-1,-4) on the baseline function turned into (-1, -2), (2,-4) on the baseline function turned into (2,-2), and so on and so forth. *86's function is therefore 1/2f(x) = 1/2(x^3 - 3x^2) *Because the y-value is what is being halved, 1/2 goes in front of the f on the left side as well as in front of the first parenthesis on the right side. On the worksheet, in each problem we were given a graph of a function [f(x)]. Also in each problem was a transformation that we had to sketch on the same coordinate axes. These problems helped us visually see multiple transformations in one formula and helped us learn how to plot and graph transformations. Now, I will do 8a from the said worksheet: Here, I recreated the graph from 8a. The blue line represents the given function f(x). The transformation formula given was 2f(x+3), meaning we must take all of the blue line's points and adjust them accordingly. The +3 after the x means that there is a horizontal shift by 3 units to the left and the 2 in front of the f means that there is a vertical stretch. Specifically, this means that every x-value must have 3 subtracted from it and every y-value must be multiplied by 2.
Baseline Function's Points Transformation's Points (-6,-2) (-9,-4) (-4, 3) (-7,6) (-1,3) (-4,6)
(2,2) (-1,4) (4,-4) (1,-8) (10,-3) (7,-6) This is what the transformation looks like once plotted and graphed:  As you can see, there is an apparent horizontal shift by 3 units to the left and a vertical stretch, just like the formula indicated. After discussing the homework we started a graded problems worksheet in class. It covers symmetry and transformations and is due this Monday. Anyways, I hope this post helped. -Wasita Links to check out if you are somehow still confused about symmetry and transformations involving functions: http://math.kennesaw.edu/~sellerme/sfehtml/classes/math1113/transformation.pdf http://www.purplemath.com/modules/symmetry3.htm