8. given ( y = x^{2}+5x - 6 ), show that the average rate of change between the points ( x = 4 ) and ( x=(4…

Explanation:

Step1: Find (y) at (x = 4)

Substitute (x = 4) into (y=x^{2}+5x - 6). [ \begin{align*} y(4)&=4^{2}+5\times4-6\ &=16 + 20-6\ &=30 \end{align*} ]

Step2: Find (y) at (x=(4 + h))

Substitute (x=(4 + h)) into (y=x^{2}+5x - 6). [ \begin{align*} y(4 + h)&=(4 + h)^{2}+5(4 + h)-6\ &=16+8h+h^{2}+20 + 5h-6\ &=h^{2}+13h+30 \end{align*} ]

Step3: Calculate the average rate of change

The formula for the average rate of change of a function (y = f(x)) from (x=a) to (x=a + h) is (\frac{f(a + h)-f(a)}{h}). Here (a = 4), (f(4)=30) and (f(4 + h)=h^{2}+13h+30). [ \begin{align*} \frac{y(4 + h)-y(4)}{h}&=\frac{(h^{2}+13h + 30)-30}{h}\ &=\frac{h^{2}+13h}{h}\ &=\frac{h(h + 13)}{h}\ &=h + 13\quad(h\neq0) \end{align*} ]

Answer:

The average rate of change of (y=x^{2}+5x - 6) from (x = 4) to (x=(4 + h)) is (h + 13) as shown above.