The general form of an exponential function is [tex]\( y = ab^x \)[/tex].

Use the regression calculator to find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] for the water lily population growth. Round to the nearest thousandth.

[tex]\( a = \square \)[/tex]

[tex]\( b = \square \)[/tex]



Answer :

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].