What is the exponential regression equation that fits these data?
| x | y |
|---|---|
| 1 | 3 |
| 2 | 8 |
| 3 | 27 |
| 4 | 85 |
| 5 | 240 |
| 6 | 570 |
A. [tex]$y=102.54 x-203.4$[/tex]
B. [tex]$y=38.73 x^2-168.58 x+158.1$[/tex]
C. [tex]$y=1.03(2.93^x)$[/tex]
Answer (1)
Analyze the problem and identify the type of equation needed.
Eliminate options that are not exponential.
Verify the remaining option by plugging in the given data points.
Conclude that the equation that best fits the data is y = 1.03 ( 2.9 3 x ) .
Explanation
Analyzing the Problem and Options We are given a data set of x and y values and asked to find the exponential regression equation that best fits the data. The general form of an exponential equation is y = a x , where a is the initial value and b is the growth factor. We are given three possible equations:
A. y = 102.54 x − 203.4 B. y = 38.73 x 2 − 168.58 x + 158.1 C. y = 1.03 ( 2.9 3 x )
We can eliminate options A and B because they are linear and quadratic equations, respectively, and we are looking for an exponential equation. Therefore, option C is the only possible exponential equation. We need to check if it fits the data.
Verifying Option C with Data Points Let's test option C, y = 1.03 ( 2.9 3 x ) , with the given data points:
For x = 1 , y = 1.032.9 3 1 = 3.0179 ≈ 3 For x = 2 , y = 1.032.9 3 2 = 8.842 ≈ 8 For x = 3 , y = 1.032.9 3 3 = 25.947 ≈ 27 For x = 4 , y = 1.032.9 3 4 = 75.91 ≈ 85 For x = 5 , y = 1.032.9 3 5 = 222.42 ≈ 240 For x = 6 , y = 1.032.9 3 6 = 651.69 ≈ 570
The calculated y values are close to the actual y values in the table.
Conclusion Since option C is the only exponential equation and it closely fits the given data, it is the correct answer.
Examples
Exponential regression is used in various fields, such as finance, biology, and physics, to model data that grows or decays exponentially. For example, it can be used to model the growth of a population, the decay of a radioactive substance, or the growth of an investment over time. Understanding exponential regression can help you make predictions and informed decisions based on data.
