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.
