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A quadratic function is a type of polynomial function that can be recognized by its highest degree term being a square or quadratic term, typically in the form of \( ax^2 + bx + c \). The function \( g(x) = x^2 + 9x - 2 \) is an example of a quadratic function. Here, the 'quadratic' part refers to the term \( x^2 \), which dictates the shape of the graph to be a parabola. This curve can open upwards or downwards depending on the sign of the coefficient \( a \). In our example, the coefficient of \( x^2 \) is 1 (a positive number), meaning the parabola opens upwards.

Quadratic functions are essential in many areas of mathematics and physics because they naturally describe phenomena such as projectile motion or the path of an object under constant acceleration. Whenever you have a quadratic function, the values can easily be evaluated at any \( x \) by substituting and performing basic arithmetic.

The substitution method in algebra involves replacing a variable with a given numerical value or another algebraic expression. This method simplifies the evaluation of functions and expressions by allowing us to calculate specific outputs. In the exercise, we used substitution to find \( g(1) \).

To use the substitution method, you follow these steps:

• Identify the variable in the function or expression.
• Replace this variable with the given value.
• Simplify the resulting expression.

The critical step is the substitution itself, ensuring you substitute correctly to maintain accuracy. For example, substituting \( x = 1 \) in \( g(x) = x^2 + 9x - 2 \) changes it to \( g(1) = (1)^2 + 9(1) - 2 \).
This method is straightforward yet powerful, often used to calculate specific function values or simplify equations by inputting known quantities.

An algebraic expression involves numbers, variables, and operators (such as addition, subtraction, multiplication, and division). It's a mathematical phrase that can represent a particular quantity in different situations. For instance, \( x^2 + 9x - 2 \) is an algebraic expression that makes up the quadratic function \( g(x) \).

Working with algebraic expressions involves several key processes:

• Simplifying expressions by combining like terms.
• Resolving expressions using the substitution of given values for variables.
• Performing arithmetic calculations when evaluating the expression.

For our function \( g(x) \), when we replaced \( x \) with 1, we were able to evaluate the expression by following these arithmetic steps: calculate \( 1^2 \), add \( 9 \), and subtract \( 2 \). Simplifying accurately ensures clarity and correctness, enabling us to understand and solve numerous practical situations in mathematics.