Let's analyze the options based on the nature of the data presented in the table and calculate the common ratios between the $y$-values.

Given data points:

• $x = 0, y = 10.000$
• $x = 1, y = 10.750$
• $x = 2, y = 11.688$
• $x = 3, y = 12.860$
• $x = 4, y = 14.325$

To verify each option, we can calculate the ratio between successive $y$-values to see if there’s a consistent ratio.

Step 1: Calculate Ratios Between Successive $y$-values

1. $\frac{10.750}{10.000} = 1.075$
2. $\frac{11.688}{10.750} \approx 1.087$
3. $\frac{12.860}{11.688} \approx 1.100$
4. $\frac{14.325}{12.860} \approx 1.114$

The ratios are not exactly consistent, meaning there isn’t a constant common ratio across all values. This rules out options indicating an exact common ratio.

Step 2: Evaluate Each Option

• Option A states that the $y$-values have a common ratio of 1.094, derived from an exponential regression. Since the ratios calculated between each successive pair are close but not exactly 1.094, this may be plausible if it's derived as an average in an exponential regression model. However, without precise exponential fitting calculations, this option is not directly verifiable here but could be true in a regression context.

• Option B mentions a common ratio of $1.075$ based on the initial ratio $\frac{10.750}{10.000}$. This ratio holds only between the first two values, but it’s not consistent across the table. This option is misleading because it implies a constant common ratio across all values, which isn’t observed.

• Option C suggests subtracting 7 from each $y$-value to obtain a common ratio of 1.25. Subtracting 7 from each $y$-value yields:

  • $y = 10.000 - 7 = 3.000$
  • $y = 10.750 - 7 = 3.750$
  • $y = 11.688 - 7 = 4.688$
  • $y = 12.860 - 7 = 5.860$
  • $y = 14.325 - 7 = 7.325$

  Then calculating ratios:

  • $\frac{3.750}{3.000} = 1.25$
  • $\frac{4.688}{3.750} \approx 1.25$
  • $\frac{5.860}{4.688} \approx 1.25$
  • $\frac{7.325}{5.860} \approx 1.25$

  This shows a consistent ratio of 1.25 after subtracting 7 from each $y$-value, so Option C is correct.

• Option D claims that there’s no common ratio, so an exponential model isn’t suitable. While the values don’t follow a strict common ratio pattern, Option C’s transformation (subtracting 7) reveals a consistent ratio, thus making this option incorrect.

Conclusion

The correct answer is Option C: "A common ratio of the $y$-values can be found as 1.25 after subtracting 7 from each $y$-value."

Would you like more details on exponential regression or the calculations?

Here are some related questions:

1. How can exponential regression be used to model non-linear data?
2. What is the significance of a common ratio in identifying exponential growth?
3. How does transformation of data (like subtraction) help in identifying patterns?
4. Could logarithmic transformations also reveal trends in similar data?
5. How would a linear regression model compare with an exponential model for this data?

Tip: When dealing with exponential patterns, checking for transformations can sometimes reveal hidden relationships in data.