Introduction to Basic Transformations: Translation of Functions

Hello friends! I hope you all enjoyed your weekend—don’t you think it was too short? Hah. Anyway, this is Kristiana and I shall attempt to summarize what we've learned in class today. 

In case you were falling asleep (not that I was) in the middle of the lecture, Mr. P was talking about Polynomial Functions. If you were really paying attention, you'll have realized by now that I was only kidding. 

Today we talked about Basic Transformation, which marks that beginning of the second unit. Basically, it's a recap of what we've done during the previous grade. I hope you all remember the meaning of function, which is for every x value, there is only one y value (...or something similar to that). In other words, y = f(x). 

Moving on, one technique that we've learned in order to figure out if something is a function or not is the use of table of values. We can also use a vertical line test. If the line intersects the graph twice, then it is not a function. 

y = x²

This graph can be considered a function because the lines only intersect once. 


This table of values could also be used. Notice that y can be repeated multuple times but x can only be used once. 

x	y
-2	4
-1	1
0	0
1	1
2	4

Mr. P also went on about stretching—not our bodies, but instead graphs! Transformation is the act of moving, flipping, stretching or compressing a base graph. 

When we graph quadratic equations, we are really translating (moving), flipping (reflecting), and stretching or compressing it. Don't worry if it doesn't make sense just yet, because today we only had time to discuss translation.

Translations are basically the horizontal and vertical movements (or shifts) using the basic shape of the parabola. Easy, right? 

Here are a couple of rules to remember:

• Horizontal Translation: affects the x values only
  y = f(x-h) : the graph shifts h units to the right (towards the positive side)
  y = f(x+h) : the graph shifts h units to the left (towards the negative side)
• Vertical Translation: affects the y values only
  y = f(x) + k : the graph shifts k units upwards (towards the positive side)
  y = f(x) + k : the graph shifts k units downwards (towards the negative side)

*IMPORTANT NOTE: Read h values as opposite and k values as they are. 
 e.g. y = f(x-3)
        h = +3 
        y = f(x) - 6
        k = -6

This is an example from the booklet we did in class:

Example 1b) implies vertical translation (since the format is f(x) + k), which means that only the value of y changes. Therefore all the points in the basic parabola will shift upwards by three units. In the table of values (for those of you that prefer using it), you will add 3 to the y-value: 

4 + 3 = 7, 1 + 3 = 4, ...etc. 

Example 1c) suggests horizontal translation, so only the values of x change. Since f(x-3) contains h, and remember, we read it as the opposite value. The basic parabola will then shift three units to the right (towards the positive side). And of course in the table of values, you will add 3 to the x-value: 

-2 + 3 = 1, -1 + 3 = 2, ...etc. 

That's pretty much it, I suppose? The topic today was sort-of mundane because it's very basic and general, but it's important to get the main idea because it'll play a big role from now on until the end of this course. If you have any questions or if you're still confused, feel free to either look over your notes from Grade 11 Pre-Calculus, ask me, or ask Mr. P (probably the best choice). 

Reminders: 

• Always identify if the question is asking for either the h or the k. 
• Remember that the h value is read as the opposite and the k value is read as it is.
• Vertical Translation: affects the y values only
• Horizontal Translation: affects the x values only

Don't forget to do your homework! It's important to keep informations fresh in our minds so that we don't forget them quickly. Oh, and our Permutations, Combinations and Binomial Theorem Assignment is due tomorrow, so make sure y'all don't forget to finish that. We also have a test on that topic this week. Good luck and may the best mathematician thrive. Haha. 

Yours Truly, 

Kristiana