### Video Transcript

In this video, weβll learn how to
identify function transformations involving horizontal and vertical stretches or
compressions. After watching this video, youβll
be able to identify graphs of horizontal and vertical dilations or enlargements and
how these transformations are described using function notation.

Weβll begin by recalling what we
mean by a function. A function is like a machine. It has an input and an output. That output is achieved by
performing an operation or a series of operations on the input. For example, take the function π
of π₯ equals two π₯ plus five. The input is π₯, and the output is
two π₯ plus five. The operations performed by the
machine are multiplying by two then adding five. Now, we want to look at how we
achieve two types of transformations. These are horizontal dilations or
enlargements β that is, enlargements parallel to the π₯-axisβ and vertical
dilations. Those are parallel to the
π¦-axis.

Letβs take a function π¦ equals π
of π₯. With this function, we take a value
of π₯, we substitute it into the function, and we get an output, a value for π¦
out. Imagine next that we multiply the
entire function and the entire output by two. So, π¦ is equal to two times π of
π₯. What effect is multiplying the
entire function by two have? Well, in this case, we substitute
the value of π₯ in as normal. But we then take our output and we
multiply that by two. This has the effect of doubling all
of the outputs or all of the values of π¦. So what will that look like on the
graph? Well, for each value of π₯, the
output will be twice as large. So, if we take this point on the
original curve, the value of π¦ will be twice as large. It will be up here somewhere.

The same can be said for this value
on the original curve; the output will be twice the size. So it would be down here
somewhere. And so, the graph will look a
little something like this. Notice that the points where our
graph crosses the π₯-axis remain unchanged. And thatβs because at these points,
the output π¦ is zero. So, if we double zero, we still get
zero. The transformation that maps the
function π¦ equals π of π₯ onto the function π¦ equals two π of π₯ is a vertical
stretch. But we really call this a dilation
by a scale factor of two parallel to the π¦-axis.

In general, we say that for a
function π¦ equals π of π₯, π¦ equals π times π of π₯ for real constants π is a
dilation by a scale factor of π parallel to the π¦-axis. In other words, itβs a vertical
stretch by a scale factor of π. But what about an alternative
transformation? Letβs imagine now we have π¦ equals
π of two π₯. This time we multiply each value of
π₯ by two and then substitute it into the formula. Weβre substituting a value twice
the size of the original into the function and then getting the relevant output. This feels a little bit like weβre
speeding up the function and completing it twice as fast.

And so, this point on the original
function will be completed in half the time. It will be here somewhere. Similarly, this point here will
also be completed in half the time. Itβs here somewhere. We can do the same with the
remaining points, and we end up with this graph shown. This is a horizontal stretch or a
dilation or enlargement by a scale factor, this time, of one-half parallel to the
π₯-axis. We can say, in general, that for a
function π¦ equals π of π₯, π¦ equals π of π times π₯ for real constants π is a
dilation by a scale factor one over π parallel to the π₯-axis. In other words, itβs a horizontal
compression. Now that we have these definitions,
letβs look at some examples.

The function π¦ equals π of π₯ is
stretched in the vertical direction by a scale factor of one-half. Write, in terms of π of π₯, the
equation of the transformed function.

Another way of saying a stretch is
to say that the function has been dilated or enlarged depending on which part of the
world youβre in. And so, we recall that for a
function π¦ equals π of π₯, π¦ equals π times π of π₯ gives us that vertical
stretch or dilation, and the scale factor is π. So, letβs compare this definition
with our question. We have the function π¦ equals π
of π₯ being stretched in the vertical direction, and the scale factor is
one-half.

So, comparing this to our
definition, we see that weβre going to let π be equal to one-half. Since π¦ equals π times π of π₯
gives us the vertical stretch by a scale factor of π, we get a vertical stretch by
a scale factor of one-half by writing π¦ equals one-half of π of π₯. Multiplying by one-half is the same
as dividing by two. And so, we can alternatively say
that, in terms of π of π₯, the equation of our transformed function is π¦ equals π
of π₯ over two.

Letβs consider a similar
example.

The function π¦ equals π of π₯ is
stretched in the horizontal direction by a scale factor of two. Write, in terms of π of π₯, the
equation of the transformed function.

Letβs recall how we achieve a
stretch or a dilation of a function π¦ equals π of π₯. We know that for a function π¦
equals π of π₯, π¦ equals π of π times π₯ gives us a horizontal stretch or
dilation by a scale factor of one over π. So, letβs compare this definition
to our question. We have a function π¦ equals π of
π₯ being stretched in the horizontal direction. The scale factor of this stretch or
enlargement is two. Comparing this to our definition,
we see that the scale factor is one over π, so weβre going to let one over π be
equal to two.

Letβs solve this equation for π by
first multiplying both sides by π. That gives us one equals two
π. Then, weβll divide both sides of
this equation by two, giving us one-half equals π or π equals one-half. This means that for a horizontal
dilation by a scale factor of two, our function becomes π¦ is equal to π of a half
π₯. Now, of course, multiplying by a
half is the same as dividing by two, so we can write a half π₯ as π₯ over two. And this means the equation of our
transformed function is π¦ equals π of π₯ over two.

Letβs now look at how to identify a
graph given the equation of the transformed function.

The figure shows the graph of π¦
equals π of π₯. Which of the following is the graph
of π¦ equals a half π of π₯?

Letβs begin by looking at the
equation of the transformed function. When we multiply by a scalar, that
is, a real constant, that represents a dilation or enlargement of some
description. In fact, when we multiply the
entire function π of π₯ by some scalar, we get a vertical dilation or enlargement
by a scale factor of that number. And so, here weβre going to stretch
the original graph vertically by a scale factor of one-half. Thatβs going to look like a
vertical compression. To identify the correct graph,
weβll identify some of the key points on our graph.

Firstly, letβs consider this point
here. It passes through the π¦-axis at
two. When we compress our graph or
stretch it vertically by a scale factor of one-half, this will now pass through a
value of π¦ half the size. Itβs going to pass through the
π¦-axis at zero, one. Similarly, letβs take the point at
1.5, negative 0.6. Weβre going to halve the value of
the π¦-coordinate. The π₯-coordinate still remains
unchanged, so itβs going to be 1.5, negative 0.3. And so, itβs going to look a little
something like this. If we compare this to the graphs
weβve been given, we see that the only one that matches this criteria and the only
one in fact that passes through the π¦-axis at one is (B). So, (B) is the graph of π¦ equals a
half π of π₯.

Letβs see if we can identify the
equations of the other graphs. Looking at graph (A), we can see
itβs actually been stretched by a scale factor of two. And so, the equation of this one
must be π¦ equals two times π of π₯. Graph (C), however, has been
compressed by a scale factor of a half. But this time, thatβs in the
horizontal direction. To achieve a horizontal dilation by
a scale factor of one-half, we need to multiply the values of π₯ by two. So, the equation of this graph is
π¦ equals π of two π₯.

Then, if we look at graph (D), we
see something similar has occurred. This time itβs stretched in a
horizontal direction but by a scale factor of two. To achieve this, we need to
multiply all the values of π₯ by one-half. So, graph (D) is π¦ equals π of a
half π₯. And graph (E) is a different beast
altogether. This represents a combination of
stretches. It stretched vertically by a scale
factor of two and horizontally by a scale factor of two. And so, its equation is, in fact, a
combination of (A) and (D). Itβs π¦ equals two times π of a
half π₯. The correct answer here, though, is
(B).

Weβll now have a look at another
example that involves recognizing dilations of functions graphically.

The red graph in the figure
represents the equation π¦ equals π of π₯, and the green graph represents the
equation π¦ equals π of π₯. Express π of π₯ as a
transformation of π of π₯.

Letβs begin by just comparing the
red graph and the green graph in our diagram. Both graphs pass through the
origin, the point zero, zero. So, letβs compare this point, this
sort of relative maximum, on the red graph to the relative maximum on the green
graph. The value of the π¦-coordinate
remains unchanged. The value of the π₯-coordinate,
however, is doubled. Itβs transformed from roughly
negative 1.25 to negative 2.5. Letβs consider this local minimum
and compare it to the local minimum on the green graph. Once again, the value of the
π¦-coordinate remains unchanged. The π₯-part of the coordinate,
though, is doubled. Itβs gone from 0.5 on the red graph
to one, roughly, on the green graph.

So, what transformation maps the
red curve onto the green curve? Well, we can see itβs been
stretched parallel to the π₯-axis. We call that a horizontal dilation
or enlargement. Since everything in the
π₯-direction has been doubled, we can say this horizontal dilation is by a scale
factor of two. So, how do we represent this using
function notation? Well, we recall that for a function
π¦ equals π of π₯, π¦ equals π of π times π₯ for real constants π represents a
horizontal stretch by a scale factor of one over π. Letβs compare our scale factor with
the scale factor in the general definition.

When we do, we see that weβre going
to let one over π be equal to two. To find the value of π then, letβs
multiply both sides of this equation by π and then divide through by two so that
one-half is equal to π or π is equal to one-half. We can therefore say that the green
graph is π¦ equals π of a half π₯. But, of course, we said that the
green graph is given by the equation π¦ equals π of π₯. So we can say that π of π₯ must be
equal to π of a half π₯. We can simplify a little by writing
a half π₯ as π₯ over two. And so, we see that π of π₯ as a
transformation of π of π₯ is π of π₯ equals π of π₯ over two.

In our next example, weβll look at
how to find coordinates of a point on a graph following a dilation or
enlargement.

The figure shows the graph of π¦
equals π of π₯ and the point π΄. The point π΄ is a local
maximum. Identify the corresponding local
maximum for the transformation π¦ equals π of two π₯.

Letβs begin by recalling the
transformation that maps the graph of π¦ equals π of π₯ onto the graph of π¦ equals
π of two π₯. We know that for a function π¦
equals π of π₯, π¦ equals π of π times π₯ is a horizontal dilation by a scale
factor of one over π. And so, if we compare our equation,
thatβs π¦ equals π of two π₯, to the general equation, π¦ equals π of ππ₯, we can
see that weβre going to have a horizontal dilation. Letβs work the scale factor out by
letting π be equal to two.

When we do, we see that for our
function π¦ equals π of π₯, π¦ equals π of two π₯ is a horizontal dilation by a
scale factor of one-half. In other words, weβre going to
compress our graph about the π¦-axis. When we do, it looks a little
something like this. Weβll call the coordinate of our
local maximum on our transformation π΄ prime as shown. We can see that its π¦-coordinate
remains unchanged, but its π₯-coordinate is halved. So, π΄ prime is two over two, one
or one, one. And so, we see that the
corresponding local maximum for the transformation π¦ equals π of two π₯ is one,
one.

In our final example, weβll look at
how these processes can help us to draw a full transformation by dilation.

The diagram shows the graph of the
function π¦ equals π of π₯ for values of π₯ greater than or equal to negative three
and less than or equal to three. Sketch the graph of π¦ equals
one-third of π of π₯ on the same set of axes.

Letβs begin by recalling what
transformation maps π¦ equals π of π₯ onto π¦ equals a third π of π₯. We know that for a function π¦
equals π of π₯, π¦ equals π times π of π₯ gives us a vertical dilation by a scale
factor of π. Comparing this with the function in
our question, we see weβre going to have a vertical dilation, but weβre going to let
π be equal to one-third. And so, for a function π¦ equals π
of π₯, π¦ is a third π of π₯ is a vertical dilation by a scale factor of
one-third. Letβs do this point by point.

Weβre going to start with the point
on the left-hand side of our diagram. This point has coordinates negative
three, three. By performing a vertical dilation
or a stretch, we find a third of the π¦-coordinate. A third of three is one, so the
corresponding coordinate on our transformed graph is negative three, one. Since itβs being stretched
vertically, it will still pass through the π₯-axis in the same place. And what about the point with
coordinates zero, negative three. Once again, the π₯-coordinate
remains unchanged, and we divide the π¦-coordinate by three to get zero, negative
one.

We can perform the same process for
the point one, negative three, and that will map onto the point with coordinates
one, negative one. It will pass through the π₯-axis in
the same place again. And finally, weβll transform the
point with coordinates three, one. We divide the π¦-coordinate by
three, and we find this point has coordinates three, a third. And so, we see that the graph of π¦
equals a third π of π₯ is as shown. Itβs a vertical dilation or a
stretch by a scale factor of one-third.

In this video, weβve learned that
we can perform dilations or enlargements in one of two ways. For a function π¦ equals π of π₯,
π¦ is π times π of π₯ for real constants π gives us a vertical dilation by a
scale factor of π. Similarly, π¦ is equal to π of
ππ₯ represents a horizontal dilation or a stretch parallel to the π₯-axis. And this time, the scale factor is
one over π.