Questions: Identify the function type that best models the data. x y [Select ] --------- 4 2 8 6 12 18 16 54 x y ------ 10 6 20 13 30 17 40 20 x y ------ 2 12 6 18 12 21 18 27

Transcript text: Identify the function type that best models the data. \begin{tabular}{|l|l|l|} \hline$x$ & $y$ & [Select ] \\ \hline 4 & 2 & \\ \hline 8 & 6 & \\ \hline 12 & 18 & \\ \hline 16 & 54 & \\ \hline \end{tabular} \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 10 & 6 \\ \hline 20 & 13 \\ \hline 30 & 17 \\ \hline 40 & 20 \\ \hline \end{tabular} \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 2 & 12 \\ \hline 6 & 18 \\ \hline 12 & 21 \\ \hline 18 & 27 \\ \hline \end{tabular}

Solution

Solution Steps

To identify the function type that best models the data, we can use Python to fit different types of functions (linear, quadratic, exponential, etc.) to the given data points and then determine which function has the best fit based on the least squares error or another fitting metric.

Step 1: Identify the Function Type for the First Data Set

Given data: \[ \begin{array}{|c|c|} \hline x & y \\ \hline 4 & 2 \\ \hline 8 & 6 \\ \hline 12 & 18 \\ \hline 16 & 54 \\ \hline \end{array} \]

We need to determine the function type that best models this data. Let's check if it fits an exponential function of the form $ y = a \cdot b^x $.

First, calculate the ratios of consecutive $ y $-values: \[ \frac{y_2}{y_1} = \frac{6}{2} = 3, \quad \frac{y_3}{y_2} = \frac{18}{6} = 3, \quad \frac{y_4}{y_3} = \frac{54}{18} = 3 \]

Since the ratios are constant, the data fits an exponential function.

Final Answer for First Data Set

$\boxed{\text{Exponential}}$

Step 2: Identify the Function Type for the Second Data Set

Given data: \[ \begin{array}{|c|c|} \hline x & y \\ \hline 10 & 6 \\ \hline 20 & 13 \\ \hline 30 & 17 \\ \hline 40 & 20 \\ \hline \end{array} \]

We need to determine the function type that best models this data. Let's check if it fits a linear function of the form $ y = mx + c $.

Calculate the differences of consecutive $ y $-values: \[ y_2 - y_1 = 13 - 6 = 7, \quad y_3 - y_2 = 17 - 13 = 4, \quad y_4 - y_3 = 20 - 17 = 3 \]

Since the differences are not constant, the data does not fit a linear function. Let's check if it fits a quadratic function of the form $ y = ax^2 + bx + c $.

Calculate the second differences: \[ (13 - 6) - (17 - 13) = 7 - 4 = 3, \quad (17 - 13) - (20 - 17) = 4 - 3 = 1 \]

Since the second differences are not constant, the data does not fit a quadratic function either. Given the pattern, it is likely a polynomial of higher degree or another type of function.

Final Answer for Second Data Set

$\boxed{\text{Other}}$

Step 3: Identify the Function Type for the Third Data Set

Given data: \[ \begin{array}{|c|c|} \hline x & y \\ \hline 2 & 12 \\ \hline 6 & 18 \\ \hline 12 & 21 \\ \hline 18 & 27 \\ \hline \end{array} \]

We need to determine the function type that best models this data. Let's check if it fits a linear function of the form $ y = mx + c $.

Calculate the differences of consecutive $ y $-values: \[ y_2 - y_1 = 18 - 12 = 6, \quad y_3 - y_2 = 21 - 18 = 3, \quad y_4 - y_3 = 27 - 21 = 6 \]

Since the differences are not constant, the data does not fit a linear function. Let's check if it fits a quadratic function of the form $ y = ax^2 + bx + c $.

Calculate the second differences: \[ (18 - 12) - (21 - 18) = 6 - 3 = 3, \quad (21 - 18) - (27 - 21) = 3 - 6 = -3 \]

Since the second differences are not constant, the data does not fit a quadratic function either. Given the pattern, it is likely a polynomial of higher degree or another type of function.

Final Answer for Third Data Set

$\boxed{\text{Other}}$

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