6. given ( f(x) = x^2 - 4x + 7 )
a. find the average rate of change of the function on the interval (-1,1).

b. describe the relationship between the rate of change and the behavior of the function on this interval.

c. how would you describe the rate of change at ( x = 2 )?

Question

6. given ( f(x) = x^2 - 4x + 7 )
a. find the average rate of change of the function on the interval (-1,1).

b. describe the relationship between the rate of change and the behavior of the function on this interval.

c. how would you describe the rate of change at ( x = 2 )?

Accepted Answer

### Part A
# Explanation:
## Step1: Recall the average rate of change formula
The average rate of change of a function $ f(x) $ on the interval $[a, b]$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $ a=-1 $ and $ b = 1 $, and $ f(x)=x^{2}-4x + 7 $.

## Step2: Calculate $ f(-1) $
Substitute $ x=-1 $ into $ f(x) $:
$$
\begin{align*}
f(-1)&=(-1)^{2}-4(-1)+7\
&=1 + 4+7\
&=12
\end{align*}
$$

## Step3: Calculate $ f(1) $
Substitute $ x = 1 $ into $ f(x) $:
$$
\begin{align*}
f(1)&=(1)^{2}-4(1)+7\
&=1-4 + 7\
&=4
\end{align*}
$$

## Step4: Calculate the average rate of change
Using the formula $\frac{f(1)-f(-1)}{1-(-1)}$:
$$
\begin{align*}
\frac{f(1)-f(-1)}{1-(-1)}&=\frac{4 - 12}{1 + 1}\
&=\frac{-8}{2}\
&=-4
\end{align*}
$$

# Answer:
The average rate of change on $[-1,1]$ is $-4$.

### Part B
# Brief Explanations:
The average rate of change we found is $-4$ (negative). For a quadratic function $ f(x)=x^{2}-4x + 7 $, the graph is a parabola opening upwards (since the coefficient of $ x^{2} $ is positive). The vertex of the parabola $ y = ax^{2}+bx + c $ is at $ x=-\frac{b}{2a}$, here $ x =-\frac{-4}{2\times1}=2 $. On the interval $[-1,1]$, which is to the left of the vertex $ x = 2 $, the function is decreasing (because as $ x $ increases from $-1$ to $ 1 $, the $ y $-values decrease from $ 12 $ to $ 4 $). A negative average rate of change indicates that the function is decreasing on the interval $[-1,1]$, so the rate of change (negative) is consistent with the function's behavior of decreasing on this interval.

# Answer:
The negative average rate of change ($-4$) indicates the function is decreasing on $[-1,1]$ (consistent with the parabola decreasing left of its vertex $ x = 2 $).

### Part C
# Explanation:
## Step1: Recall the concept of instantaneous rate of change
The rate of change at a point $ x = a $ for a function $ f(x) $ is the instantaneous rate of change, which is given by the derivative of the function at $ x = a $, i.e., $ f^{\prime}(a) $.

## Step2: Find the derivative of $ f(x) $
Given $ f(x)=x^{2}-4x + 7 $, using the power rule $(x^{n})^\prime=nx^{n - 1}$, the derivative $ f^{\prime}(x) $ is:
$$
f^{\prime}(x)=2x-4
$$

## Step3: Evaluate the derivative at $ x = 2 $
Substitute $ x = 2 $ into $ f^{\prime}(x) $:
$$
f^{\prime}(2)=2(2)-4=4 - 4=0
$$

# Answer:
The rate of change at $ x = 2 $ (instantaneous rate) is $ 0 $. This is the slope of the tangent line at $ x = 2 $, and since $ x = 2 $ is the vertex of the parabola, the function has a minimum (or maximum, but here it's a minimum as the parabola opens upwards) at this point, so the tangent line is horizontal (slope $ 0 $).

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QUESTION IMAGE

Question

Explanation:

Part A

Step1: Recall the average rate of change formula

Step2: Calculate \( f(-1) \)

Step3: Calculate \( f(1) \)

Step4: Calculate the average rate of change

Step1: Recall the concept of instantaneous rate of change

Step2: Find the derivative of \( f(x) \)

Step3: Evaluate the derivative at \( x = 2 \)

Answer:

Part B