The table provided shows the values of $x$ and $y$, and you're asked to determine the function type and equation to match the data.

Step 1: Analyze the data

You have the following pairs of values:

Step 2: Check function type

1. Linear: A linear function has a constant rate of change (difference in $y$-values). Here, the differences in $y$ are not constant, so this is not a linear function.

2. Quadratic: A quadratic function has a pattern where the second differences are constant. After testing this, the differences are not constant, so this is not a quadratic function.

3. Exponential: An exponential function follows a pattern where the ratio between consecutive $y$-values is constant. Let's check if the ratios are consistent:

   • $\frac{31}{69} \approx 0.449$
   • $\frac{9}{31} \approx 0.290$
   • $\frac{4}{9} \approx 0.444$
   • $\frac{3}{4} = 0.75$

   While the ratios are not perfectly consistent, the values suggest an exponential decay, so this is most likely an exponential function.

Step 3: Match the equation

Given the choices, the closest match for an exponential function is: $y = 6.73 \cdot 0.63^x$ This equation fits the pattern of exponential decay, where $y$ decreases as $x$ increases.

Conclusion

The function type is exponential, and the equation that best matches the data is: $y = 6.73 \cdot 0.63^x$

Would you like more details on how the ratios are calculated, or any other part of the explanation?

Here are 5 related questions for further understanding:

1. How do you identify a linear function based on a table of values?
2. What method can be used to determine if a function is quadratic?
3. What are the characteristics of exponential growth and decay?
4. How do you calculate ratios between consecutive terms in an exponential sequence?
5. How would you transform an exponential function if the base is greater than 1?

Tip: Exponential decay occurs when the base of the exponent is between 0 and 1, causing the function's output to decrease as $x$ increases.