Substitute \( x = 1 \) into the function: \( f(1) = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1 \).
["Understanding Substitute ( x = 1 ) into the Function: A Guide to Evaluating Polynomial Functions", "When working with polynomial functions, evaluating the function at a specific value of ( x ) is a fundamental concept in algebra. One common substitution is replacing ( x ) with ( 1 ), which often simplifies calculations and helps in understanding function behavior at particular points.", "Let’s explore what it means to substitute ( x = 1 ) into the function:", "[\nf(x) = 2(1)^2 - 4(1) + 1\n]", "### Step-by-Step Evaluation", "Step 1: Substitute ( x = 1 )\nReplace every instance of ( x ) with 1:", "[\nf(1) = 2(1)^2 - 4(1) + 1\n]", "Step 2: Apply exponent rules\nCalculate the exponent:\n[\n(1)^2 = 1\n]", "So the expression becomes:", "[\nf(1) = 2(1) - 4(1) + 1\n]", "Step 3: Perform multiplication\nMultiply each term:\n[\n2(1) = 2, \quad -4(1) = -4\n]", "Now the equation looks like:", "[\nf(1) = 2 - 4 + 1\n]", "Step 4: Simplify the expression\nAdd and subtract from left to right:\n[\n2 - 4 = -2\n]\n[\n-2 + 1 = -1\n]", "### Final Result", "[\nf(1) = -1\n]", "This result shows that when the input ( x = 1 ) is applied to the quadratic function, the output is ( -1 ). Such evaluations are essential in graphing, analyzing function behavior, solving equations, and applying algebra in real-world modeling.", "### Why Evaluate at ( x = 1 )?", "- Quick checks: Substituting small integers like 1 helps verify function values without full expansion.\n- Function behavior: Evaluations reveal whether the function crosses the x-axis, peaks, or reaches a minimum/maximum near ( x = 1 ).\n- Practical applications: In physics, economics, or engineering, plugging in observed values (like time ( t = 1 )) lets us predict outcomes using mathematical models.", "### Summary", "Substituting ( x = 1 ) into ( f(x) = 2(1)^2 - 4(1) + 1 ) simplifies neatly to ( f(1) = -1 ). This basic yet powerful computation forms the building blocks of function analysis and is a key skill in algebra and applied mathematics.", "Keywords: substitute x=1, function evaluation, algebra practice, polynomial substitution, evaluate f(1), math tutorial, quadratic function, function simplification."]
