To determine which type of function would best model the given data, we can consider the relationship between the side length of the square pyramid and its surface area. Let's summarize and analyze the provided data points:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Side Length} (cm) & \text{Surface Area} (cm^2) \\
\hline
0 & 0 \\
\hline
1 & 11 \\
\hline
2 & 24 \\
\hline
3 & 39 \\
\hline
4 & 56 \\
\hline
\end{array}
\][/tex]
To determine the type of function, we observe the data points and check their behavior. A common approach is to fit a polynomial to the data.
In fitting the data points to a polynomial function, we find the following coefficients for the polynomial:
[tex]\[
\text{Coefficients: } [1.0, 10.0, 9.53293115 \times 10^{-15}]
\][/tex]
These coefficients suggest the following polynomial equation:
[tex]\[
\text{Surface Area} = 1.0 \times (\text{Side Length})^2 + 10.0 \times (\text{Side Length}) + (9.53293115 \times 10^{-15})
\][/tex]
Given that the coefficient for the quadratic term is significant and the coefficients for the linear and constant terms also play a notable role, we observe that the data fits a quadratic function well.
[tex]\[
\text{Surface Area} = \text{Side Length}^2 + 10 \times \text{Side Length}
\][/tex]
This polynomial equation is a quadratic function. Therefore, the type of function that best models the data is:
[tex]\[ \boxed{\text{Quadratic}} \][/tex]
Now, let’s analyze the graph of the data. Given that the surface area grows quadratically with the side length, the graph will be a parabola opening upwards.
So, the description of the graph of the data would be:
[tex]\[ \boxed{\text{An upward-opening parabola}} \][/tex]
Thus, the best description of the graph of the data is an upward-opening parabola, and the type of function that would best model the data is quadratic.
