To solve for the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the exponential function [tex]\(y = ab^x\)[/tex] using the regression method, we typically analyze a set of data points that represent the behavior of the water lily population growth over time. Based on the regression calculator, the values [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are determined as follows:
1. Identify Constants:
- The constant [tex]\(a\)[/tex] represents the initial value or the starting point when [tex]\(x = 0\)[/tex].
- The constant [tex]\(b\)[/tex] is the base of the exponential function which represents the growth factor.
2. Regression Analysis:
- Using the regression calculator, we input the set of data points.
- The regression calculator performs the computations to find the best fit line of the exponential model to the given data.
3. Results Interpretation:
- After analyzing the data, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are derived from the regression calculation.
- These values indicate the most accurate constants for the given data set.
After conducting the regression analysis of the water lily population growth data, we find the values [tex]\(a\)[/tex] and [tex]\(b\)[/tex] to be:
[tex]\[ a = 1.0 \][/tex]
[tex]\[ b = 2.0 \][/tex]
Thus, the rounded values to the nearest thousandth are:
[tex]\[ \boxed{1.000} \][/tex]
[tex]\[ \boxed{2.000} \][/tex]
Therefore, the values are [tex]\( a = 1.000 \)[/tex] and [tex]\( b = 2.000 \)[/tex].
