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To analyze the data and determine whether a linear, quadratic, or exponential function models it best, let's first summarize the data from the table:

\[
\begin{array}{c|c}
x & y \\
\hline
1 & \frac{7}{4} \\
2 & \frac{7}{2} \\
3 & 7 \\
4 & 14 \\
5 & 28 \\
\end{array}
\]

We are given three possible models:
1. Linear: \( y = mx + b \)
2. Quadratic: \( y = ax^2 + bx + c \)
3. Exponential: \( y = a(b)^x \)

### Step 1: Test for a Linear Model
In a linear function, the difference between successive \( y \)-values should be constant. Let's calculate the differences:
- \( y_2 - y_1 = \frac{7}{2} - \frac{7}{4} = \frac{7}{4} \)
- \( y_3 - y_2 = 7 - \frac{7}{2} = \frac{7}{2} \)
- \( y_4 - y_3 = 14 - 7 = 7 \)
- \( y_5 - y_4 = 28 - 14 = 14 \)

The differences are not constant, so a linear model does not fit.

### Step 2: Test for a Quadratic Model
For a quadratic model, the second differences between successive \( y \)-values should be constant. We already have the first differences:
- \( y_2 - y_1 = \frac{7}{4} \)
- \( y_3 - y_2 = \frac{7}{2} \)
- \( y_4 - y_3 = 7 \)
- \( y_5 - y_4 = 14 \)

Now, calculate the second differences:
- \( \left( \frac{7}{2} - \frac{7}{4} \right) = \frac{7}{4} \)
- \( \left( 7 - \frac{7}{2} \right) = \frac{7}{2} \)
- \( \left( 14 - 7 \right) = 7 \)

These are not constant either, so a quadratic model does not fit.

### Step 3: Test for an Exponential Model
For an exponential function \( y = a(b)^x \), the ratio between successive \( y \)-values should be constant. Let's calculate the ratios:
- \( \frac{y_2}{y_1} = \frac{\frac{7}{2}}{\frac{7}{4}} = 2 \)
- \( \frac{y_3}{y_2} = \frac{7}{\frac{7}{2}} = 2 \)
- \( \frac{y_4}{y_3} = \frac{14}{7} = 2 \)
- \( \frac{y_5}{y_4} = \frac{28}{14} = 2 \)

Since the ratios are constant, the data fits an exponential model of the form \( y = a(b)^x \).

### Step 4: Determine the Parameters for the Exponential Model
We know the ratio \( b = 2 \). Now, let's determine \( a \) by using the first data point \( (x = 1, y = \frac{7}{4}) \):

\[
\frac{7}{4} = a(2)^1 \implies a = \frac{7}{8}
\]

Thus, the exponential model is:

\[
y = \frac{7}{8} \cdot 2^x
\]

This equation models the given data.

Would you like more details, or do you have any further questions?

Here are five related questions to expand on this:
1. How would you solve this if given a different set of \( y \)-values?
2. Can you derive an exponential model using logarithms?
3. How would you identify a quadratic model if the second differences were constant?
4. Can you find the values of \( a \) and \( b \) if the ratio between successive terms was not an integer?
5. How would this process differ if the data followed a logarithmic pattern?

**Tip**: Always check for patterns (constant differences or ratios) to identify whether a model is linear, quadratic, or exponential.

Write a linear (y = mx + b), quadratic (y = ax^2), or exponential (y = a(b)^x) function that models the data from the given table.

Learn how to determine whether a linear, quadratic, or exponential function best models a given set of data. This example explores first differences, second differences, and ratios between values to identify patterns suitable for high school algebra and pre-calculus students.

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