Question

If the function  g(x)=6x+2 models the number of leaves on a plant x weeks after being planted, which of the following accurately calculates the average rate of change in leaves between weeks 6 and 10?(1 point)Responsesf(b)−f(a)b+a=62−3810+6=1.5 leavesStart Fraction f left parenthesis b right parenthesis minus f left parenthesis a right parenthesis over b plus a End Fraction equals Start Fraction 62 minus 38 over 10 plus 6 End Fraction equals 1.5  leavesf(b)+f(a)=62+38=100 leavesf left parenthesis b right parenthesis plus f left parenthesis a right parenthesis equals 62 plus 38 equals 100 leavesf(b)+f(a)b−a=62+3810−6=25 leavesStart Fraction f left parenthesis b right parenthesis plus f left parenthesis a right parenthesis over b minus a End Fraction equals Start Fraction 62 plus 38 over 10 minus 6 End Fraction equals 25  leavesf(b)−f(a)b−a=62−3810−6=6 leaves

Answer 1

To calculate the average rate of change of the function \( g(x) = 6x + 2 \) between weeks 6 and 10, we will use the formula for the average rate of change:

\[ \text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a} \]

where \( a = 6 \) and \( b = 10 \).

Now, we will first evaluate \( g(6) \) and \( g(10) \):

1. Calculate \( g(6) \): \[ g(6) = 6(6) + 2 = 36 + 2 = 38 \]

2. Calculate \( g(10) \): \[ g(10) = 6(10) + 2 = 60 + 2 = 62 \]

Now that we have \( g(6) = 38 \) and \( g(10) = 62 \), we substitute these values into the average rate of change formula:

\[ \text{Average Rate of Change} = \frac{62 - 38}{10 - 6} = \frac{24}{4} = 6 \]

So the average rate of change in leaves between weeks 6 and 10 is \( 6 \) leaves per week.

The correct choice from the given responses is:

\[ f(b) - f(a) \over b - a = \frac{62 - 38}{10 - 6} = 6 \text{ leaves} \]