Question

what is the functions average rate of change over each interval?
graph of f(x) on a coordinate grid with x-axis from -10 to 9 and y-axis from -10 to 10

a. the average rate of change over the interval ( x = -3 ) to ( x = -2 ) is (\boxed{quad}).

b. the average rate of change over the interval ( x = -2 ) to ( x = 1 ) is (\boxed{quad}).

c. the average rate of change over the interval ( x = 0 ) to ( x = 1 ) is (\boxed{quad}).

Explanation:

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

Step1: Find \( f(-3) \) and \( f(-2) \)

From the graph, at \( x = -3 \), \( f(-3) = 0 \); at \( x = -2 \), \( f(-2) = 2 \).

Step2: Apply average rate of change formula

The formula for average rate of change is \( \frac{f(b) - f(a)}{b - a} \). Here, \( a = -3 \), \( b = -2 \). So, \( \frac{f(-2) - f(-3)}{-2 - (-3)} = \frac{2 - 0}{-2 + 3} = \frac{2}{1} = 2 \).

Step1: Find \( f(-2) \) and \( f(1) \)

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

Step2: Apply average rate of change formula

Using \( a = -2 \), \( b = 1 \), we get \( \frac{f(1) - f(-2)}{1 - (-2)} = \frac{-4 - 2}{1 + 2} = \frac{-6}{3} = -2 \).

Step1: Find \( f(0) \) and \( f(1) \)

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

Step2: Apply average rate of change formula

With \( a = 0 \), \( b = 1 \), we have \( \frac{f(1) - f(0)}{1 - 0} = \frac{-4 - (-4)}{1} = \frac{0}{1} = 0 \).

Answer:

\( 2 \)

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