IDNLearn.com provides a comprehensive platform for finding accurate answers. Our platform provides trustworthy answers to help you make informed decisions quickly and easily.
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.
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.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Find clear answers at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.