The function *f* is defined for all real values. The average rate of change of *f* on the interval [*a*, *a* + 1] is given by the expression *g* = –3*a* + 2. Which of the following statements is correct?

 The average rate of change is negative, so the function *f* is linear with a negative slope. The average rate of change decreases at a constant rate, so the function *f* is a parabola that opens down. The average rate of change is negative, so the function *f* is a parabola that opens down. The average rate of change decreases at a constant rate, so the function *f* is linear and decreasing.





The average rate of change of a function \( f \) over the interval \([a, a + 1]\) is given by the expression \( g = -3a + 2 \).

We can analyze each statement based on the given expression for the average rate of change:

1. **"The average rate of change is negative, so the function \( f \) is linear with a negative slope."**
   - **Incorrect**. While the average rate of change can be negative, the given expression \( g = -3a + 2 \) is not constant, which means the function \( f \) cannot be linear. A linear function would have a constant rate of change, but here it depends on \( a \), indicating that the function \( f \) is not linear.

2. **"The average rate of change decreases at a constant rate, so the function \( f \) is a parabola that opens down."**
   - **Correct**. The expression \( g = -3a + 2 \) represents a linear function of \( a \), which means the rate of change of \( g \) with respect to \( a \) is constant (i.e., the derivative of \( g \) with respect to \( a \) is -3). This suggests that the function \( f \) could be quadratic (parabolic), and since the average rate of change decreases (the coefficient of \( a \) is negative), the function likely opens downward.

3. **"The average rate of change is negative, so the function \( f \) is a parabola that opens down."**
   - **Incorrect**. The average rate of change is not always negative. Depending on the value of \( a \), the expression \( g = -3a + 2 \) can be positive or negative. Thus, the average rate of change is not always negative, so this statement is not necessarily true.

4. **"The average rate of change decreases at a constant rate, so the function \( f \) is linear and decreasing."**
   - **Incorrect**. The average rate of change decreases at a constant rate, but this implies that the function \( f \) is quadratic (parabolic), not linear. If the rate of change were constant, the function would be linear, but here it depends on \( a \), so \( f \) cannot be linear.

Thus, the correct statement is:

**"The average rate of change decreases at a constant rate, so the function \( f \) is a parabola that opens down."**

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Would you like more details or further clarification on this? Here are five related questions for deeper understanding:

1. How do you compute the average rate of change for a specific function over an interval?
2. What does it mean when the average rate of change is a linear expression?
3. Can a function with a decreasing average rate of change be linear?
4. How does the sign of the coefficient in a linear rate of change affect the function’s graph?
5. What are the key characteristics of a parabolic function?

**Tip:** The average rate of change is the slope of the secant line between two points on the function. For quadratic functions, this slope changes linearly with the input variable.

The function f is defined for all real values. The average rate of change of f on the interval [a, a + 1] is given by the expression g = –3a + 2. Which of the following statements is correct?

The average rate of change is negative, so the function f is linear with a negative slope. The average rate of change decreases at a constant rate, so the function f is a parabola that opens down. The average rate of change is negative, so the function f is a parabola that opens down. The average rate of change decreases at a constant rate, so the function f is linear and decreasing.

Explore how the average rate of change, given by g = –3a + 2, suggests that the function f is quadratic and opens downward. Learn how this rate of change decreases at a constant rate and what that implies about the graph of f. Suitable for students in Grades 9-11.

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