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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 \(f(x) = ax^2 + bx + c\). Here, 'a', 'b', and 'c' are constants and 'x' is the variable. In our exercise, we evaluated \(f(x) = 5x^2\) for different values of x. Trusting our solution steps, we substitute the given value of x into the function and simplify. Quadratic functions produce a parabola when graphed. For example, when x=2, the function \(f(x) = 5x^2\) simplifies to \(20\). Similarly, for x=-2, it also simplifies to \(20\) because squaring a negative number results in a positive number. The vertex form of a quadratic function is another useful form, \(f(x) = a(x-h)^2 + k\), which gives a clear view of the vertex of the parabola and its direction.

Cubic functions are another type of polynomial function where the highest degree of the variable is 3. The general form is \(g(x) = ax^3 + bx^2 + cx + d\). In the given exercise, the function is \(g(x) = 5x^3\). Evaluating a cubic function involves substituting the given x value into the equation and solving. When substituting x=2, \(g(2) = 5(2)^3 = 40\). For \(x=-2\), \(g(-2) = 5(-2)^3 = -40\). Unlike quadratic functions, cubic functions have a characteristic 'S' shaped curve when graphed. These functions show different transformations based on the sign of the coefficients and the constant term.

The substitution method is a fundamental technique in algebra for solving equations and evaluating functions. It involves replacing the variable in the expression with a given value. In the exercise, we used this method to evaluate expressions like \(f(x) = 5x^2\) and \(g(x) = 5x^3\) by substituting \(x = 2\) and \(x = -2\). The steps include:

• Identify the given value of x
• Replace all instances of x in the function with the given value
• Simplify the resulting expression step-by-step

This method simplifies the process and helps in understanding what the function produces for particular input values. It is crucial for learning how to handle more complex algebraic and polynomial expressions.

Polynomial expressions are combinations of variables and coefficients involving operations like addition, subtraction, and multiplication, but never division by a variable. They take forms like \(ax^n + bx^{n-1} + ... + k\), where 'n' is a non-negative integer. Polynomial expressions can be classified based on their degree. For example, a quadratic polynomial has a degree of 2, while a cubic polynomial has a degree of 3. In this exercise, we worked with quadratic \(f(x) = 5x^2\) and cubic \(g(x) = 5x^3\) expressions. To evaluate these polynomials, substitute the given values and simplify. Polynomials form the foundation of many algebraic operations and understanding them is crucial for advancing in math. They are also essential in calculus where they are used in differentiation and integration.