8. given ( y = x^{2}+5x - 6 ), show that the average rate of change between the points ( x = 4 ) and ( x=(4 + h) ) is ( h + 13 )

Question

Accepted Answer

# 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 $f(4 + h)=h^{2}+13h+30$.
$$
\begin{align*}
\frac{y(4 + h)-y(4)}{h}&=\frac{(h^{2}+13h + 30)-30}{h}\
&=\frac{h^{2}+13h}{h}\
&=\frac{h(h + 13)}{h}\
&=h + 13\quad(h
eq0)
\end{align*}
$$

# 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.

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QUESTION IMAGE

Question

Explanation:

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

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

Step3: Calculate the average rate of change

Answer: