Question

Use the table to answer the question.

0 375,000
1
2
3
4
5
The Diaz family purchased a home for $375,000. They decide to sell their home five years later. If the value of the home is increasing at a rate of 6% each year, the situation can be modeled using the equation f(x)=375,000(1.06)x Finish the table to determine which graph displays the correct plotted points for the situation.(1 point)
Responses

An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis is labeled Years and ranges from 350,000 to 550,000 in increments of 25,000. The y-axis is labeled Home Value and ranges from 0 to 7 in increments of 1.
Image with alt text: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis is labeled Years and ranges from 350,000 to 550,000 in increments of 25,000. The y-axis is labeled Home Value and ranges from 0 to 7 in increments of 1.

A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 0 to 350,000 in increments of 50,000.

Answer 1

To complete the table and determine which graph correctly represents the scenario given, we need to calculate the home values for each year based on the model \( f(x) = 375,000(1.06)^x \).

Let's compute the values for \( x \) from 0 to 5:

1. **When \( x = 0 \)**:
\[
f(0) = 375,000(1.06)^0 = 375,000(1) = 375,000
\]

2. **When \( x = 1 \)**:
\[
f(1) = 375,000(1.06)^1 = 375,000(1.06) = 397,500
\]

3. **When \( x = 2 \)**:
\[
f(2) = 375,000(1.06)^2 = 375,000(1.1236) \approx 421,350
\]

4. **When \( x = 3 \)**:
\[
f(3) = 375,000(1.06)^3 = 375,000(1.191016) \approx 446,257
\]

5. **When \( x = 4 \)**:
\[
f(4) = 375,000(1.06)^4 = 375,000(1.26247696) \approx 472,859
\]

6. **When \( x = 5 \)**:
\[
f(5) = 375,000(1.06)^5 = 375,000(1.338225) \approx 501,091
\]

Now, we can summarize the values in the table:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 375,000 \\
1 & 397,500 \\
2 & 421,350 \\
3 & 446,257 \\
4 & 472,859 \\
5 & 501,091 \\
\hline
\end{array}
\]

Now that we have the completed values, let's summarize them:
- At year 0: $375,000
- At year 1: $397,500
- At year 2: $421,350
- At year 3: $446,257
- At year 4: $472,859
- At year 5: $501,091

Pay attention to the description of the graphs:
1. **Increasing curve** - This would reflect the positive growth in home value.
2. **Decreasing curve** - This would reflect a decline in home value.

Since the value of the home is increasing at a rate of 6% per year, the correct graph that corresponds to this data is the **increasing curve**.

Therefore, the correct response is the graph that shows an increasing curve on a coordinate plane for the home values over time.