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Look at this table:

[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
1 & 3 \\
\hline
2 & 9 \\
\hline
3 & 27 \\
\hline
4 & 81 \\
\hline
5 & 243 \\
\hline
\end{tabular}
\][/tex]

Write a linear [tex]\((y = mx + b)\)[/tex], quadratic [tex]\(\left(y = ax^2\right)\)[/tex], or exponential [tex]\(\left(y = a(b)^x\right)\)[/tex] function that models the data.

[tex]\[ y = \][/tex]

[tex]\(\square\)[/tex]


Sagot :

To model the given data points with an appropriate function, we start by assessing the patterns in the data.

The data points are:
[tex]\[ (1, 3), (2, 9), (3, 27), (4, 81), (5, 243) \][/tex]

Step-by-Step Solution:

1. Identify the Pattern:
- Notice how y changes as x increases. For each increment in x, y seems to increase multiplicatively.
- Rapidly increasing values suggest an exponential relationship.

2. Form of the Exponential Model:
- Let's assume the data follows the model form: [tex]\( y = a \cdot b^x \)[/tex]

3. Determine Constants [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- By observing the pattern:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 9 \)[/tex]
- We will fit the model [tex]\( y = a \cdot b^x \)[/tex] using the given data:

4. Use the first data point to find [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- For [tex]\( x = 1 \)[/tex]: [tex]\( 3 = a \cdot b^1 \implies a \cdot b = 3 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( 9 = a \cdot b^2 \implies a \cdot b^2 = 9 \)[/tex]

5. Solve for [tex]\(b\)[/tex]:
- From the equations formed:
- [tex]\( a \cdot b = 3 \)[/tex]
- [tex]\( a \cdot b^2 = 9 \)[/tex]
- Dividing the second equation by the first:
[tex]\( \frac{a \cdot b^2}{a \cdot b} = \frac{9}{3} \)[/tex]
[tex]\( b = 3 \)[/tex]

6. Solve for [tex]\(a\)[/tex]:
- Substitute [tex]\( b = 3 \)[/tex] into [tex]\( a \cdot b = 3 \)[/tex]:
[tex]\( a \cdot 3 = 3 \implies a = 1 \)[/tex]

7. Finalize the Exponential Model:
- With [tex]\( a = 1 \)[/tex] and [tex]\( b = 3 \)[/tex], we get the exponential function:
[tex]\[ y = 1 \cdot 3^x = 3^x \][/tex]

Therefore, the model that fits the data is given by:
[tex]\[ y = 3^x \][/tex]

This exponential function [tex]\( y = 3^x \)[/tex] accurately represents the pattern observed in the given data points.