Question

the data in the following table indicate that between the ages of 1 and 11, the human brain does not grow linearly, or steadily. a scatter plot for the data is shown below the table. the graphing calculator screen to the right displays the percentage of an adult brain, y, for a child at age x, where 1 ≤ x ≤ 11. use this information to complete parts (a) through (c) below. a. explain why a logarithmic function was used to model the data. choose the correct answer below. a. the data increase rapidly and then continue to increase even more rapidly. b. the data increase rapidly and then begin to level off. c. the data increase at a steady rate. d. the data increase rapidly and then begin to decrease. b. use the graphing calculator screen to express the model in function notation, with numbers rounded to the nearest whole number. f(x) =

Explanation:

Step1: Analyze the characteristics of logarithmic function

The general form of a logarithmic function \(y = a + b\ln x\) (\(b>0\)) has a graph that increases rapidly at first and then levels off as \(x\) gets larger.

Step2: Analyze each option

• Option A: A function that increases rapidly and then continues to increase even more rapidly is more like an exponential function (\(y = a\cdot b^{x}\), \(b > 1\)) or a power function (\(y=ax^{n}\), \(n>1\)).
• Option B: Since the logarithmic function \(y=a + b\ln x\) (\(b>0\)) has the property of increasing rapidly at first and then leveling off, this option is correct.
• Option C: A linear function \(y=mx + c\) (\(m>0\)) has a steady - rate of increase.
• Option D: A quadratic function \(y = ax^{2}+bx + c\) (\(a<0\)) or some other non - logarithmic functions may increase and then decrease.

Step3: Write the function in function notation

Given \(y=a + b\ln x\), where \(a = 31.96101054\approx32\) and \(b=28.84887617\approx29\). In function notation \(f(x)=32 + 29\ln x\)

Answer:

a. B. The data increase rapidly and then begin to level off.
b. \(f(x)=32 + 29\ln x\)