Question

f(x)=6x+5
Find the average rate of the change from x=1 to x=5 . Only the value.

Find the average rate of change from x=-2 to x=6. Only the value.

Answer 1

$\bf slope = m = \cfrac{rise}{run} \implies \cfrac{ f(x_2) - f(x_1)}{ x_2 - x_1}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array}\\\\[-0.35em] \rule{31em}{0.25pt}\\\\ f(x)= 6x+5\qquad \begin{cases} x_1=1\\ x_2=5 \end{cases}\implies \cfrac{f(5)-f(1)}{5-1} \\\\\\ \cfrac{[6(5)+5]~~-~~[6(1)+5]}{4}\implies \cfrac{35-11}{4}\implies \cfrac{24}{4}\implies \boxed{6}$

$\bf ~\hspace{8em} \begin{cases} x_1=-2\\ x_2=6 \end{cases}\implies \cfrac{f(6)-f(-2)}{6-(-2)} \\\\\\ \cfrac{[6(6)+5]-[6(-2)+5]}{6+2}\implies \cfrac{41-(-7)}{8}\implies \cfrac{41+7}{8}\implies \boxed{6}$