### Video Transcript

Which of the following graphs
represents the equation π¦ equals three to the power of π₯?

Our equation π¦ equals three to the
power of π₯ represents an exponential equation. So letβs recall what we know about
exponential functions. Firstly, we know that an
exponential function of the form π of π₯ equals π to the power of π₯, where π is
a positive real constant, passes through at zero, one. In other words, it passes through
the π¦-axis at one. So letβs see if we can eliminate
any of our graphs from our question. Graph B passes through at zero. Graph C doesnβt seem to intersect
the π¦-axis at all. Graph E intersects at negative
one. And that leaves us with Graph A and
Graph D, which both intersect the π¦-axis at one.

Thereβs 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. Letβs let π₯ be equal to one, since
the π₯-coordinate is one, and see if the π¦-coordinate is indeed three. If π₯ is equal to one, π¦ is equal
to three to the power of one, which is indeed three. And so we can infer that the graph
of the equation π¦ equals three to the power of π₯ 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 π is greater than
one, our graph represents exponential growth. In other words, itβs always
increasing. Whereas if π is between zero and
one, it represents exponential decay; itβs always decreasing. We can see that the graph of D is
decreasing over its entire domain. Itβs always sloping downwards. And so the value of π, the base if
you will, needs to be between zero and one. So this could be π¦ equals
one-third to the power of π₯, for example. The correct answer here then is
A.