Putting It All Together

Hey guys, This is Angelo!
This blog is going to talk about how to put all the transformations, translations, reflections and stretching together. 

Let's start with the equation.

 "a" and "d" are both outside and must read as it is.

 "b" and "c" are both inside and must be read as opposites.

Now that you know the equation, let's work on some examples.

Given the following graph of f(x), graph the transformation "-f(x+1)-3".

This transformation formula has just about everything. There's a left-shift of one (the "+1" inside), a move-down by three (the "–3" outside), and a flip-upside-down (the "minus" sign out front). And, worse yet, I have no formula for f(x), so I can't cheat; I have to do the transformation.

Let's work on the first point (-3,-2).

I'll shift the point by 1 unit, making it (-4,-2).
Then I will flip the point over the x-axis making it (-4,2).
I will then move the point down 3 units to (-4,-1).

We're done 1 out of 4 points.
On to the next point.

I'll shift the point by 1 unit, making it (-3,-4).
Flip the point across the x-axis making it (-3,4) then making the point go down 3 units making it (-3,1).

2 points out of 4 done.
Repeat the steps for the next 2 points which I will skip for length purposes.

...And you should end up with this. 

This is pretty much how you do everything all at ones. I hope you enjoyed reading this and I hope it actually helped you out! I'm sorry I don't have much funny pictures to put on here, maybe next time I'd think of more clever way of putting pictures in. 


- Angelo signing out, Peace out A-Town.

Reflections and Stretches

Hii guys
this is a blog about our lesson on reflections and stretches

Reflections:
There are two types of reflections, the reflection over the x-axis and the reflection over the y-axis.

• The reflection over x-axis, makes the y values negative
• The reflection over y-axis, makes the x values negative

Can A Mirror Line Be Vertical? Yes.

In fact Mirror Lines can be in any direction. Imagine turning the photo at the top in different directions ... ... the reflected image is always the same size, it just faces the other way:

A reflection is a flip over a line

Reflection Transformation: Example 1

The coordinates of LMNO are: (-7,5); (0,5); (-2,1); (-5,1) LMNO is reflected over the X-axis making the coordinates of L1M1N1O1: (-7,-5); (0,-5); (-2,-1); (-5,-1) Note how the x-coordinates remain the same but the y-coordinates change to their opposite integer (i.e. the sign changes). This is always the case with reflections over the X-axis.

Reflection Transformation: Example 2

The coordinates of PQSR are: (-8,-3); (-2,-3); (-2,-6); (-8,-6) PQSR is reflected over the Y-axis making the coordinates of P1Q1S1R1: (8,-3); (2,-3); (2,-6); (8,-6) Note how the y-coordinates remain the same but the x-coordinates change to their opposite integer (i.e. the sign changes). This is always the case with reflections over the Y-axis.

y = -f(x)	Reflects it about x-axis
y = f(-x)	Reflects it about y-axis

y = -f(x) makes y opposite

y = f(-x) makes x opposite

what is the reflected ordered pair if  (5,-2) is reflected on both axis

• x-axis (5,-2) -> (5,2)
• y-axis (5,-2) -> (-5,-2)

OR

1. Measure from the point to the mirror line

2. Measure the same distance again on the other side and place a dot.

3. Then connect the new dots up!

1.)                    2.)                 3.)

Stretches:

There are two types of stretches: the vertical stretches and the horizontal stretches

Vertical stretches:

y= af(x) and y= 1/af(x) stretches the entire graph vertically by a factor of |a| units.

~The coordinates of the points (x,y) translates to (x,ay) and (x,1/ay). The value of x does not change.

Horizontal stretches:

y= f(b x)b and y= f (1/b x) stretches the entire graph horizontally  by a factor of |b| units

~the coordinates of the point  change to (x,y) to (1/b x, y) and (bx,y). The  value of y does not change.

• f(x)= f(2x)

multiply x values by 1/2 or divide by 2

• f(x)= f (1/2x)

multiply x values by 2 or divide by 1/2

example:

The graph of f(x) = 2 sin 2x is a stretch of sin x by scale factor 2 in the y-axis, and a stretch scale factor 1/a in the x-axis.

f(x) = sin x

The original function...

f(x) = 2 sin x

Stretches the original function in the y axis...

Horizontal Stretch A horizontal stretching is the stretching of the graph away from the y-axis.  A horizontal compression is the squeezing of the graph towards the y-axis.If the original (parent) function is y = f (x), the horizontal stretching or compressing of the function is the function f (ax). if 0 < a < 1 (a fraction), the graph isstretched horizontally by a factor of a units.

if a > 1, the graph is compressed horizontally by a factor of a units. if a should be negative, the horizontal compression or horizontal stretching of the graph is followed by a reflection of the graph across the y-axis. Vertical Stretch A vertical stretching is the stretching of the graph away from the x-axis. A vertical compression is the squeezing of the graph towards the x-axis.If the original (parent) function is y = f (x), the vertical stretching or compressing of the function is the function a f(x). if 0 < a < 1 (a fraction), the graph iscompressed vertically by a factor of a units. if a > 1, the graph is stretched verticallyby a factor of a units. If a should be negative, then the vertical compression or vertical stretching of the graph is followed by a reflection across the x-axis.

End =)

Sources:

• http://www.biology.arizona.edu/biomath/tutorials/transformations/reflections.html
• http://www.mathsisfun.com/sets/function-transformations.html
• http://www.regentsprep.org/Regents/math/algtrig/ATP9/funclesson1.htm

Test

I am ready for the test mr piatek:)

Translations

Hi Guys! I found some additional information about Translations and I hope it helps! ;)

Introduction to Translations and Special Functions

Calculus students work with only a few families of functions—absolute value, n th root, cubic, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions. Two or more of these functions might be combined arithmetically or by using function composition. In this chapter, we will look at the absolute value function (whose graph is in Figure 5.1), the square root function (whose graph is in Figure 5.2), and the cubic function (whose graph is in Figure 5.3).

Fig. 5.1

Fig. 5.2

Fig. 5.3

We will also look at how these functions are affected by some simple changes. Knowing the effects certain changes have on a function will make sketching its graph by hand much easier. This understanding will also help you to use a graphing calculator. One of the simplest changes to a function is to add a number. This change will cause the graph to shift vertically or horizontally.

Shifting Left and Up on a Graph - Adding 1 to x and y

What effect does adding 1 to a function have on its graph? It depends on where we put “+1.” Adding 1 to x will shift the graph left one unit. Adding 1 to y will shift the graph up one unit.

• y = | x + 1|, 1 is added to x , shifting the graph to the left 1 unit. See Figure 5.4.

Fig. 5.4

For the graphs in this chapter, the solid graph will be the graph of the original function, and the dashed graph will be the graph of the transformed function.

• y = | x | + 1, 1 is added to y (which is | x |) shifting the graph up 1 unit. See Figure 5.5.

Fig. 5.5

•  , 2 is added to x , shifting the graph to the left 2 units. See Figure 5.6.

Fig. 5.6

•  , 2 is added to y (which is  ) shifting the graph up 2 units. See Figure 5.7.

Fig. 5.7

Shifting Right and Down on a Graph - Subtracting 1 from x and y

Subtracting a number from x will shift the graph to the right while subtracting a number from y will shift the graph down.

• y = ( x − 1) ³ , 1 is subtracted from x , shifting the graph to the right 1 unit. See Figure 5.8.

Fig. 5.8

• y = x ³ − 1, 1 is subtracted from y (which is x ³ ), shifting the graph down 1 unit. See Figure 5.9.

Fig. 5.9

Stretching and Compressing a Graph - Multiplying x and y by 1 and -1

Multiplying the x -values or y -values by a number changes the graph, usually by stretching or compressing it. Multiplying the x -values or y -values by −1 will reverse the graph. If a is a number larger than 1 ( a > 1), then multiplying x by a will horizontally compress the graph, but multiplying yby a will vertically stretch the graph. If a is positive but less than 1 (0 < a < 1), then multiplying xby a will horizontally stretch the graph, but multiplying y by a will vertically compress the graph.

• , the graph is horizontally compressed. See Figure 5.10.

Fig. 5.10

• , the graph is vertically stretched. See Figure 5.11.

Fig. 5.11

• , the graph is horizontally stretched. See Figure 5.12.

Fig. 5.12

• , the graph is vertically compressed. See Figure 5.13.

Fig. 5.13

For many functions, but not all, vertical compression is the same as horizontal stretching, and vertical stretching is the same as horizontal compression.

Reflecting the Graph Across the x and y-Axis

Multiplying the x -values by −1 will reverse the graph horizontally. This is called reflecting the graph across the y-axis . Multiplying the y-values by −1 will reverse the graph vertically. This is calledreflecting the graph across the x-axis .

Fig. 5.14

Fig. 5.15

When a function is even, reflecting the graph across the y -axis does not change the graph. When a function is odd, reflecting the graph across the y -axis is the same as reflecting it across the x -axis.

We can use function notation to summarize these transformations.

y = af ( x + h ) + k

• If h is positive, the graph is shifted to the left h units.
• If h is negative, the graph is shifted to the right h units.
• If k is positive, the graph is shifted up k units.
• If k is negative, the graph is shifted down k units.
• If a > 1, the graph is vertically stretched. The larger a is, the greater the stretch.
• If 0 < a < 1, the graph is vertically compressed. The closer to 0 a is, the greater the compression.
• The graph of − f ( x ) is reflected across the x -axis.
• The graph of f (− x ) is reflected across the y -axis.

The graphs below are various transformations of the graph of y = | x |.

Fig. 5.16

Fig. 5.17

Fig. 5.18

Fig. 5.19

Here are some examples:

The graph of y = f ( x ) is given in Figure 5.20. Sketch the transformations. We will sketch the graph by moving the points (−4, 5), (−1, −1), (1, 3), and (4, 0).

Fig. 5.20

• y = f ( x + 1) − 3
   Table 5.1

  Original point	Left 1 x − 1	Down 3 y − 3	Plot this point
  (−4, 5)	−4 − 1 = −5	5 − 3 = 2	(−5, 2)
  (−1,−1)	−1 − 1 = − 2	−1 − 3 = −4	(−2,−4)
  (1, 3)	1 − 1 = 0	3 − 3 = 0	(0, 0)
  (4, 0)	4 − 1 = 3	0 − 3 = −3	(3,−3)

  
  
  
  Fig. 5.21
• y = − f ( x )
  Table 5.2

  Original point	x does not changex	Opposite of y      −y	Plot this point
  (−4, 5)	−4	−5	(−4,−5)
  (−1,−1)	−1	−(−1) = 1	(−1, 1)
  (1, 3)	1	−3	(1,−3)
  (4, 0)	4	−0 = 0	(4, 0)

  
  Fig. 5.22
• y = 2 f ( x − 3)
  Table 5.3

  Original point	Right 3 x + 3	Stretched 2 y	Plot this point
  (−4, 5)	−4 + 3 = −1	2(5) = 10	(−1, 10)
  (−1,−1)	−1 + 3 =2	2(−1) = −2	(2,−2)
  (1, 3)	1 + 3 = 4	2(3) = 6	(4, 6)
  (4, 0)	4 + 3 = 7	2(0) = 0	(7, 0)

  
  
  
  Fig. 5.23
• 
  Table 5.4

  Original point	Opposite of x −x	Compressed and up 2  y+ 2	Plot this point
  ( − 4, 5)	− (−4) = 4	(5) + 2 =	(4,  )
  ( − 1, − 1)	−(−1)= 1	(−1) + 2 =	(1,  )
  (1, 3)	− 1	(3) + 2 =	(− 1,  )
  (4, 0)	− 4	(0) + 2 = 2	(−4, 2)

  
  
  
  Fig. 5.24
  
  
  
  
  - END - 
  
  
  
  
  SOURCES: 
  http://www.education.com/study-help/article/pre-calculus-help-translations-special/#heading1
  http://www.education.com/study-help/article/pre-calculus-help-translations-special/?page=2#heading2