Chapter P: Graphing Functions

1. First, I chose the function y=lxl and would represent our "start" or original function. This is shown in BLACK.
2. Next, I performed a vertical translation. The function would be y=lxl-3, where you go 3 units down from the original function. This is shown in RED.
3. Then, I performed a horizontal translation. The function would be y=lx+4l, where you move 4 units to the left of the original function. This is shown in BLUE.
4. After, I combined both the vertical and horizontal translation, and transformed the original function into y=lx+4l-3 where the graph moved 4 units to the left of the x-axis then 3 units down the y-axis. This is shown in GREEN.
5. Then, I performed a reflection where the original function would be flipped 180 degrees (over x-axis) This is shown in PURPLE.
6. Finally, I made a transformation by combining step 4 and 5. This made the function y=-lx+4l-3 where the original function was reflected over the x-axis and moved 4 units to the left of the x-axis and 3 units down the y-axis from the original function. This is shown in BLACK.

Ch. P Transforming a Function

The parent function of the original graph is f(x)=x (as indicated with the light pencil).

The first transformation of the graph would be a vertical transformation by moving the original graph of f(x)=x down 5 units, transforming the graph to f(x)=x-5 (as indicated with the red graph).

The next transformation of the graph would be a horizontal transformation by moving the original graph of f(x)=x to the left 2 units, transforming the graph to f(x)=x+2 (as indicated with the blue graph).

The next transformation of the graph would be a horizontal-vertical transformation by moving the original graph of f(x)=x to the left 2 units and down 4 units, transforming the graph to f(x)=x+2-4 (as indicated by the green graph).

The next transformation of the graph would be a reflection. To reflect across the x-axis, you have to multiply the original graph by (-1). Thus the new graph, f(x)=-x (as indicated by the purple graph).

The final transformation of the graph would be a horizontal-vertical reflection. The graph would be moved 2 units to the left, 8 units up, and the whole function would be multiplied by (-1), f(x)=-x+2+8 (as indicated by the black graph).

Ch P: How to do Graphing Functions

Graphing Functions / Transformations
Back to School Night Extra Credit

Well, since my mom didn't completed her education when she was young, she had no clue what she was asked to do on this assignment. So, I have to show her how to do a vertical and a horizontal translation combined.

First, I asked her to graph the Absolute Value parent function in black. I had to show her how to draw the function, because she seems clueless on what to do.
Then, I asked her to translate the graph 3 units to the right and use red to trace. This is the horizontal translation.
Lastly, I asked her to translate the graph 6 units up and use purple/violet to trace. This is the vertical translation.

When we were finished, my mom feels kind of happy that she drew a graph, but she still ís clueless about the purpose of the graphing functions. Overall, I'm proud to see her at least try to sketch the graph.

Ch. P - Graphing Functions

Black: y= |x| Parent

Red: Vertical translation y= |x| + 8

Blue: Horizontal translation y= |x-5|

Green: Vertical + Horizontal translation y= |x-7| - 2

Purple: Reflection over x-axis y= -|x|

Black: y= -|x+10| - 4 Vertical + Horizontal Translation + Reflection over x-axis

1. My starting(parent) equation was y=|x|

2. I performed a vertical translation on graph 1 which shifted the graph up by 8

3. I performed a horizontal translation on graph 1 which shifted the graph to the left by 5

4. I performed a vertical AND horizontal translation on graph 1 which shifted it to the right by 7 and down by 2

5. I performed a reflection over the x-axis on graph 1 which changed all the y values to opposite

6. I performed a vertical AND horizontal translation on graph 1 and also did a reflection over the x-axis which moved the graph down by 4, to the left by 10 and changed the y values to opposite.

Graphing Functions

1. The original graph i used is the one in color "Black" . It's original funtion is y= [x] (y is equal to the absolute value of X. That is what you get if you were to graph it.

2. I wanted to verticle translate this original graph so what i do is i move it down vertically 4 units. This vertical translation gave us the new funtion y=[x] -4 which is shown is the red graph above.

3. Then, Said i want make a horizontal translation it's very similar to verticle translation but this time you would have to go horizontally across the x-intercept instead. In blue graph above i horizontal translate it 8 units to the left which gave me the new function of y= [x-8]

4. Now if you want to recflect this across the x-axis you would have to do the reflection by flipping the graph over (shown in purple graph above). This new funtion is y= -[x] (where negative is on the outside)

5. After that, you can also combine both the vertical "and" horizontal translation. This means that you are moving the graph in 2 directions, "vertically" and "horizontally". In this funtion i move the graph 8 units to the right" horizontally" and 4 units down "vertically". It's shown in the green graph above.

6. Finally, WHAT'S Awesome! is that you can actually combine the whole thing by creating this tranformation just by putting together the ( verical (4 down) +horizonal (8 right) translation and our reflection across the x-axis) When you put them together your result will be the same as in my green graph. This new funtion gives us y= -[x-8]-4

Graphing Functions / Transformations

1) First, I graphed the Absolute Value parent function.
y = "Absolute Value of X"

2) Then, I did a vertical translation in the color red. I translated the parent function 10 units up along the y-axis and got the new graph,
y = "The Absolute Value of X" plus 10

3) Next, I did a horizontal translation in the color blue. I translated the parent function 6 units to the left along the x-axis and got the new graph,
y = "The Absolute Value of X plus 6"

4) I already did a vertical and a horizontal translation, now I did a vertical and horizontal translation combined in the color green. First, I translated the parent function 10 units to the right and 2 units up, and got the new graph,
y = "Absolute Value of X minus 10" plus 2


5) Next, I did a reflection over the x-axis in the color purple. All I did was flipped the Absolute Value parent function upside down, to get the new graph,
y = - "Absolute Value of X"

6) Lastly, I combined a reflection, vertical translation, and horizontal translation in the color black. First, I translated the graph 10 units to the right, 2 units up, and then reflected the graph over the x-axis to get the final graph,
y = - "Absolute Value of X minus 10" plus 2

Altogether, I did five transformations.

Chapter P - Graphing Functions

In this post I shall demonstrate how to do transformations.

1. The function I used in this graph is y=x^3.

2. I graphed this function in black at the origin.

3. Now I did a vertical translation in red and moved up by five in the y-axis. Up means add and down means subtract in the y-axis. So I have to add by 5, y=x^3+5

4. Then I did a horizontal translation in blue and moved right by 5 in the x-axis. Moving left is adding and moving right is subtracting in the x-axis. So I subtracted x by 5, y=(x-5)^3

5. In green, I did both a horizontal and vertical translation. So I moved up 5 and right 5, y=(x-5)^3+5

6. In purple, I did a reflection by adding a negative sign to the x. This flips the shape upside down,  some shapes look the same even after being reflected.

7. Finally, in black I did a reflection, vertical, and horizontal translation. I moved the shape down 5 and right 5 then flipped the image. So the equation is y=(-x-5)^3-5

Ch P - Graphing Functions

The parent function is x^3 (black). We can change this function by using translations. To translate vertically we must add to y. Here, I translated by 1 and got y=x^3 +1 (red). To translate horizontally I shifted 5 to the right and subtracted from x, resulting in (x-5)^3 (blue). Combining both will result in (x-5)^3+1 (green). As for reflections, I set x to -x. I got a reflection across the x-axis using y=-x^3 (purple). We can also combine reflections and translations, as seen in y=-(x-5)^3-1 (Sharpie). There will not be a reflection across the y-axis on the original function since that would have no effect.

Chapter P-Graphing Functions

1. The original parent function of this graph was y=|x|

2. Then, I wanted to do a vertical translation 8 units up, which led me to get the equation of y=|x|+8. As you can see, this function was in red and the vertex moved 8 units up on the y-axis.

3. Next, to do a horizontal translation, I decided that I wanted to move the graph 8 units to the right along on x-axis, so the equation became y=|x-8|, the function is shown in blue.

4. After that, I wanted to do a reflection across the x-axis, so my equation was y = -|x|, which was shown in purple. All of my y point from my (x,y) coordinates had become negative.

5. In green, I had done a horizontal translation 2 units to the left, followed by a vertical translation 4 units down(y=|x+2|-4.

6. Finally, for my last black graph, equation I had reflected my original function across the x-axis and then did a vertical translation 10 units down. The equation i ended up with was y= -|x-3|-10.




For Back to School Night, I brought my mom. She laughed at the idea of me teaching her Math, because she had barely been able to get a decent education growing up. As the night proceeded and as she began trying the translations it made a little more sense to her. She got the idea of doing vertical translations and horizontal translations.

Ch. P: Graphing Functions

So what I did here was interpret the original function y=|x| (shown facing upwards in black) in these few different changes you see here in various colors. I started off with a vertical translation (as you can see in red), which is y=|x| + 3. It is still the same function, but it only moved 3 spaces up on the y-axis. The next one was a horizontal translation (as shown in blue) as y=|x - 1|. This here is moved to the right one unit of the x-axis. In green, I performed both a horizontal and vertical translation which is y=|x-2| + 3 (right two units, up 3 units). The next one was a reflection shown in purple, y=-|x|. As you can see, the original function is just simply reflected, so it's facing downwards. The last transformation that I graphed was all of these put together -- a vertical translation, a horizontal translation, and a reflection. And so, this is y=-|x-1|-1.Back to School Night Extra Credit:
For Back to School Night I had to bring my brother (my parents were unavailable for the time being -- my brother is old enough to be my dad though! Haha) and so I told him to choose a function. He chose the cubic root function because he said it looked cool, so I helped him through it. He said he does not even remember doing these translations in his high school years, but that was about 15 years ago. Overall it was fun teaching him, because he understands most of the stuff that I told him.

Chapter P: Graphing Functions

Back to school night Extra Credit: My mom never got a proper education so she was rather delighted to knoe that she was learning some math. She got confused alot since it was new to her and it was pretty complicated since I had to translate into another language. She eventually got the hang of it and got pretty much the main point of it. Even though she didn't finish, she at least learned something new.

First, I had to draw the original function(in black) which is y= l x l. Next, I drew the vertical translation(in red) where I moved up the y-axis 7 units up and the equation was then y= l x l +7. Afterwards, I drew the horizontal translation(in blue) where I moved 4 units to the left on the x-axis, and the equation would then be y= l (x+4)^2 l. Then, I drew the vertical and horizontal translation(in green) where I moved 7 units up the y-axis and 4 units to the left, x-axis and the equation would be y= l (x+4)^2 l +7. Then, I drew the vertical reflection of the original function across the x-axis and the equation would be y=-l x l. Lastly, I drew the reflection (across the x-axis) of the vertical and horizontal translation. I would first reflect across the x-axis, moving 4 units to the left on the x-axis, and going down 7 units on the y-axis, the equation would equal y= l -x+4)^2 l -7. In total, I did 5 transformations excluding the original function.

Ch. P - Graphing Functtions

1. I started by graphing the original (parent) function of y = ³√x (cubic root of X), which is the graph (black) intersecting the origin.
2. Next, I translated the function y = ³√x, 9 units up along the y-axis to create a vertical translation of y = ³√x+9. The translation is shown in red intersecting at (0,9).
3. Then I created a horizontal translation, by moving the function 9 units to the right along the x-axis to create y = ³√(x-9). The translation is shown in blue intersecting at (9,0).
4. Then I combined both vertical and horizontal translation, I transformed the original function into y = ³√(x-9)+9, where the graph moved 9 units to the right of the x-axis then 9 units up the y-axis. The translation is shown in green marker in the first quadrant and intersecting at (9,9).
5. Then I did a reflection where I reflected the original function across the x-axis to create y = -³√x which is shown in purple intersecting at the origin.
6. Finally, I graphed a transformation by combining the vertical translation, horizontal translation and the reflection over the x-axis. I moved the original function reflected across the x-axis, translated 9 units to the right and 9 units down to make y= -³√(x-9)-9. The transformation is shown in black marker in the fourth quadrant intersecting at (9,-9).

CH. P - How to Do Graphing Functions

1.) Describe what you did using mathematical vocabulary. The following words must be used correctly, and highlighted or boldfaced in your description.

- First off, I sketched the basic function which was y=x^2 also known as the parent graph (colored in black.) Then, I sketched a vertical translation where I moved up 5 on the y-axis (colored in red) and the equation came out to be y=x^2+6. After that, I sketched a horizontal translation where I moved 4 to the right on the x-axis (colored in blue) and the equation came out to be y=(x-4)^2. Then, I sketched a vertical and horizontal translation where I moved up 6 and 4 to the right (colored in green) and the equation came out to be y=(x-4)^2+6. Finally, I sketched a vertical reflection across the x-axis and the equation came out to me y=-x^2. Overall, I made 5 new transformations.

2.) Scan your graphed functions and include the the picture in your blog.

3.) Back to School Night Extra Credit - Scan your Parent's classwork into your blog. Write a few sentences about what you and your parents thought about learning transformations.

- Well, first off, my mom thought that this project was somewhat hard. LOL. I don't know why but she had trouble figuring out what to do so I had to explain everything very explicitly and descriptive to her. She started to get the whole idea of graphing the functions and putting them into equations. My mom's favorite subject is math so even though it was a challenge at first, she kind of enjoyed it because now she knows how to sketch the graphs and write the equations that goes along with it.

Ch. P - Graphing Function

1. For this graph, I used the original function of y=√x (square root of x) which is shown in black marker with the original function boxed in the first quadrant.
2. Next, I translated the function (y=√x) 3 units down along the y-axis to create a vertical translation of y=√x-3. The translation is shown in red marker in the fourth quadrant.
   
3. Then to create a horizontal translation, I moved the function 4 units to the right along the x-axis to create y=√(x-4). The translation is shown in blue marker in the first quadrant.
4. And to combine both vertical and horizontal translation, I transformed the original function into y=√(x+6)+3 where the graph moved 6 units to the left of the x-axis then 3 units up the y-axis. The translation is shown in green marker in the second quadrant.
5. Then I did a reflection where I reflected the original function across the x-axis to create y= -√x which is shown in purple marker in the fourth quadrant.
6. Lastly, I created a transformation by combining the vertical and horizontal translation and the reflection across the x-axis to move the original function reflected across the x-axis, translated 8 units to the left and 2 units down to make y= -√(x+8)-2. The transformation is shown in black marker in the third quadrant.

Graphing Functions

I drew the function y=x^2, which is the parent graph and the one in black. I made five different transformation to the original one. I made a horizontal shift 3, to the right in green. And a vertical shift 2, downwards in blue. I combined the two translations to get y=(x-3)^2-2, which is in red. I then, made a reflection of the graph, which is basically a transformation over the x-axis.

During back to school night, I taught my mom how to make the same y=x^2 graph, but with different translation. The first thing I told her to do was to trace the parent graph on a patty paper, immediately she recognized it as a parabola. I taught her how to do a vertical and horizontal shift and how to read the functions. At the end i was surprised she actually understood all of it. Throughout my whole life, I had always asked my dad for help on homework, i guess i should ask my mom more often.

CH. P - How to Do Graphing Functions

1. First I sketched the basic function which was y=x^2 also known as the parent function. It is drawn with the color black.
2. Then I did a vertical translation by moving six units up the y-axis resulting in the equation y=x^2+6. The graph of this equation is in red.
3. Next I did a horizontal translation by moving four units to the right on the x-axis and that resulted in y=(x-4)^2. The graph of this equation is in blue.
4. Then I did a reflection across the x-axis and my equation turned up to be y=-x^2. The graph of this equation is in purple.
5. Then I combined my vertical and horizontal translations and so my new equation turned up to be y=(x-4)^2+6. The graph of this equation was in green.
6. Finally I combined my vertical and horixontal translations with my reflection across the x-axis and I got this equation y=-(x-4)^2+6. The graph is in black.
Overall I performed five transformations since I changed the basic function five times and got five differnt equations from the basic function: y=(x-4)^2, y=-x, y=x^2+6, y=(x-4)^2+6, and y=-(x-4)^2+6.