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.