Video Transcript
Observe the given graph and then
answer the following questions. Find the 𝑦-intercept in the shown
graph. As this graph represents an
exponential function, every 𝑦-value is multiplied by 𝑏 when 𝑥 increases by
Δ𝑥. Find 𝑏 for Δ𝑥 equals one. Find the equation that describes
the graph in the form 𝑦 equals 𝑎𝑏 to the power of 𝑥 over Δ𝑥.
Let’s begin by finding the graph’s
𝑦-intercept. We can see that the line of the
graph passes through the point zero, 10. Therefore, the 𝑦-intercept is
10. Now, let’s find the value of 𝑏 in
the equation of the graph. In the problem, we are told that
the graph represents an exponential function for which every 𝑦-value is multiplied
by 𝑏 when 𝑥 increases by Δ𝑥. And we’re asked to find 𝑏 when Δ𝑥
equals one. We’ve already seen that the graph
passes through the point zero, 10. And we can also see that it passes
through the point one, 20. The change in 𝑥 between these two
points is one, so it satisfies Δ𝑥 equals one.
We are told that the 𝑦-value is
multiplied by 𝑏 when 𝑥 increases by Δ𝑥. If the 𝑦-value of our first point
is 𝑦 one and the 𝑦-value of our second point is 𝑦 two, then 𝑏 equals 𝑦 two over
𝑦 one which is equal to 20 divided by 10, which is equal to two. And finally, we need to find the
equation that describes the graph in the form 𝑦 equals 𝑎𝑏 to the power of 𝑥 over
Δ𝑥. We are already given Δ𝑥 equals
one, and we have found the value of 𝑏 equal to two. Therefore, 𝑦 equals 𝑎 times two
to the 𝑥 over one, which is just equal to 𝑎 times two to the 𝑥. To find the value of 𝑎, we can
substitute the 𝑥- and 𝑦-values of points we know to be on the graph.
We’ve already found the
𝑦-intercept to be 10, so we know that the point zero, 10 lies on the graph. This means we can substitute the
values of 𝑥 equals zero and 𝑦 equals 10 into the equation which gives us 10 equals
𝑎 times two to the zero. By the zero-exponent rule, two to
the zero is just equal to one. Therefore, this gives 𝑎 equals
10. This gives us our equation and the
final answer 𝑦 equals 10 times two to the 𝑥.
