Welcome to the first of several videos on function transformations. This video focuses on horizontal
and vertical stretches and compressions. So, for comparing f of x to a function in this form,
we'll be looking at how the values of "a" and "b" affect the graph of the original function.
So, if we have y equals "a" times f of x, where "a" is greater than one, this will stretch the
graph of f of x vertically by a factor of "a." And if we have y equals "a" times f of x, where
"a" is between zero and one, this will compress the graph of f of x vertically by a factor of "a."
So, let's compare f of x to one-half times f of x and two times f of x by
completing T-tables. For f of x equals x squared, to find y, we’ll just square x,
so we'd have one squared, two squared, three squared, and four squared.
Let's first take a look at two times f of x, which would be two times x squared. Well,
if this is x squared and we want to find two times x squared, we can just multiply two times these
y-values. So, we'd have two times one, two times four, two times nine, and two times sixteen. And
this would be a vertical stretch because notice for these same x-coordinates,
these y-coordinates are much larger, and therefore stretching it vertically.
Now, if we take a look at one-half times f of x, or one-half times x squared, we can just multiply
one-half times these y-values to obtain one-half times x squared. So, we'd have one-half times one,
one-half times four, one-half times nine, and one-half times sixteen. And notice for
the same x-values, these y-values are less and therefore would compress the graph vertically.
So, what you may have noticed is once you identify the value of "a," to find
the y-coordinates of the transformed function with the same x-coordinates,
you would just multiply each y-coordinate by "a" and leave the x-coordinates the same.
If we take a look at these three graphs on the same coordinate plane, we'll notice the
original function is in blue. And if we stretch the blue function vertically by a factor of two,
we would have h of x, which is equal to two times f of x. Versus if we take
the blue function and compress it vertically by a factor of one-half,
we would have g of x, which is equal to one-half times f of x.
Let's take a look at an animation of this. So, if "a" is greater than one,
we can see a vertical stretch by a factor of "a." Let's take a look at what happens
when "a" is between zero and one. So, here we have "a" equals one, "a" equals 0.9, 0.8,
and so on. So, we have a vertical compression, in this case by a factor of three-tenths, and so on.
Let's now take a look at a horizontal stretch and horizontal compression. If
we have f of b times x, where "b" is greater than one, this will compress
the graph of f of x horizontally. And if we have y equals f of b times x, where
"b" is between zero and one, this will stretch the graph of f of x horizontally.
So, let's go ahead and compare T-tables for f of x, f of two times x,
and f of one-half times x. So, we already know this would be one squared,
two squared, three squared, and four squared. Notice now that we're multiplying
x by two and then squaring it. So, if x is one-half, two times one-half would be one,
and then squared we'd have one. Then two times one would be two, squared would be four. Two times
one and a half squared would be three squared or nine, and two times two squared would be sixteen.
Notice, when "b" is equal to two, we leave the y-coordinates the same, and then either
multiply the original x-coordinates by one over "b" or just divide by "b."
Let's now take a look at f of one-half times x,
which would equal one-half times x squared. So, "b" is equal to one-half. Well,
multiplying by one over one-half would be the same as multiplying by two. So, in this case,
if we multiply the original x-coordinates by two, we are going to get the same y-coordinates.
Let's take a look at the first couple: if x is equal to two, one-half times two is one,
squared that'd be one. One-half times four would be two squared,
that'd be four. And this would be considered a horizontal stretch.
If we take a look at these three graphs again, we have the original function in blue,
and when "b" is equal to two, we have a horizontal compression,
so we're compressing it this way to obtain the red function, f of two times x. And if we stretch
the blue function horizontally when "b" is equal to one-half, we would have the green function.
Let's take a look at an animation of this as well. So, when "b" is greater than one,
we have a horizontal compression, as we see here. And when "b" is between zero and one,
we have a horizontal stretch, as we see here.
Okay, let's go ahead and take a look at some of our own examples. The first thing we need
to be able to do is recognize the parent function for f of x. If f of x is equal
to three times the absolute value of x, let's let our parent function, g of x,
equal the absolute value of x. And we should be able to make a nice graph of this without
using our graphing calculators. We should all know this forms a V-shape, very similar to this graph.
Let's go ahead and identify some key points on this graph, like the origin, (2, 2),
and how about the (-2, 2). Now, looking at the original function, we should be able to
make the connection that f of x is equal to three times g of x, which means "a" is equal to three,
which means that our graph is going to be a graph that's vertically stretched by a factor
of three from this original red graph. So, to find points on this transformed function,
we can just multiply each of these y-coordinates by three. Since two times three would be six, one
point on the transformed graph would be (-2, 6). Zero times three would still be zero, so we have
the origin. And here we have another y-coordinate of two, so this would become (2, 6). And so, very
quickly and easily, we can graph the transformed function based on determining the value of "a."
Let's take a look at this one. Now we have f of x equal to the square root of two times x. Again,
we should be able to recognize that the parent function, which we'll call g of x, is equal to
the square root of x. Let's go ahead and graph that function. If x is zero, if x is zero, y would
be zero. If x is one, y is one. If x is four, the square root of four would be two. If x is nine,
we have the square root of nine, that'd be three. Sixteen, four. So, this is the parent function.
What we're going to do is graph the transformed function. Let's go ahead and make a T-table for
some of the key points on this graph like (0, 0), (1, 1), (4, 2), (9, 3), and (16, 4).
Now, we need to be able to recognize that f of x is equal to g of two times x, which is equal to
the square root of two times x. So, that tells us that "b" is equal to two. So, since "b" is equal
to two, we have a horizontal compression, which means to find points on the transformed function,
we can just multiply the x-coordinate by one over "b" or just divide by "b." So,
if "b" is equal to two, we'll just take each of these x-coordinates, divide them by two,
and keep the y-coordinates the same. So, the origin would still be on the
transformed function. The next point would be (1/2, 1), (4 divided by 2 would be 2),
so we have the point (2, 2). Nine divided by two would be four and a half, or 4.5,
(3). And then instead of (16, 4), we'll have sixteen divided by two, that'd be (8, 4).
This should be enough points to make a nice graph of the transformed function in green. So,
this has been horizontally compressed, meaning pushed back in this direction.
Okay, I hope you found this video helpful. Thank you for watching.