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Quadratic functions are a type of polynomial function where the highest degree of the variable is 2.
The general form of a quadratic function is given by:

• \( f(x) = ax^2 + bx + c \)

Here, \( a \), \( b \), and \( c \) are constants, and \( a \) should not be zero for the function to remain quadratic. These functions are known for their distinctive parabolic graphs.
A graph of this function shape-forming a parabola that opens upward if \( a > 0 \) and downward if \( a < 0 \).

• The vertex of the parabola is a significant point, often representing the maximum or minimum value of the function.
• The axis of symmetry is a vertical line passing through the vertex.
• The roots or zeroes of the quadratic function are the points where the graph crosses the x-axis, which can be found using the quadratic formula.

All these aspects make the quadratic function an essential part of differential calculus for examining changes and behaviors in curves.

The concept of a difference quotient helps in understanding how a function changes between two points. It is a crucial tool in differential calculus that leads to the derivative of a function.
The difference quotient for a function \( f(x) \) is expressed as:

• \( \frac{f(x+h) - f(x)}{h} \)

Here, \( h \) represents a small change in \( x \). As \( h \) approaches zero, the difference quotient converges to the derivative of the function, which describes how \( f(x) \) changes at an infinitesimal level.
The original problem demonstrates this fundamental process applied to a quadratic function, displaying how specific calculations on the difference quotient simplify to form its derivative.
Understanding the difference quotient is akin to understanding the core of differentiation, leading to discoveries about the slope of tangents to graphs.

Polynomial expansion refers to the process of expressing a polynomial, especially in nested formats, into a simple, extended form. The exercise features this in Step 2 and Step 3 by expanding \( (x+h)^2 \) for a complete expression.
When expanding \( (x+h)^2 \), you apply the distributive property to obtain:

• \( x^2 + 2xh + h^2 \)

With this expansion done, it's easily plugged back into the function to further simplify and evaluate expressions.
Polynomial expansion is important as it breaks down complex expressions into linearity, making it more manageable to work through in calculations like differentiation.
With quadratic functions, expansion simplifies terms and clarifies how each component behaves under algebraic manipulation—vital for deeper calculus operations like finding limits or derivatives.