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A quadratic function is a polynomial function of degree two. This means the highest power of the variable, typically denoted as \(x\), is two. The general form of a quadratic function is

• \( f(x) = ax^2 + bx + c \)

where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
In our exercise, the function given is \( f(x) = 3x^2 \). This is a specific case where the linear term and the constant term (\(b\) and \(c\)) are zero. Quadratic functions are characterized by their U-shaped graphs, called parabolas, which can open upwards or downwards depending on the sign of the coefficient \(a\).
For \( f(x) = 3x^2 \), since \(3\) is positive, the parabola opens upwards. Understanding how a quadratic function behaves is crucial for evaluating and expanding it, as seen in function evaluations and polynomial expansions.

Function evaluation involves calculating the output of a function for a specific input value. It is an essential part of understanding how functions work and is often the first step in solving function-related problems.
In the given exercise, the function \( f(x) = 3x^2 \) is evaluated at various points. Initially, it is evaluated at \( x = 1 \), which simplifies to \( f(1) = 3(1)^2 = 3 \). This means, when \(x\) equals 1, the function outputs 3.
Next, to find \( f(c+h) \) (a fundamental part of difference quotients), we substitute \( x = c + h \) into the function.

• This gives us \( f(c+h) = 3(c+h)^2 \).

After expanding \((c+h)^2\), we substitute back to find \( f(c+h) = 3 + 6h + 3h^2 \). Function evaluation is a powerful tool, allowing us to explore how functions behave and change when inputs vary.

In mathematics, polynomial expansion involves expressing a polynomial as a sum involving terms of lower degree. This process is especially useful in simplifying complex expressions.
In the context of the exercise, the goal was to expand \((c+h)^2\) in order to evaluate \( f(c+h) \). The expansion of \((c+h)^2\) follows the identity

• \((c+h)^2 = c^2 + 2ch + h^2\)

This is a straightforward application of the binomial theorem for expanding powers of sums.
Substituting \( c = 1 \) into the expanded polynomial gives \( (1+h)^2 = 1 + 2h + h^2 \). Expanding polynomials helps transform expressions into forms amenable to further operations, like addition and subtraction, which were used to find \( f(c+h) - f(c) \) and analyze changes in function values.