Question

interpreting values within the regression equation

the regression equation you found for the water lilies is \\(y = 3.915(1.106)^x\\).

in terms of the water lily population change, the value 3.915 represents:

the value 1.106 represents:

Explanation:

Identify the model structure

Using the Exponential Function Properties knowledge point

$$ y = a \cdot b^x $$

where \(a = 3.915\) and \(b = 1.106\).

Interpret the initial value

Using the Exponential Growth and Decay Behavior knowledge point

$$ y(0) = 3.915 \cdot (1.106)^0 = 3.915 $$

This represents the initial population of water lilies (at \(x = 0\)).

Interpret the growth factor

Using the Exponential Growth and Decay Behavior knowledge point

$$ b = 1.106 = 1 + 0.106 $$

This represents the growth factor, meaning the population is multiplied by \(1.106\) each unit of time (or increases by \(10.6\%\) per unit of time).

Answer:

The regression equation you found for the water lilies is \(y = 3.915(1.106)^x\).

In terms of the water lily population change, the value 3.915 represents: <blank>the initial population of water lilies</blank>

The value 1.106 represents: <blank>the growth factor (or the factor by which the population is multiplied each unit of time)</blank>