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A quadratic function is a type of polynomial function with the highest degree of 2, typically expressed in the form \(f(x) = ax^2 + bx + c\). In this standard form, \(a\), \(b\), and \(c\) are constants, and \(a eq 0\), ensuring that the graph is a parabola rather than a straight line.
The graph of a quadratic function is a curve called a parabola, which can open upwards or downwards depending on the sign of \(a\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
In the given exercise, \(f(x) = 3x^2 + 2x - 7\) is the function. Here, the coefficient \(a\) which is 3 is positive, so the parabola opens upwards. Understanding the basic nature of the quadratic function helps in analyzing how changes in \(x\) affect \(f(x)\). It is fundamental when considering how the average rate of change operates over a specific interval.

An interval in mathematics represents a range of values. For quadratic functions, identifying the correct interval is crucial for calculating changes. An interval \([a, b]\) includes all values starting from \(a\) and ending at \(b\), including \(a\) and \(b\) themselves.
In this particular problem, the interval given is \([-4, 2]\). This selection forms the basis of our calculation for change analysis.
Think of the interval as a section of the function's journey. It frames the start and end points, creating a clearer focus for analysis on specific segments of the graph. This allows us to calculate how the function behaves specifically between these two values.

Calculating function values at specific points is like taking a snapshot of the function at a given \(x\). It's essential to plug these values to see how \(f(x)\) behaves.
To find the function value for \(x = -4\), we substitute into the function:

• \(f(-4) = 3(-4)^2 + 2(-4) - 7\)
• Simplifying gives: \(48 - 8 - 7 = 33\)

For \(x = 2\), substitute similarly:

• \(f(2) = 3(2)^2 + 2(2) - 7\)
• Simplify to get: \(12 + 4 - 7 = 9\)

These values are critical as they provide the endpoints for the interval \([-4, 2]\) that we need for calculating the average rate of change. Precise calculation ensures accuracy in understanding how much the function changes.

The average rate of change formula is a straightforward yet powerful tool in analyzing the behavior of functions over an interval. It tells you how much the function changes per unit change in \(x\).
The formula for the average rate of change of a function \(f(x)\) over an interval \([a, b]\) is given by:

• \[ \frac{f(b) - f(a)}{b - a} \]

For this case, substituting \(a = -4\) and \(b = 2\), we computed:

• \(f(2) = 9\) and \(f(-4) = 33\)
• \[ \frac{9 - 33}{2 - (-4)} = \frac{-24}{6} = -4 \]

This outcome, \(-4\), means the function decreases on average by 4 units for each increase of 1 unit in \(x\). Understanding this formula allows for precise insights into both linear and nonlinear functions, providing a snapshot of overall behavior over specific sections.