Video Transcript
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.
