Problem 55 - Algebra 2

Question

Given \(f(x)=x^{2}-5 x+6,\) find each value. $$ f(2 a) $$

Step-by-Step Solution

Answer

The expression for \( f(2a) \) is \( 4a^2 - 10a + 6 \).

Step 1: Substitute the Expression

To find the value of \( f(2a) \), we need to substitute \( 2a \) into the function \( f(x) = x^2 - 5x + 6 \). Replace every occurrence of \( x \) with \( 2a \) in the expression. This gives us \( f(2a) = (2a)^2 - 5(2a) + 6 \).

Step 2: Simplify Exponents

First, calculate the square of \( 2a \). Recall that \((2a)^2 = 4a^2\). Substitute back into the expression to get \( f(2a) = 4a^2 - 5(2a) + 6 \).

Step 3: Distribute and Simplify

Now, distribute \( -5 \) over \( 2a \), resulting in \( -10a \). Update the expression: \( f(2a) = 4a^2 - 10a + 6 \).

Step 4: Combine the Terms

Finally, we've reached the simplified expression. There are no like terms to combine further, so the function evaluated at \( 2a \) is \( f(2a) = 4a^2 - 10a + 6 \).

Key Concepts

Quadratic Functions

A quadratic function is one of the simplest polynomial functions, and it forms the backbone of many algebraic concepts. At its core, a quadratic function is defined by a second-degree polynomial, typically expressed through the standard form, \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). This form indicates that the variable \( x \) is raised to the power of two.

• The graph of a quadratic function is a parabola.
• It either opens upwards or downwards, depending on whether "\( a \)" is positive or negative.
• The vertex of the parabola is its highest or lowest point, and it's directly influenced by all three coefficients \( a \), \( b \), and \( c \).

Understanding quadratic functions involves recognizing their algebraic form and geometric representation. This includes figuring out how changes in the coefficients affect the shape and position of the parabola in the coordinate plane.

Variable Substitution

Variable substitution is a method used to simplify or evaluate expressions by replacing variables with given values or expressions. In the context of evaluating functions, it involves directly substituting the variable within the original function.
To approach variable substitution, you follow these broad steps:

• Identify the variable to test or replace in the expression.
• Directly substitute the chosen value or expression in place of the variable wherever it appears.
• Ensure correct replacement by double-checking each term of the expression.

In the given exercise, we substitute \( 2a \) for \( x \) in the quadratic function, transforming \( f(x) = x^2 - 5x + 6 \) into \( f(2a) = (2a)^2 - 5(2a) + 6 \). This step is crucial as it allows the recalculation of the function's output based on new inputs. After substitution, what's left is to simplify the resulting expression.

Polynomial Simplification

Polynomial simplification entails reducing expressions to their simplest and most manageable form. After performing a substitution, simplification helps to consolidate terms and provides a clearer picture of the resulting polynomial expression.
Here’s how you can simplify polynomials effectively:

• First, perform any necessary operations such as exponents. For example, calculate \((2a)^2\) which results in \(4a^2\).
• Carry out distribution, such as multiplying coefficients through parentheses: \(-5(2a)\) equates to \(-10a\).
• Combine all like terms. Look for terms that share the same variable raised to the same power and combine them. In our example, there are no terms we can further combine in \(4a^2 - 10a + 6\).

Simplifying polynomials is all about performing logical arithmetic operations until no further reduction is possible. This allows for a cleaner, more readable expression, useful in both application and further mathematical manipulation.