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Understanding quadratic functions is a fundamental part of algebra. A quadratic function is a nonlinear function that is typically written in the form of \(ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) represent constants, while \(x\) is the variable. The key characteristic of quadratic functions is that the highest degree of the variable \(x\) is 2. This gives the function its 'U' shaped curve, known as a parabola, when graphed.

Let's take the function \(g(x) = x^2 + 9x - 2\). In this function, the coefficient of \(x^2\) (which is 1) determines the direction of the parabola. Since it is positive, the parabola will open upwards. Quadratic functions are used in various real-life applications such as calculating areas, modeling the path of objects in projectile motion, and optimizing business profits.

The substitution method is a useful technique in algebra, especially for evaluating functions. It involves replacing a variable with a given value and simplifying the resulting expression. This process helps to find specific outputs of functions, which is particularly helpful in understanding how the function behaves for particular inputs.

For example, consider the function \(g(x) = x^2 + 9x - 2\). If we want to find \(g(0)\), we simply substitute \(x = 0\) into the function. This gives us the expression \((0)^2 + 9(0) - 2\). By substituting in this way, we convert the algebraic expression with a variable into a numerical expression that can easily be evaluated to yield \(g(0) = -2\). Thus, substitution simplifies problems greatly and aids in understanding how changes in the input \(x\) affect the output.

Evaluating algebraic expressions is a crucial task in mathematics. It involves substituting values for variables and performing arithmetic operations to simplify the expression. This process lets us find the numerical value of the expression and understand it more concretely.

When we evaluate \(g(x) = x^2 + 9x - 2\) at \(x=0\), it becomes \(0^2 + 9\times0 - 2\). The first step is to compute the squares and products, such as \(0^2\) (which is 0) and \(9\times0\) (which is also 0). Then, the simplified expression \(0 + 0 - 2\) results in \(-2\). Thus, algebraic evaluation effectively turns a problem involving unknowns (variables) into a straightforward arithmetic calculation.

This skill is not only vital for solving theoretical math problems but also for various practical applications in science, technology, and everyday life, where we often need to understand the effect of specific inputs on outcomes.