Let's analyze the problem step by step using the given functions and values to fill in the blanks correctly.
First, let's determine the initial trade-in value for each phone when [tex]\( x = 0 \)[/tex]:
- For Tonya's phone [tex]\( f(x) = 490(0.88)^x \)[/tex]:
[tex]\[
f(0) = 490(0.88)^0 = 490 \text{ dollars}
\][/tex]
- For Leo's phone, from the table, [tex]\( g(0) = 480 \text{ dollars} \)[/tex].
Between Tonya and Leo, Tonya's phone has the greater initial trade-in value (\[tex]$490 vs. \$[/tex]480).
Now, let's calculate the average rate of decrease in trade-in value over the first four months for both phones.
### Tonya's Phone
The trade-in values at [tex]\( x = 0, 2, 4 \)[/tex] for Tonya's phone:
[tex]\[
f(0) = 490(0.88)^0 = 490 \text{ dollars}
\][/tex]
[tex]\[
f(2) = 490(0.88)^2 = 490 \times 0.7744 = 379.456 \text{ dollars}
\][/tex]
[tex]\[
f(4) = 490(0.88)^4 = 490 \times 0.59969536 = 293.8507264 \text{ dollars}
\][/tex]
From [tex]\( x = 0 \)[/tex] to [tex]\( x = 4 \)[/tex] months, let's compute the average rate of decrease:
[tex]\[
\text{Average rate of decrease} = \frac{f(0) - f(4)}{4 - 0} = \frac{490 - 293.8507264}{4} \approx 49.0373184 \text{ dollars per month}
\][/tex]
### Leo's Phone
The trade-in values from the table:
[tex]\[
g(0) = 480 \text{ dollars}
\][/tex]
[tex]\[
g(2) = 360 \text{ dollars}
\][/tex]
[tex]\[
g(4) = 270 \text{ dollars}
\][/tex]
From [tex]\( x = 0 \)[/tex] to [tex]\( x = 4 \)[/tex] months, let's compute the average rate of decrease:
[tex]\[
\text{Average rate of decrease} = \frac{480 - 270}{4 - 0} = \frac{210}{4} = 52.5 \text{ dollars per month}
\][/tex]
### Conclusion
- Initial Trade-in Value: Tonya's phone has the greater initial trade-in value.
- Rate of Decrease: The trade-in value of Tonya's phone decreases at an average rate slower than the trade-in value of Leo's phone (49.0373184 vs. 52.5 dollars per month).
So the filled-in statements would be:
[tex]\[
\boxed{\text{Tonya's}} \ \text{phone has the greater initial trade-in value.}
\][/tex]
[tex]\[
\text{During the first four months, the trade-in value of Tonya's phone decreases at an average rate} \ \boxed{\text{slower than}} \ \text{the trade-in value of Leo's phone.}
\][/tex]
