Question

find the functions average rate of change over each interval.
a. from x = -3 to x = -2
b. from x = -2 to x = 1
c. from x = 0 to x = 1
d. from x = 1 to x = 2
e. from x = -1 to x = 0
f. from x = -1 to x = 2
(there is a graph of the function f(x) on the right side of the text, with x-axis and y-axis labeled, and the curve of f(x) plotted.)

Explanation:

To find the average rate of change of a function \( f(x) \) over the interval \([a, b]\), we use the formula:

$$ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} $$

We first identify \( f(a) \) and \( f(b) \) from the graph for each interval.

Part a: From \( x = -3 \) to \( x = -2 \)

Step 1: Identify \( f(-3) \) and \( f(-2) \)

From the graph, at \( x = -3 \), \( f(-3) = -4 \); at \( x = -2 \), \( f(-2) = 2 \).

Step 2: Apply the formula

$$ \text{Average Rate of Change} = \frac{f(-2) - f(-3)}{-2 - (-3)} = \frac{2 - (-4)}{-2 + 3} = \frac{6}{1} = 6 $$

Part b: From \( x = -2 \) to \( x = 1 \)

Step 1: Identify \( f(-2) \) and \( f(1) \)

From the graph, \( f(-2) = 2 \); at \( x = 1 \), \( f(1) = -4 \).

Step 2: Apply the formula

$$ \text{Average Rate of Change} = \frac{f(1) - f(-2)}{1 - (-2)} = \frac{-4 - 2}{1 + 2} = \frac{-6}{3} = -2 $$

Part c: From \( x = 0 \) to \( x = 1 \)

Step 1: Identify \( f(0) \) and \( f(1) \)

From the graph, \( f(0) = -3 \); \( f(1) = -4 \).

Step 2: Apply the formula

$$ \text{Average Rate of Change} = \frac{f(1) - f(0)}{1 - 0} = \frac{-4 - (-3)}{1} = \frac{-1}{1} = -1 $$

Part d: From \( x = 1 \) to \( x = 2 \)

Step 1: Identify \( f(1) \) and \( f(2) \)

From the graph, \( f(1) = -4 \); at \( x = 2 \), \( f(2) = 0 \).

Step 2: Apply the formula

$$ \text{Average Rate of Change} = \frac{f(2) - f(1)}{2 - 1} = \frac{0 - (-4)}{1} = \frac{4}{1} = 4 $$

Part e: From \( x = -1 \) to \( x = 0 \)

Step 1: Identify \( f(-1) \) and \( f(0) \)

From the graph, at \( x = -1 \), \( f(-1) = 0 \); \( f(0) = -3 \).

Step 2: Apply the formula

$$ \text{Average Rate of Change} = \frac{f(0) - f(-1)}{0 - (-1)} = \frac{-3 - 0}{0 + 1} = \frac{-3}{1} = -3 $$

Part f: From \( x = -1 \) to \( x = 2 \)

Step 1: Identify \( f(-1) \) and \( f(2) \)

From the graph, \( f(-1) = 0 \); \( f(2) = 0 \).

Step 2: Apply the formula

$$ \text{Average Rate of Change} = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{0 - 0}{2 + 1} = \frac{0}{3} = 0 $$

Answer:

s:
a. \( \boldsymbol{6} \)
b. \( \boldsymbol{-2} \)
c. \( \boldsymbol{-1} \)
d. \( \boldsymbol{4} \)
e. \( \boldsymbol{-3} \)
f. \( \boldsymbol{0} \)