Video Transcript
In this video, we’ll learn how to
sketch and identify the graphical transformations of exponential functions. An exponential function is one of
the form 𝑓 of 𝑥 equals 𝑏 to the power of 𝑥. 𝑏 is a positive real number not
equal to one and the variable 𝑥 occurs as an exponent. These functions are hugely
important within mathematics as they have all sorts of applications. We use them to model exponential
growth and decay. For example, we might use an
exponential function to model population growth or the amount of money in an
investment account given specific compound interest requirements. We’ll begin by looking at the shape
of such graphs.
Which graph demonstrates
exponential growth?
First, let’s recall what we mean by
an exponential function. It’s a function of the form 𝑓 of
𝑥 equals 𝑏 to the power of 𝑥, where 𝑏 is a positive real number not equal to one
and in which the variable 𝑥 occurs as an exponent. Let’s look at what happens if we
try to plot two types of this function. We’ll plot the function 𝑓 of 𝑥
equals two to the power of 𝑥. In other words, one where 𝑏 is
greater than one and the function 𝑔 of 𝑥 equals a half to the power of 𝑥. In this case, we’re looking at the
behaviour where 𝑏 is in the open interval from zero to one.
We’ll use a table for each. When 𝑥 is negative two, 𝑓 of 𝑥
is two to the power of negative two. That’s one over two squared, which
is one over four or 0.25. When 𝑥 is negative one, 𝑓 of 𝑥
is two to the power of negative one, which is a half or 0.5. In the same way, 𝑓 of zero is one,
𝑓 of one is two, 𝑓 of two is four, and 𝑓 of three is two cubed, which is
eight. Similarly, for 𝑔 of 𝑥, we get 𝑔
of negative two to be four, 𝑔 of negative one to be two, and so one. Let’s plot these on the same
axes. When we plot the function 𝑓 of 𝑥
and join it with a smooth curve, we see it’s increasing over its entire domain. In other words, it’s always sloping
upwards. Whereas 𝑔 of 𝑥 is decreasing;
it’s always sloping downwards.
We say, in fact, that a function of
the form 𝑓 of 𝑥 equals 𝑏 to the power of 𝑥, where 𝑏 is a real constant greater
than one, represents exponential growth. Whereas when 𝑏 is greater than
zero and less than one, the function represents exponential decay. So which of our graphs demonstrates
exponential growth? In other words, it looks a little
bit like the function 𝑓 of 𝑥. Well, we see that that function is
𝑏.
Now, in fact, we can infer another
property of these functions from the graphs we plotted. Notice how these parts of the lines
seem to get closer and closer to the 𝑥-axis. They’ll never actually reach the
𝑥-axis, though. And that’s because the number gets
fractionally smaller each time since we’re halving the value of the function. But it will never get to zero. We call this line, the 𝑥-axis or
the line 𝑦 equals zero, a horizontal asymptote.
We can therefore say the
following. An exponential function is one of
the form 𝑓 of 𝑥 equals 𝑏 to the power of 𝑥, where 𝑏 is a positive real number
not equal to one. If 𝑏 is greater than one, the
function models exponential growth. And if it is greater than zero and
less than one, it models exponential decay. The 𝑥-axis, or the line 𝑦 equals
zero, is a horizontal asymptote to such functions. There’s actually one further
property that we can establish, so let’s look at an example.
Determine the point at which the
graph of the function 𝑓 of 𝑥 equals six to the power of 𝑥 intersects the
𝑦-axis.
We recall that the 𝑦-axis is the
vertical line whose equation is 𝑥 equals zero. We can therefore find the point of
intersection of the graph with the 𝑦-axis by letting 𝑥 equal to zero and solving
for 𝑦. When we do, when we set 𝑥 equal to
zero, we get 𝑦 equals six to the power of zero. But of course, we know that
anything to the power of zero is one. This means that the graph
intersects the 𝑦-axis at 𝑦 equals one. Now, that’s of course when 𝑥
equals zero. So the coordinate of intersection
is zero, one. In fact, if we take the general
graph of a function 𝑓 of 𝑥 equals 𝑏 to the power of 𝑥, where 𝑏 is a real
constant greater than zero and not equal to one, we know that 𝑓 of zero is 𝑏 to
the power of zero, which is also one.
Remember, no matter the value of
𝑏, as long as it’s a real constant, 𝑏 to the power of zero will always be one. We can therefore say that an
exponential function of the form 𝑓 of 𝑥 equals 𝑏 to the power of 𝑥 intersects
the 𝑦-axis at one or the point with coordinates zero, one.
In our next example, we’ll consider
how we can identify the correct graph of exponential functions using the features
we’ve established and a bit of substitution.
Which of the following graphs
represents the equation 𝑦 equals three to the power of 𝑥?
Our equation 𝑦 equals three to the
power of 𝑥 represents an exponential equation. So let’s recall what we know about
exponential functions. Firstly, we know that an
exponential function of the form 𝑓 of 𝑥 equals 𝑏 to the power of 𝑥, where 𝑏 is
a positive real constant, passes through at zero, one. In other words, it passes through
the 𝑦-axis at one. So let’s see if we can eliminate
any of our graphs from our question. Graph B passes through at zero. Graph C doesn’t seem to intersect
the 𝑦-axis at all. Graph E intersects at negative
one. And that leaves us with Graph A and
Graph D, which both intersect the 𝑦-axis at one.
There’s two ways that we can check
which one of our graphs is correct. We could pick a point and test
this. For example, our first graph passes
through the point with coordinates one, three. Let’s let 𝑥 be equal to one, since
the 𝑥-coordinate is one, and see if the 𝑦-coordinate is indeed three. If 𝑥 is equal to one, 𝑦 is equal
to three to the power of one, which is indeed three. And so we can infer that the graph
of the equation 𝑦 equals three to the power of 𝑥 must pass through the point one,
three. And so our graph is A.
There is, however, another way we
could have tested this. We know that if 𝑏 is greater than
one, our graph represents exponential growth. In other words, it’s always
increasing. Whereas if 𝑏 is between zero and
one, it represents exponential decay; it’s always decreasing. We can see that the graph of D is
decreasing over its entire domain. It’s always sloping downwards. And so the value of 𝑏, the base if
you will, needs to be between zero and one. So this could be 𝑦 equals
one-third to the power of 𝑥, for example. The correct answer here then is
A.
Let’s have a look at another
example.
Which of the following graphs
represents the equation 𝑦 equals a quarter to the power of 𝑥.
It’s useful to begin by spotting
that this is an exponential equation. An exponential equation is one of
the form 𝑦 equals 𝑏 to the power of 𝑥, where 𝑏 is a real positive constant not
equal to one. Now, we know several things about
the graphs of exponential equations. We know that their 𝑦-intercepts
for a start are one. They pass through the point zero,
one. And so we can instantly eliminate
three of our graphs. We can eliminate A, B, and C. Graph A actually intersects at zero
as does graph C, whereas graph B doesn’t appear to intersect the 𝑦-axis at all.
Now, we also know something about
the shape of these curves. If our value for 𝑏 is greater than
one, then we’re representing exponential growth. And the graph looks a little
something like this. Notice that the 𝑥-axis represents
a horizontal asymptote of our graph. It gets closer and closer but never
quite touches it. Now, if 𝑏 is greater than zero and
less than one, we have exponential decay. Our graph is decreasing over its
entire domain. The 𝑥-axis is still a horizontal
asymptote to our graph, but this time it looks a little like this.
So what’s our value of 𝑏? Well, the equation is 𝑦 equals a
quarter to the power of 𝑥. So 𝑏 is equal to a quarter which
is greater than zero and less than one. That tells us that our graph
represents exponential decay. It’s going to be decreasing over
its entire domain. We can see that that’s graph D.
In our final example, we’ll look at
how to identify the graph of a more complicated exponential equation.
Which of the following graphs
represents the equation 𝑦 equals two times three to the power of 𝑥.
Now, whilst it may not look like
it, this is an example of an exponential equation. It’s essentially a multiple of its
general form 𝑦 equals 𝑏 to the power of 𝑥, where 𝑏 is a positive real constant
not equal to one. This time, though, it’s of the form
𝑎𝑏 to the power of 𝑥. Remember, according to the order of
operations, we apply the exponent before multiplying. So this is three to the power of 𝑥
times two. And this means we’re going to need
to recall what we know about the transformations of graphs. Well, for a graph of the function
𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of 𝑥 plus some constant 𝑎 is a translation by
zero 𝑎. It moves 𝑎 units up.
The graph of 𝑦 equals 𝑓 of 𝑥
plus 𝑏 is a translation by negative 𝑏 zero. This time it moves 𝑏 units to the
left. Now, if we look at our equation, we
see that we haven’t added a constant at all. So we recall the other rules we
know. 𝑦 is equal to some constant 𝑎
times 𝑓 of 𝑥 is a vertical stretch or enlargement by a scale factor of 𝑎. Whereas 𝑦 equals 𝑓 of 𝑏𝑥 is a
horizontal stretch by scale factor one over 𝑏. Now going back to our equation, we
have three to the power of 𝑥. And we’re timesing the entire
function by two. And so we’re looking at a vertical
stretch. In fact, we need to perform a
vertical stretch of the function 𝑦 equals three to the power of 𝑥 by a scale
factor of two.
So what does the graph of 𝑦 equals
three to the power of 𝑥 look like. It’s an exponential function, and
the base is greater than one. That means our function represents
exponential growth. This means we can eliminate graphs
A and B. They actually represent exponential
decay, since they’re decreasing; they’re sloping downwards. So we need to choose from C, D, and
E. And so we also recall that the
function 𝑦 equals 𝑏 to the power of 𝑥 passes through the 𝑦-axis at one. Our function 𝑦 equals three to the
power of 𝑥 will do the same. It’ll pass through zero, one. But it’s been stretched vertically
by a scale factor of two. This means our function 𝑦 equals
two times three to the power of 𝑥 must pass through at zero, two.
Out of C, D, and E, the only
function that does so is E. C passes through at one and D
passes through at three. And so the graph that represents
the equation 𝑦 equals two times three to the power of 𝑥 is E.
In this video, we learned that an
exponential function is of the form 𝑓 of 𝑥 equals 𝑏 to the power of 𝑥, where 𝑏
is a positive real number not equal to one. We saw that for values of 𝑏
greater than one, our function models exponential growth; it slopes upwards. And that if zero is less than 𝑏,
which is less than one, the function models exponential decay; it slopes
downwards. Note that the reason we disregarded
𝑏 equals one is if 𝑏 is equal to one, the function gives a simple horizontal line,
which is not exponential growth. Finally, we saw that these graphs
pass through the 𝑦-axis at one and they have a horizontal asymptote given by the
𝑥-axis or the line 𝑦 equals zero.
