Question

Use the table to answer the question. x-values y-values 0 5 1 35 2 245 3 1715 4 12005 5 84035 Given the following xy 𝑥 𝑦 -chart, create a function model. Does this illustrate a linear function, an exponential function, or a quadratic function to represent the dataset? (1 point) Responses f(x)=5(7)x 𝑓 ( 𝑥 ) = 5 ( 7 ) 𝑥 and exponential function f left parenthesis x right parenthesis equals 5 left parenthesis 7 right parenthesis superscript x baseline and exponential function f(x)=5(7)x 𝑓 ( 𝑥 ) = 5 ( 7 ) 𝑥 and quadratic function f left parenthesis x right parenthesis equals 5 left parenthesis 7 right parenthesis superscript x baseline and quadratic function f(x)=5x+30 𝑓 ( 𝑥 ) = 5 𝑥 + 30 and linear function f left parenthesis x right parenthesis equals 5 x plus 30 and linear function f(x)=7(5)x 𝑓 ( 𝑥 ) = 7 ( 5 ) 𝑥 and exponential function

Answer 1

To analyze the dataset provided, we can look at the values of \( y \) for each corresponding \( x \):

- \( x = 0 \), \( y = 5 \)
- \( x = 1 \), \( y = 35 \)
- \( x = 2 \), \( y = 245 \)
- \( x = 3 \), \( y = 1715 \)
- \( x = 4 \), \( y = 12005 \)
- \( x = 5 \), \( y = 84035 \)

First, let's observe how the values of \( y \) change with respect to \( x \).

### Change in \( y \) Values
To understand whether the function is linear, exponential, or quadratic, we can look at the ratios of successive \( y \) values.

1. From \( x = 0 \) to \( x = 1 \):
\[ \frac{35}{5} = 7 \]

2. From \( x = 1 \) to \( x = 2 \):
\[ \frac{245}{35} = 7 \]

3. From \( x = 2 \) to \( x = 3 \):
\[ \frac{1715}{245} = 7 \]

4. From \( x = 3 \) to \( x = 4 \):
\[ \frac{12005}{1715} = 7 \]

5. From \( x = 4 \) to \( x = 5 \):
\[ \frac{84035}{12005} = 7 \]

Each ratio is constant (7), which indicates that the function is exponential.

### Conclusion
This dataset can be modeled by an exponential function of the form:
\[
f(x) = 5(7^x)
\]
Thus, the best fit for the function based on the given options clearly shows that this is an exponential function.

### Final Answer
The function model is \( f(x) = 5(7^x) \) and it represents an exponential function.