If the function \(f(x) = 2x^2 - 3x + 1\), determine \(f(3)\).

["# Evaluating the Quadratic Function: How to Calculate ( f(3) ) for ( f(x) = 2x^2 - 3x + 1 )", "Understanding how to evaluate a function at a specific value is a fundamental skill in algebra and calculus. Whether you're solving for output, analyzing behavior, or working on complex equations, knowing how to compute ( f(3) ) for a given function like ( f(x) = 2x^2 - 3x + 1 ) is essential. In this article, we’ll break down step-by-step how to determine ( f(3) ) for this quadratic function, explaining key concepts and why this evaluation matters.", "## What Does It Mean to Evaluate ( f(3) )?", "The expression ( f(x) = 2x^2 - 3x + 1 ) defines a mathematical function that assigns a value to each input ( x ). When we write ( f(3) ), we are asking: What value does the function return when the input is 3? This means substituting ( x = 3 ) into the formula and simplifying the expression.", "## Step-by-Step Evaluation of ( f(3) )", "Let’s calculate ( f(3) ) using the given function:", "[\nf(x) = 2x^2 - 3x + 1\n]", "### Step 1: Substitute ( x = 3 )", "Replace every ( x ) in the function with 3:", "[\nf(3) = 2(3)^2 - 3(3) + 1\n]", "### Step 2: Compute the exponent and multiplications", "Calculate ( (3)^2 = 9 ), then multiply by 2:", "[\n2 \ imes 9 = 18\n]", "Next, compute ( -3 \ imes 3 = -9 ).", "Now the expression becomes:", "[\nf(3) = 18 - 9 + 1\n]", "### Step 3: Perform the addition and subtraction", "Add and subtract from left to right:", "[\n18 - 9 = 9\n]\n[\n9 + 1 = 10\n]", "Thus,", "[\nf(3) = 10\n]", "## Answer: ( f(3) = 10 )", "This means the function ( f(x) = 2x^2 - 3x + 1 ) outputs 10 when ( x = 3 ).", "## Why Calculating ( f(3) ) Matters", "Evaluating functions at specific points is critical in many applications, including:", "- Finding the output of a model given an input variable\n- Solving real-world problems like projectile motion or cost functions\n- Understanding function behavior and graphing\n- Preparing for higher-level math, including derivatives and integrals", "Knowing how to compute ( f(3) ) forms the foundation for analyzing and applying quadratic functions in science, engineering, economics, and more.", "## Final Thoughts", "Evaluating functions like ( f(x) = 2x^2 - 3x + 1 ) at specific values is a straightforward yet powerful skill. With clear steps—substitution, substitution of powers, arithmetic—you can quickly determine outputs like ( f(3) = 10 ). Practice this methodic approach, and you’ll build confidence in handling polynomials and other mathematical expressions.", "Next time you encounter a quadratic or polynomial function, remember: evaluating it at any number is simply plugging in that number and simplifying. And in this case, ( f(3) = 10 )."]