Question

average rate of change quick check
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)
( \frac{f(10)-f(6)}{10 - 6}=\frac{62 - 38}{10+6}=1.5 ) leaves
( f(6)+f(4)=62 + 38=100 ) leaves
( \frac{f(10)-f(6)}{10 - 6}=\frac{62 - 38}{10 - 6}=6 ) leaves
( \frac{f(10)+f(6)}{10 - 6}=\frac{62+38}{10 - 6}=25 ) leaves

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 6\), \(b=10\), and \(f(x)=6x + 2\).

Step2: Calculate \(f(10)\) and \(f(6)\)

• For \(x = 10\), \(f(10)=6\times10+2=60 + 2=62\).
• For \(x = 6\), \(f(6)=6\times6+2=36+2 = 38\).

Step3: Apply the average - rate - of - change formula

Substitute \(f(10) = 62\) and \(f(6)=38\) into \(\frac{f(b)-f(a)}{b - a}\), we get \(\frac{f(10)-f(6)}{10 - 6}=\frac{62-38}{4}\).

Answer:

\(\frac{f(10)-f(6)}{10 - 6}=\frac{62 - 38}{4}=6\) leaves. So the correct option is \(\frac{f(10)-f(6)}{10 - 6}=\frac{62-38}{4}=6\) leaves.