I was not in class on Friday morning, as I had to go to "Doors Open," an extracurricular program for history class. But despite my absence, I asked around and determined that we didn't learn anything in the morning. All we were told was that exercise nine was assigned for homework over the weekend, but we weren't taught about stretches yet so we could not complete the exercise. So basically, the entire class was spent working on other subject work or "working" on the smart board. Nothing new was learned in this class.
I was present for the afternoon class, but then again there really wasn't any point to be there. We again didn't learn anything or accomplish any work, so we were either completing work concerning other subjects or (as the majority of the class did) played hangman about "math" on the board. Yeah...that was the main topic...math. Well, that was what the afternoon pretty much consisted of. But now on to where the most recent information that he taught us.
y=-f(x) <-- This function will be reflected over the x-axis, as the negative value is changing the y-values of the function. That negative sign is -1, so if you multiply all y-values by -1 the graph will appear to be flipped over the x-axis.
y=f(-x) <-- The same fits for this function, but it will be reflected over the y-axis. The negative 1 is multiplied by all the x-values of the function, causing them to be flipped horizontally over the y-axis.
He then ended it by saying that inverse functions will be explained in class, which was later proved to be true.
Today's class, we learned a lot useful and very interesting information about our new unit. I came into class and I saw that Mr. K had the aforementioned video up on the smart board, and I naturally assumed that we were going to go over it. This was true. So we began the class by watching the video, as much of the class did not get to watch it over the weekend. He paused the video at certain points and elaborated upon and fixed what he said in the video. The first time he paused it was to explain stretches and compressions.
When the function y=f(x) is altered by adding a value to multiply the output, in the case he showed us, y=2f(x), we can find the new y-values of the function easily. The 2 in front of the function alters y-coordinates, so all we must do is multiply the y values in the original graph by 2 to find the new functions y-coordinates. This causes a vertical stretch, and he showed this stretch by using the following graph:
The red line represents y=2sin(x).
The graph shows how once the graph was transformed by multiplying all y-values by 2, the graph was stretched vertically. But if the value is decreased, say to less than 1, such as 1/2, how will the graph look?
In the graph, the transformation is compressed vertically, which shows what happens to the y-values of the original function when the output is multiplied by a number less than 1. The graph becomes compressed, and the y-values are multiplied by, in this case, 1/2, which causes the new function to have lower y-values, causing it's compression. But this is not the only type of stretch that he showed us, as he also showed us horizontal stretch.
The graph now shows y=sin(x) compared to y=sin(2x). The graph is compressed, and Mr. K went in depth into his explanation of why this is. He explained that:
- In this case, you're doubling the input before it was calculated by the sin function. Before you do this, you multiply the x value by 2, doubling the original value before you find the sin.
- This makes the function happen faster, as you're multiplying the value before it is input into the function.
- To determine the x-values of the transformed function, you multiply all x-values by the recipricol of the value. In this case, to find any x-coordinate in the new function, you must multiply the original functions value by 1/2.
Each of the above graphs help portray how functions are stretched and compressed and then once we were shown all of these graphs, we were given a general equation to represent these parameters. In general we see:
The value of a represents the vertical stretch of the functions transformation. The value of a is used to multiply the y-values of the original function to find the new function. A alters the y-values of a function.
The value of b represents the horizontal stretch of the functions transformation. Though, the value of b is not the value used to determine the new function's coordinates, we must use the reciprocal of b. We must do this since b alters the values being input into the equation before it is actually calculated. For example, in the function y=sin(x), the transformation of y=sin(2x) causes the values of x to be doubled before it is calculated into the function of sine. In order to counteract this, we must use the reciprocal so that the values will end up equivalent before going into the sine function. In this case, if x=Π/2.
Then we partially reviewed translations from previous classes:
The value of a represents the horizontal slide of the function.
- If a is greater than 0 then the function is shifted to the right a units. In this case, the x-coordinates increase.
- If a is less than 0 then the function is shifted to the left a units. In this case, the x-coordinates decrease.
The value of b represents the vertical slide of the function.
- If b is greater than 0 then the function is shifted up b units. In this case, the y-coordinates increase.
- If b is less than 0 then the function is shifted down b units. In this case, the y-coordinates decrease.
After we finished reviewing translations, we then worked on graphing some of these newly taught stretches and compressions, as well as translations. He showed us a graph on the smart board, and then we had to alter it accordingly. These questions can be found on slide 2 here. The last question was composed of both stretches and translations, and from this question he taught us an important rule:
[Stretches come before translations]
Always work out stretches before translations. Well, now that all that was done, he quickly taught us about inverse functions.
He began by outright telling us that an inverse undoes whatever has been done to something. He then contrasted this to the "inverse" relationship between his daughter and his wife and himself. Every morning, his daughter goes into her clean room, then she transforms the room into a messy room. At night, the super inverse parents come into the messy room to clean up the mess, and trasnform the room back into a clean room. He showed a diagram of this:
Then we went more into the work involving the inverse function. The -1 above the inverse function's F does not mean it's to the exponent -1, that would be written as [f(x)]-1 . An inverse function "undoes" the transformations of a function through a routine, and the function of a regular function switches y and x values.
f --> f-1
(a,b) --> (b,a)
If you consider Mr. K's example, the messy room is the domain of f(x), and the clean room is the range. In other words, in the function f(x), the messy room represents all the x-coordinates, while the clean room represents all the y-coordinates. In the inverse of the function, however, we switch the messy and clean rooms around. By doing this, we're switching the domain and range around, which is exactly what happens in an inverse function.
(a, b) = (messy room, clean room)
(b, a) = (clean room, messy room)
He then showed us the inverse function of f-1(x) through different methods.
The inverse of a function can be determined algebraically. In the equation, switch the x and y variables, then solve for y to receive the naturalform of a function. We must due this so that we can graph it. Consider the function f(x).
f(x) --> y=√2x3-4
f-1(x) --> x=√2y3-4
y=√((x2+4)/2) <-- we are now able to graph this.
We can also display the inverse of a function by undoing whatever has been done to the function. Whatever order is taken to transform the function, the inverse will undo these transformations by reversing the order and reversing the operations. An example is shown in here on slide 3.
In order to graph the inverse of a function, you must graph the function then exchange all x values with all y values. You can also check here on the third slide to see an example of a graph, or for any of the inverse information.
And that concludes my scribe post for today. To reiterate my earlier statement, tomorrow's scribe is: