Lesson 5.4


You may have noticed that many equations
can be solved by undoing the order of
operations.
This strategy also applies for more complex
power equations that arise in real-world
problems.

Rita wants to deposit $500 into a savings
account so that its doubling time will be 8
years. What annual percentage rate is
necessary for this to happen? (Assume the
interest on the account is compounded
annually.)
Rita will need to find an
account with an annual
percentage rate of
approximately 9.05%.

When an exponential graph models decay,
the graph approaches the horizontal axis as x
gets very large.

When the context is growth, the graph
approaches the horizontal axis as x gets
increasingly negative.
A horizontal line through any long-run
value is called a horizontal asymptote.

A motion sensor is used to measure the
distance between itself and a swinging
pendulum. A table records the greatest
distance for every tenth swing. At rest, the
pendulum hangs 1.25 m from the motion
sensor. Find an equation that models the data
in the table below.

Plot these data. The graph shows a curved
shape, so the data are not linear. As the
pendulum slows, the greatest distance
approaches a long-run value of 1.25 m. The
graph appears to have a horizontal asymptote
at y=1.25. The pattern looks like a shifted
decreasing geometric sequence, so an
exponential decay equation is a good choice
for the best model.

An exponential decay function in point-ratio
x-x
y

y
(b
) , has the horizontal
form,
1
asymptote y=0.
Because these data approach long-run value
of 1.25, the exponential function must be
translated up 1.25 units. To do so, replace y
with y -1.25. The coefficient, y1, is also a yvalue, so you must also replace y1 with
y1- 1.25 in order to account for the
translation.
1

y-1.25  (y1  1.25)(bx-x1 )


To find the value of b, select one point for
(x1, y1) and a second point (not too close) for
(x, y).
If you choose (10, 1.97) and (50, 1.36) you
will find


So a model for this data is
y =1.25 + 0.72(0.9451)x-10 .
Using different pairs of points will generate
different values for b, but all of the values
should be similar.
Graph the model to check whether the points
chosen here resulted in a good-fitting model.