Problem 40 - Algebra 2

Question

Find each value if \(f(x)=x^{2}-3 x+2\) $$ f(3) $$

Step-by-Step Solution

Answer

The value of \(f(3)\) is 2.

Step 1: Understand the Function

We are given a quadratic function \(f(x) = x^2 - 3x + 2\). This is a polynomial function of degree 2.

Step 2: Substitute the Value

In this exercise, we need to find the value of \(f(3)\). To do this, substitute \(x = 3\) into the function: \(f(3) = (3)^2 - 3(3) + 2\).

Step 3: Simplify the Expression

Now, calculate each term in the expression: \((3)^2 = 9\), \(-3(3) = -9\), and the constant \(+2 = 2\). Substitute these values back into the function: \(f(3) = 9 - 9 + 2\).

Step 4: Calculate the Result

Perform the final arithmetic: \(9 - 9 + 2 = 2\). Therefore, the value of \(f(3)\) is \(2\).

Key Concepts

Polynomial Functions

Polynomial functions are a special kind of mathematical expression that consist of variables and coefficients. They are called 'polynomials' because they are 'many terms' expressions.
A polynomial like the quadratic function given, uses these basic components:

• **Variable**: A symbol used to represent numbers (e.g., x).
• **Coefficient**: A number placed in front of the variable (e.g., in 3x, 3 is the coefficient).
• **Degree**: The highest exponent to which the variable is raised (for example, 2 in x^2).

In our example, the quadratic function is:\[f(x) = x^2 - 3x + 2\]This expression is called a **quadratic** because the highest degree is 2. Each term in this polynomial is either a constant, or formed by multiplying a constant with one or more variables raised to non-negative integer powers. Understanding polynomial functions is crucial because they form the foundation for algebraic operations and calculus.

Substitution Method

The substitution method is a straightforward way to replace a variable with a number or another expression. This helps us evaluate expressions or solve equations.
In this case, we were asked to find the value of the quadratic function at a specific point, which is given as \( f(3) \).
Here's how substitution works step by step:

• Identify the variable you need to substitute (here, it's \( x \)).
• Replace every instance of that variable in the function with the number provided, in this instance, 3.
• Apply the standard order of operations, i.e., parentheses, exponents, multiplication and division, addition and subtraction (PEMDAS).

By substituting 3 for \( x \) in our function \( x^2 - 3x + 2 \), we have \( f(3) = (3)^2 - 3(3) + 2 \), enabling us to evaluate the expression accurately.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that involves reducing the expression to its simplest form. This process can include combining like terms and performing basic arithmetic calculations.
In the context of our example, once the substitution is complete:

• Calculate each part of the expression: \( 3^2 = 9 \), \(-3 \times 3 = -9\) and the constant term is 2.
• Replace these values back into the expression to simplify: \( f(3) = 9 - 9 + 2 \).
• Perform any addition or subtraction to reach the final outcome.

By following these steps, you arrive at the result: \( f(3) = 2 \).
Simplifying helps to make expressions easier to handle and allows us to confidently arrive at the correct solutions.