To determine which function family best represents the data provided, we can visually or mathematically analyze the pattern of the $y$ values corresponding to the $x$ values.

Here are the $x$ and $y$ pairs from your dataTo determine which function family best represents the data provided, we can visually or mathematically analyze the pattern of the $y$ values corresponding to the $x$ values.

Here are the $x$ and $y$ pairs from your

\hline x & y \\ \hline -6.1 & -8.4 \\ -5.9 & -7.6 \\ -4.5 & -3.2 \\ -3.1 & -1.0 \\ 1.1 & 0 \\ 1.2 & 0.1 \\ 1.9 & 0.4 \\ 4.8 & 3.8 \\ 5.3 & 5.9 \\ 6.0 & 8.1 \\ \hline \end{array}$$ ### Steps to determine the best function family: 1. **Linear Function**: A linear function has the form $$y = mx + b$$, where the relationship between $$x$$ and $$y$$ is a straight line. To check if the data is linear, we could plot the points or calculate the difference between consecutive $$y$$ values. If the difference is constant, it suggests a linear relationship. 2. **Quadratic Function**: A quadratic function has the form $$y = ax^2 + bx + c$$. If the data appears to follow a parabolic curve, it might fit a quadratic function. Here, the difference between consecutive $$y$$ values would change at a constant rate (second difference is constant). 3. **Exponential Function**: An exponential function has the form $$y = a \cdot b^x$$. This suggests rapid increase or decrease in $$y$$ values as $$x$$ changes. 4. **Logarithmic Function**: A logarithmic function has the form $$y = a \cdot \ln(x) + b$$. This function increases or decreases rapidly at first and then slows down. 5. **Cubic Function**: A cubic function has the form $$y = ax^3 + bx^2 + cx + d$$. This would suggest an "S" shaped curve if the data follows a cubic pattern. ### Analysis of the Data: - **Initial Observation**: The data appears to have a general increasing trend. - **Non-Linearity**: The data does not seem to follow a straight line, as the increments between $$y$$ values are not consistent. - **Second Differences**: If we calculate the second differences between $$y$$ values, and they are constant, this might suggest a quadratic relationship. Given the gradual increase in $$y$$ values and potential non-linear pattern, the data might fit well with a quadratic function, suggesting that the best fit could be: **Quadratic Function**. Would you like me to perform a detailed calculation or plotting to confirm this hypothesis? --- ### Relative Questions: 1. How can the second difference help in identifying a quadratic function? 2. What are the properties of a quadratic function that distinguish it from other function families? 3. Can an exponential function also fit this data? If yes, under what circumstances? 4. How do you calculate and interpret the first and second differences of a dataset? 5. What tools can be used to fit a function to a given dataset? **Tip**: For datasets that aren't clearly linear, checking second differences or plotting the data can often reveal the underlying pattern.