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A quadratic function is a type of polynomial that has the highest degree of 2. It usually takes the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. This particular shape of a curve is known as a parabola. They come in a U-shape, which can either open upwards or downwards depending on the sign of \( a \).
Parabolas have a main characteristic point called the vertex, and they are symmetric around a vertical line known as the axis of symmetry. In our exercise, the function given is \( f(x) = x^2 - 1 \). Here, \( a = 1 \), \( b = 0 \), and \( c = -1 \). This particular parabola opens upwards.
Quadratic functions have various applications in real life, from physics to finance, and understanding them is important for grasping many mathematical concepts.

Function evaluation means finding the output of a function for a specific input value. For example, if you have a function \( f(x) = x^2 - 1 \), and you need to find \( f(2) \), you replace every \( x \) in the equation with 2. So it becomes \( (2)^2 - 1 = 4 - 1 \), which simplifies to 3.
Function evaluation is a simple but crucial step in calculus and algebra because it allows us to know the value of the function at specific points, which is necessary for analyzing the behavior of the function. In the original problem, we found both \( f(1) \) and \( f(2) \), and these values are used in finding the average rate of change.
It’s like checking a function’s response at specific instances to understand its general behavior better.

The difference quotient is a formula used to find the average rate of change of a function over a specific interval. The formula is \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \). This helps in determining how much a function's output changes with respect to changes in its input.
In simpler terms, it’s similar to calculating the slope of the line connecting two points on the graph of a function. This concept is essential when learning about derivatives and calculus. It helps in finding how a function behaves between two points, which is especially helpful when working with nonlinear functions.
In the problem discussed, the difference quotient is used to find the average rate of change between \( x_1 = 1 \) and \( x_2 = 2 \) for the function \( f(x) = x^2 - 1 \). The outcome was a difference quotient (or average rate of change) of 3, showing how the function increases over that interval.