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.