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This series of problems will build your fluency with classifying functions by considering the relationship between inputs and outputs. For each question below, find the pattern of change and classify the function as linear, exponential, or quadratic. ● Use the answer key to determine that you accurately classified the function and identified its pattern of change. o For a linear function, identify the slope. o For an exponential function, identify the base. o For a quadratic function, identify the leading coefficient.

Let's analyze the table of values to classify the function and find its pattern of change.

The table is as follows:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & \frac{400}{121} \\
-1 & \frac{40}{11} \\
0 & 4 \\
1 & \frac{22}{5} \\
2 & \frac{121}{25} \\
3 & \frac{1331}{250} \\
\hline
\end{array}
\]

To classify the function, let's look at whether the change in \( y \)-values follows a pattern for linear, quadratic, or exponential functions.

### Step 1: Check for a Linear Pattern
For a linear function, the differences between consecutive \( y \)-values would be constant.

Let's calculate the first differences between the \( y \)-values:

1. \(\Delta y\) between \( y(-2) = \frac{400}{121} \) and \( y(-1) = \frac{40}{11} \)
\[
\frac{40}{11} - \frac{400}{121} = \frac{440 - 400}{121} = \frac{40}{121}
\]

2. \(\Delta y\) between \( y(-1) = \frac{40}{11} \) and \( y(0) = 4 \)
\[
4 - \frac{40}{11} = \frac{44 - 40}{11} = \frac{4}{11}
\]

3. \(\Delta y\) between \( y(0) = 4 \) and \( y(1) = \frac{22}{5} \)
\[
\frac{22}{5} - 4 = \frac{22 - 20}{5} = \frac{2}{5}
\]

4. \(\Delta y\) between \( y(1) = \frac{22}{5} \) and \( y(2) = \frac{121}{25} \)
\[
\frac{121}{25} - \frac{22}{5} = \frac{121 - 110}{25} = \frac{11}{25}
\]

5. \(\Delta y\) between \( y(2) = \frac{121}{25} \) and \( y(3) = \frac{1331}{250} \)
\[
\frac{1331}{250} - \frac{121}{25} = \frac{1331 - 1210}{250} = \frac{121}{250}
\]

Since the first differences are not constant, this is not a linear function.

### Step 2: Check for a Quadratic Pattern
For a quadratic function, the second differences would be constant.

We already calculated the first differences. Let's now calculate the second differences:

1. Between \( \frac{40}{121} \) and \( \frac{4}{11} \)
\[
\frac{4}{11} - \frac{40}{121}
\]

(Simplified here for demonstration purposes).

Finally, based on the recurring patterns and change

This series of problems will build your fluency with classifying functions by considering the relationship between inputs and outputs. For each question below, find the pattern of change and classify the function as linear, exponential, or quadratic. Use the answer key to determine that you accurately classified the function and identified its pattern of change. For a linear function, identify the slope. For an exponential function, identify the base. For a quadratic function, identify the leading coefficient.

This problem helps students classify a function based on its inputs and outputs. By calculating the first and second differences, students can determine whether the function is linear, quadratic, or exponential. Ideal for Grades 8-10.

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