Find the average rate of change of the function over the given interval.

To find the average rate of change of the function over the given interval, let's follow these steps:

### Given:
- **Function:** \( y = x^2 + 4x - 7 \)
- **Interval:** \( x = 0 \) to \( x = 7 \)

### Formula:
The average rate of change of the function \( f(x) \) from \( x_1 \) to \( x_2 \) is given by:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

### Step-by-Step Solution:

1. **Evaluate the function at \( x = 0 \) and \( x = 7 \):**
   - \( f(0) = 0^2 + 4(0) - 7 = -7 \)
   - \( f(7) = 7^2 + 4(7) - 7 = 49 + 28 - 7 = 70 \)

2. **Substitute these values into the formula:**
   - \( x_1 = 0 \), \( x_2 = 7 \)
   - \( \text{Average Rate of Change} = \frac{f(7) - f(0)}{7 - 0} = \frac{70 - (-7)}{7} = \frac{70 + 7}{7} = \frac{77}{7} = 11 \)

### Answer:
The average rate of change of the function \( y = x^2 + 4x - 7 \) from \( x = 0 \) to \( x = 7 \) is **11**.

Would you like more details or have any questions? Here are some related questions:

1. How does the average rate of change compare to the slope of the tangent line at a point?
2. How would the average rate of change be affected if the interval was \( x = 0 \) to \( x = 10 \)?
3. What does the average rate of change tell you about the function's behavior between \( x = 0 \) and \( x = 7 \)?
4. How can you calculate the average rate of change if the function is given as a table of values?
5. Can the average rate of change of a function be zero over an interval? Under what conditions?

**Tip:** The average rate of change is useful for understanding the overall behavior of a function over a specific interval, providing insight into whether the function is increasing or decreasing on average.

Learn how to find the average rate of change of the function y = x^2 + 4x - 7 from x = 0 to x = 7. This problem involves evaluating the function at two points, applying the average rate of change formula, and understanding its significance. Suitable for high school students.

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