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Hi everyone. Today we’re going to talk about how to describe the transformations of a function
algebraically. To complete this problem, we will first identify the transformations we
need to describe and then show algebraically how each of these transformations will affect
our original function.
Let’s take a look. In this particular problem, we’ve been given the function or the theoretical
function y equals f of x and asked to algebraically describe the following transformations of
y equals f of x.
So what they’re basically saying here is that we haven’t been given f of x explicitly.
It could be anything. It could be x plus 7. It could be x cubed plus x squared minus 4.
It could be any function but they’re saying given any function, perform the following
transformations and indicate what that transformation would look like in conjunction here with f
of x.
So this will be just a great overview for how to perform the following transformations
and should give you a better idea of how to look for transformations when you see them.
So given y equals f of x, if we want to shift that graph no matter what it is, but we want
to shift it up three units, then our function would be y equals f of x plus 3. That’s
easy to understand because no matter what we plug in for x here, the resulting y value
would be three units more if we just add 3.
So it would be a vertical shift up 3 and the same will go for shifting down 3. We will
just get y equals f of x minus 3. This would shift the graph down vertically three units.
When we’re talking about shifting left and right, we have a similar concept except that
this plus 3 and negative 3 here need to be included in the function. So instead of having
it be separate, right? Here we’ve kind of just separately added on a plus 3 and separately
added on a negative 3. But in this case, if we want to shift right three units, then our
function will be y equals f of x minus 3.
So instead of just adding 3 or subtracting 3 separate from and outside of this f of x
function, it’s integrated inside of the function which means that if our original
function were let’s say x cubed and we wanted to shift it right three units, we would have
to plug in x minus 3 for x and get the quantity x minus 3 cubed.
So we really have to incorporate that negative 3 and it’s somewhat counterintuitive that
it’s a minus 3 when we’re shifting to the right but it is opposite like that from
what you would think. It’s minus 3 here. Shifting left three units is y equals f of
x plus 3. Maybe we can go through and do this here.
If we had a function y equals let’s say x squared, right? Then our transformation
here to shift up 3 would just be x squared plus 3. To shift down 3 would be x squared
minus 3. To shift to the right, would be y equals x minus 3 squared and to shift to the
left would be y equals the quantity x plus 3 squared.
So you can see the difference there. If we’re going to reflect about the x-axis, that means
we’re going to flip it over the x-axis exactly where it is, we will get y equals negative
f of x. So we just add that negative out in front of the f of x or multiply it by negative
1. If we want to reflect about the y-axis, we have to incorporate that negative inside
of our f of x and we will get f of negative x.
So that’s the difference here and the transformation looks like this. Again, if we have the function
x squared, then we would get y equals negative x squared to flip it over the x-axis. We would
get y equals negative x squared to flip it over the y-axis.
Finally, if we want to stretch vertically by a factor of 3 or shrink vertically by a
factor of 3, then our functions look like this. To stretch vertically, we want to say
y equals 3 times f of x. That means every y coordinate on our original function x squared
will now be three times what it was in the original function. If we shrink vertically,
we want to say y equals one-third f of x and every y coordinate that was on our original
graph will now be one-third what it was originally.
What that looks like here of course is y equals 3 x squared and y equals one-third x squared
and we won’t take the time to do it here but if you started with the function x squared
and you graphed all of these transformations, you would see visually how the graph would
get shifted up three units down, three units to the right, to the left, et cetera. That’s
it. That’s just a basic overview of transformations of functions up, down, right, left and flipping
over the x and y-axis and stretching and shrinking.
So I hope you found that video helpful. If you did, like this video down below and subscribe
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