91Ó°ÊÓ

A quadratic function is a second-degree polynomial function of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \). Understanding the general shape and behavior of parabolas is crucial when evaluating a quadratic function at a certain point.

When finding values for a given quadratic function, such as \( f(3a) \), it involves substituting the variable \( x \) with the given input (in this case, \( 3a \)), and then simplifying the resulting expression using algebraic rules. The importance of recognizing the form of a quadratic allows students to anticipate the structure of the solution, possibly involving a perfect square or the need for factoring.

Function substitution is a core concept in algebra that involves replacing the variable in a function's formula with a specific value or another expression. This technique is frequently used to evaluate functions at a particular point or to simplify complex expressions.

In the context of quadratic functions, to evaluate \( f(3a) \), you would replace each occurrence of \( x \) with \( 3a \) in the original function \( f(x) = 2x^2 - 5x + 2 \). This step allows you to determine the function's value at the input \( 3a \), which is essential in understanding how functions behave for different variable values. It is important to follow the order of operations correctly when substituting, ensuring that you first square the \( 3a \), then multiply by 2, and proceed with the rest of the operations accordingly.

Algebraic simplification refers to the process of reducing expressions to their simplest form, which often makes them easier to understand or further manipulate. This can involve combining like terms, factoring polynomials, or expanding expressions using the distributive property among other methods.

After substituting \( 3a \) into the quadratic function, the expression \( f(3a) = 2(3a)^2 - 5(3a) + 2 \) needs to be simplified. This involves squaring \( 3a \) to get \( 9a^2 \), then multiplying by 2, and finally subtracting the product of 5 and \( 3a \). The result, \( 18a^2 - 15a + 2 \), is the simplified form of this particular function evaluation. Mastery of algebraic simplification is crucial, as it enhances problem-solving skills and helps students simplify complex algebraic expressions efficiently.