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Quadratic functions are a type of polynomial function with the form \( ax^2 + bx + c \). The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \). These functions are fundamental in mathematics because they model various physical phenomena and have applications in diverse fields, including physics and finance.
The function \( g(x) = 2x^2 - 4x \) is a standard quadratic function. Here, the coefficient \( a \) is 2, which means the parabola opens upwards and is relatively narrow compared to other parabolas where \( a \) might be smaller. The function's vertices and intercepts are calculated by evaluating or solving for \( x \) when \( g(x) = 0 \).
Since quadratic functions are smooth and continuous, they are perfect candidates to analyze using calculus principles such as finding the average rate of change. Understanding these functions provides a foundation to solve more complex problems.

In calculus, intervals help us define the specific section of a function that we are interested in examining. An interval can be closed, open, or half-open based on whether or not their endpoints are included. In our example, the interval is \([-1, 3]\), a closed interval that includes both endpoints -1 and 3.
Intervals are crucial when evaluating functions, like determining the average rate of change. On the interval \([-1, 3]\), we assess the change by calculating the difference in the function's values at these endpoints. Using a defined interval helps us focus our analysis on relevant segments of the graph, capturing the function's behavior in that space.
Being able to work with intervals not only improves understanding of single-variable functions but also lays the groundwork for studying more advanced topics like integrals and differential equations in calculus.

Function evaluation is the process of finding the function's output for specific inputs. It involves substituting a value for \( x \) in the equation and performing the operations to get the result. For the function \( g(x) = 2x^2 - 4x \), evaluating at a particular point means replacing \( x \) with that point's value and solving.
In our exercise, we evaluate \( g \) at the endpoints of the interval \([-1, 3]\). First, we substitute \( x = -1 \) into the function to compute \( g(-1) \), which results in 6. Similarly, evaluating at \( x = 3 \) yields \( g(3) = 6 \). These evaluations give us the function values needed to apply formulas, like those for calculating average rate of change.
Understanding how to evaluate functions is vital as it serves as the first step in many calculus problems and aids in graphing functions to predict their behavior accurately.