Question
Use formula f(b)-f(a)/b-a average rate of change given the interval (-2,0)
f(x)= x^2 + 6x +8
f(x)= x^2 + 6x +8
Answers
GPT-4o mini
To find the average rate of change of the function \( f(x) = x^2 + 6x + 8 \) over the interval \((-2, 0)\), we will use the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = -2 \) and \( b = 0 \).
First, we need to compute \( f(-2) \) and \( f(0) \):
1. **Calculate \( f(-2) \)**:
\[
f(-2) = (-2)^2 + 6(-2) + 8
\]
\[
= 4 - 12 + 8
\]
\[
= 0
\]
2. **Calculate \( f(0) \)**:
\[
f(0) = (0)^2 + 6(0) + 8
\]
\[
= 0 + 0 + 8
\]
\[
= 8
\]
Now we can substitute \( f(-2) \) and \( f(0) \) into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(0) - f(-2)}{0 - (-2)} = \frac{8 - 0}{0 + 2} = \frac{8}{2} = 4
\]
Thus, the average rate of change of the function \( f(x) = x^2 + 6x + 8 \) over the interval \((-2, 0)\) is \( \boxed{4} \).
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = -2 \) and \( b = 0 \).
First, we need to compute \( f(-2) \) and \( f(0) \):
1. **Calculate \( f(-2) \)**:
\[
f(-2) = (-2)^2 + 6(-2) + 8
\]
\[
= 4 - 12 + 8
\]
\[
= 0
\]
2. **Calculate \( f(0) \)**:
\[
f(0) = (0)^2 + 6(0) + 8
\]
\[
= 0 + 0 + 8
\]
\[
= 8
\]
Now we can substitute \( f(-2) \) and \( f(0) \) into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(0) - f(-2)}{0 - (-2)} = \frac{8 - 0}{0 + 2} = \frac{8}{2} = 4
\]
Thus, the average rate of change of the function \( f(x) = x^2 + 6x + 8 \) over the interval \((-2, 0)\) is \( \boxed{4} \).
f(b)-f(a)/ b-a = ___— ___/0-2=___
GPT-4o mini
It looks like you want to structure the average rate of change formula in a specific way while substituting values for the variable \(b\) and \(a\) with an interval of \((0, -2)\) instead of \((-2, 0)\).
Let’s fill in the information using the average rate of change formula \( \frac{f(b) - f(a)}{b - a} \):
Given the interval \( (0, -2) \), we have:
- \( a = 0 \)
- \( b = -2 \)
### Step 1: Calculate \( f(0) \) and \( f(-2) \)
We already calculated these values before:
1. **For \( f(0) \)**:
\[
f(0) = 8
\]
2. **For \( f(-2) \)**:
\[
f(-2) = 0
\]
### Step 2: Apply to the formula
Now we substitute these values into the average rate of change formula:
\[
\frac{f(-2) - f(0)}{-2 - 0} = \frac{0 - 8}{-2 - 0}
\]
This simplifies to:
\[
\frac{-8}{-2} = \frac{-8}{-2} = 4
\]
### Final Form
So, filling in the structure you provided:
\[
f(b) - f(a) = f(-2) - f(0) = 0 - 8 = -8
\]
\[
b - a = -2 - 0 = -2
\]
The entire expression looks like:
\[
\frac{f(b) - f(a)}{b - a} = \frac{-8}{-2} = 4
\]
Therefore, the completed expression into the form you provided is:
\[
f(b) - f(a) = -8 \quad , \quad b - a = -2 \quad , \quad \frac{f(b) - f(a)}{b - a} = 4
\]
This means the answer remains \( \boxed{4} \).
Let’s fill in the information using the average rate of change formula \( \frac{f(b) - f(a)}{b - a} \):
Given the interval \( (0, -2) \), we have:
- \( a = 0 \)
- \( b = -2 \)
### Step 1: Calculate \( f(0) \) and \( f(-2) \)
We already calculated these values before:
1. **For \( f(0) \)**:
\[
f(0) = 8
\]
2. **For \( f(-2) \)**:
\[
f(-2) = 0
\]
### Step 2: Apply to the formula
Now we substitute these values into the average rate of change formula:
\[
\frac{f(-2) - f(0)}{-2 - 0} = \frac{0 - 8}{-2 - 0}
\]
This simplifies to:
\[
\frac{-8}{-2} = \frac{-8}{-2} = 4
\]
### Final Form
So, filling in the structure you provided:
\[
f(b) - f(a) = f(-2) - f(0) = 0 - 8 = -8
\]
\[
b - a = -2 - 0 = -2
\]
The entire expression looks like:
\[
\frac{f(b) - f(a)}{b - a} = \frac{-8}{-2} = 4
\]
Therefore, the completed expression into the form you provided is:
\[
f(b) - f(a) = -8 \quad , \quad b - a = -2 \quad , \quad \frac{f(b) - f(a)}{b - a} = 4
\]
This means the answer remains \( \boxed{4} \).