Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

The value of some cell phones changes exponentially after initial release. Here are graphs showing the depreciation of two phones 1, 2, and 3 years after they were released.

Phone A

A graph showing exponential decay with points (0, 1000), (1, 600), (2, 360), and (3, 216). The x-axis is labeled time (years) and the y-axis is labeled value (dollars).

Phone B

A scatter plot shows decreasing dollar values over time in years. Points labeled: (0, 840), (1, 630), (2, 472.5), and (3, 354.38). The x-axis is time (years), the y-axis is value (dollars).

1. Which phone is more expensive to buy when it is first released?

Compare your answer:

Phone A costs $1,000 and Phone B costs $840, so Phone A is more expensive.

2. Which graph depicts the value of the cell phone falling in value more quickly? Explain how you know.

Compare your answer:

Phone A is depreciating more quickly than phone B. This was determined by comparing the value of the phones each year. For instance, after one year, phone A was only worth $600 compared to phone B's value of $630. And, again, after three years, phone A was valued at $216 compared to phone B's value of $354.38.

3. How does the value of phone A change with every passing year?

Compare your answer:

Phone A loses $400 in the first year, which is $\frac{2}{5}$ (or 0.4 or 40%) of the original value. It loses $240 in the second year, which is $\frac{2}{5}$ (or 0.4 or 40%) of $600.

4. How does the value of phone B change with every passing year?

Compare your answer:

Phone B loses $210 in the first year, which is $\frac{1}{4}$ (or 0.25 or 25%) of the original value, and $157.50 in the second year, which is $\frac{1}{4}$ of $630.

5. If the phones continue to depreciate by the same factor each year, what will the value of each phone be 4 years after its initial release?

a. Enter your answer for Phone A.

Compare your answer:

$\frac{3}{5} \cdot $ 216 = $ 129.60$

b. Enter your answer for Phone B.

Compare your answer:

$\frac{3}{4} \cdot ($ 354.38) \approx $ 265.79$

6. For each cell phone, write an equation that relates the value of the phone in dollars to the years since release, $t$ . Use $v$ for the value of Phone A and $w$ for the value of Phone B.

a. Enter your answer for Phone A.

Compare your answer:

$v = 1, 000 \cdot (\frac{3}{5})^{t}$

b. Enter your answer for Phone B.

Compare your answer:

$w = 840 \cdot (\frac{3}{4})^{t}$

Are you ready for more?

Extending Your Thinking

When given data, it is not always clear how to best model it. In this case, we were told the value of the cell phones was changing exponentially. Suppose, however, we were instead just given the initial values of the cell phones when released and the values after each of the first three years.

1. Use technology to compute the best fit line for each cell phone. Round any numbers to the nearest dollar.

a. Enter your answer for Phone A.

Compare your answer:

$v = 933 - 259 t$

b. Enter your answer for Phone B.

Compare your answer:

$w = 816 - 161 t$

2. Explain why, in this situation, an exponential model might be more appropriate than the linear model you just created.

Compare your answer:

A linear model would quickly lead to negative phone values. For example, Phone A would cost about –$103 four years after its initial release. It makes more sense that the phone would lose a fraction of its value each year.

Self Check

For the following exercise, consider this scenario: For each year $t$ , the population of trees in Forest A is represented by the function $A(t)=115(1.025)^t$ , (red curve). In a neighboring forest, Forest B, the population of the same type of tree is represented by the function $B(t)=82(1.029)^t$ , (blue curve). Looking at the graph provided, which forest had a greater number of trees initially?

$GRAPH OF TWO EXPONENTIAL FUNCTIONS. THE FUNCTION A OF T IN RED HAS A $y$-intercepts OF 115. THE FUNCTION OF B OF T IN BLUE HAS A $y$-intercepts OF 82.$

Forest B, because it has a greater growth rate.
Forest A, because it has a greater growth rate.
Forest A, because it starts higher on the $y$ -axis.
Both forests started with the same number of trees.

Additional Resources

Analyzing $a$ and $b$ in the Exponential Function $y = a b^{x}$

Six differently colored arrows labeled A–F curve and cross on a grid with x- and y-axes. Arrows point in various directions, illustrating different equations or trajectories. Each label is color-matched to its arrow.

Which graph has the largest value for $b$ ?
Which graph has the smallest value for $b$ ?
Which graph has the largest value for $a$ ?
Which graph has the smallest value for $a$ ?

When using the form $y = a b^{x}$ , it is important to remember that $a$ is the initial value, or the $y$ -intercept, and $b$ is the growth factor. The growth factor reflects an exponential growth function when the value is greater than one. The growth factor reflects an exponential decay function when the value is between zero and one.

For question 1, you are looking for the largest growth factor, which would mean an exponential growth function that is increasing the most rapidly. First, identify the graphs that reflect exponential growth, which would be D, E, and F, and then decide which is the steepest. Graph D increases much faster than both graphs E and F.

For question 2, you are looking for the steepest rate of exponential decay, so you need the graph that has the smallest $b$ -value. The graphs that reflect exponential decay are graphs A, B, and C. The graph that has a $b$ -value that is closest to 0 is graph C. Graph C has the steepest exponential decay because its $b$ -value is the closest to 0 and, therefore, the smallest.

For question 3, you are looking for the largest initial value or $y$ -intercept. This could be with a growth or decay function; it just needs to cross the $y$ -axis higher than the other graphs. Graph C has the highest $a$ value.

For question 4, you are looking for the smallest initial value or $y$ -intercept. This could be a growth or decay function; it just needs to cross the $y$ -axis lower than the other graphs. Graph F has the lowest $a$ value.

Try it

Try It: Analyzing $a$ and $b$ in the Exponential Function $y = a b^{x}$

1. Which graphs represent exponential growth?

2. Which graphs represent exponential decay?

3. Which graph has the highest value for $b$ ?

4. Which graph has the highest value for $a$ ?

Compare your answers:

Here is how to analyze and compare graphs of exponential functions:

Graphs indicate exponential growth if they are increasing from left to right. Graphs D, E, and F represent exponential growth.
Graphs indicate exponential decay if they are decreasing from left to right. Graphs A, B, and C represent exponential decay.
Graph E has the highest value of $b$ , or growth rate, since it is increasing faster than graphs D and F.
Graphs C and D have the highest value for $a$ because they have the same $y$ -intercept.

5.7.2 Comparing Graphs

Activity

Are you ready for more?

Extending Your Thinking

Additional Resources

Analyzing a a and b b in the Exponential Function y = a b x y = a b x

Try It: Analyzing a a and b b in the Exponential Function y = a b x y = a b x

Analyzing $a$ and $b$ in the Exponential Function $y = a b^{x}$

Try It: Analyzing $a$ and $b$ in the Exponential Function $y = a b^{x}$