Consider the function:  
g
(
x
)
=
−
4
x
2
+
5
x
+
4
 

Find the average rate of change of  
g
(
x
)
  between the points  
x
=
−
2
  and  
x
=
2
 . Give your answer as an integer or reduced fraction.

The average rate of change of a function between two points is given by the formula:

\[
\frac{g(b) - g(a)}{b - a}
\]

where \( g(x) \) is the function, and \( a \) and \( b \) are the two given points.

Given \( g(x) = -4x^2 + 5x + 4 \), \( a = -2 \), and \( b = 2 \), we need to:

1. Calculate \( g(-2) \)
2. Calculate \( g(2) \)
3. Find the difference \( g(2) - g(-2) \)
4. Divide the result by \( 2 - (-2) = 4 \)

### Step 1: Calculate \( g(-2) \)

\[
g(-2) = -4(-2)^2 + 5(-2) + 4 = -4(4) - 10 + 4 = -16 - 10 + 4 = -22
\]

### Step 2: Calculate \( g(2) \)

\[
g(2) = -4(2)^2 + 5(2) + 4 = -4(4) + 10 + 4 = -16 + 10 + 4 = -2
\]

### Step 3: Find \( g(2) - g(-2) \)

\[
g(2) - g(-2) = -2 - (-22) = -2 + 22 = 20
\]

### Step 4: Divide by \( b - a = 2 - (-2) = 4 \)

\[
\frac{g(2) - g(-2)}{2 - (-2)} = \frac{20}{4} = 5
\]

Thus, the average rate of change of \( g(x) \) between \( x = -2 \) and \( x = 2 \) is:

\[
\boxed{5}
\]

Would you like more details on any step or have any questions?

Here are 5 related questions to expand on this:

1. What is the interpretation of the average rate of change in the context of functions?
2. How would you calculate the instantaneous rate of change at a specific point?
3. Can the average rate of change be negative? If so, what does that imply?
4. How is the average rate of change related to the slope of a secant line?
5. How would the average rate of change differ if we were given different points?

**Tip**: The average rate of change is similar to finding the slope between two points on a graph of the function.

Consider the function: g(x) = −4x^2 + 5x + 4. Find the average rate of change of g(x) between the points x = −2 and x = 2. Give your answer as an integer or reduced fraction.

Discover how to find the average rate of change of the quadratic function g(x) = −4x^2 + 5x + 4 between x = -2 and x = 2. This solution involves calculating values at specific points and applying the formula for rate of change. Perfect for students in Grades 9-10.

Math Problem Statement

Consider the function:
g ( x )

Find the average rate of change of
g ( x ) between the points
x

− 2 and
x

Solution

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