Question

modeling with exponential and logarithmic equations assignment active 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: data x 0 5 10 15 20 25 30 y 4 7 10 15 33 51 79 exponential regression y ≈ 3.915(1.106)^x, r ≈ 0.996

Explanation:

Step1: Analyze the exponential model

The general form of an exponential function is \( y = a(b)^x \), where \( a \) is the initial value and \( b \) is the growth factor.

Step2: Identify the role of 3.915

In the equation \( y = 3.915(1.106)^x \), when \( x = 0 \), \( y = 3.915(1.106)^0 = 3.915 \). So 3.915 represents the initial number of water lilies (when \( x = 0 \), the starting population).

Step3: Identify the role of 1.106

In the exponential growth model \( y = a(b)^x \), \( b \) is the growth factor. If \( b>1 \), it represents growth. Here, \( 1.106>1 \), so \( 1.106 = 1 + r \), where \( r \) is the growth rate. So 1.106 represents the growth factor (or the factor by which the population multiplies each time period, indicating a 10.6% growth rate since \( 1.106 - 1 = 0.106 \) or 10.6%).

Answer:

• The value 3.915 represents the initial number of water lilies (the population when \( x = 0 \)).
• The value 1.106 represents the growth factor (the factor by which the water lily population grows each time period, corresponding to a 10.6% growth rate).