IDNLearn.com provides a reliable platform for finding accurate and timely answers. Our community provides accurate and timely answers to help you understand and solve any issue.
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.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.