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To solve the question "Qual é a taxa de variação média de \( f \) ao longo do intervalo \([-3, 9]\)?" (translated as "What is the average rate of change of \( f \) over the interval \([-3, 9]\)?"), we need to use the formula for the average rate of change of a function:

\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]

Where:
- \( a = -3 \) and \( b = 9 \)
- \( f(a) = f(-3) \) and \( f(b) = f(9) \) are the function values at \( x = -3 \) and \( x = 9 \), respectively.

From the graph:
- \( f(-3) = 7 \)
- \( f(9) = 1 \)

Now, we can plug these values into the formula:

\[
\text{Average rate of change} = \frac{f(9) - f(-3)}{9 - (-3)} = \frac{1 - 7}{9 + 3} = \frac{-6}{12} = -\frac{1}{2}
\]

Thus, the average rate of change of \( f \) over the interval \([-3, 9]\) is \( -\frac{1}{2} \).

Would you like more details or have any questions?

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Here are 5 related questions for practice:
1. How would the average rate of change change if the interval was \([-3, 3]\)?
2. What is the meaning of the average rate of change being negative in this context?
3. Can you find the instantaneous rate of change at \( x = 0 \)?
4. How would you estimate the derivative of the function at \( x = 6 \)?
5. What is the behavior of the function at points where it reaches local maxima?

**Tip:** The average rate of change is equivalent to the slope of the secant line connecting two points on the graph.

Qual é a taxa de variação média de f ao longo do intervalo [-3, 9]?

Solve for the average rate of change of the function f(x) over the interval [-3, 9] using the formula (f(b) - f(a)) / (b - a). This problem covers key algebraic concepts and function analysis, suitable for students in Grades 9-11.

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