["Finding the Quadratic Polynomial ( f(x) ) That Passes Through Key Points: A Step-by-Step Guide", "When tasked with finding a quadratic polynomial ( f(x) ) that satisfies specific function values—such as ( f(0) = 4 ), ( f(1) = 1 ), and ( f(2) = -4 )—mathematicians rely on fundamental principles from algebra. Quadratic polynomials take the general form:", "[
\nf(x) = ax^2 + bx + c
\n]", "where ( a ), ( b ), and ( c ) are constants to be determined. In this article, we’ll walk through a clear, step-by-step method to find the unique quadratic polynomial matching the given conditions, and explore why this approach works.", "---", "### Step 1: Write the General Form and Use Given Points", "Since ( f(x) = ax^2 + bx + c ), we substitute the known values into this equation to form a system of equations:", "1. ( f(0) = 4 ) → ( a(0)^2 + b(0) + c = 4 )
\n → ( c = 4 )", "2. ( f(1) = 1 ) → ( a(1)^2 + b(1) + c = 1 )
\n → ( a + b + c = 1 )", "3. ( f(2) = -4 ) → ( a(2)^2 + b(2) + c = -4 )
\n → ( 4a + 2b + c = -4 )", "---", "### Step 2: Substitute ( c = 4 ) Into the Other Equations", "Using ( c = 4 ), update the remaining equations:", "- ( a + b + 4 = 1 ) → ( a + b = -3 ) (Equation A)
\n- ( 4a + 2b + 4 = -4 ) → ( 4a + 2b = -8 ) (Equation B)", "---", "### Step 3: Solve the System of Two Equations", "From Equation A:
\n[
\nb = -3 - a
\n]", "Substitute into Equation B:", "[
\n4a + 2(-3 - a) = -8 \
\n4a - 6 - 2a = -8 \
\n2a - 6 = -8 \
\n2a = -2 \
\na = -1
\n]", "Now substitute ( a = -1 ) back into ( b = -3 - a ):", "[
\nb = -3 - (-1) = -2
\n]", "---", "### Step 4: Write the Final Polynomial", "With ( a = -1 ), ( b = -2 ), and ( c = 4 ), the desired quadratic polynomial is:", "[
\n\boxed{f(x) = -x^2 - 2x + 4}
\n]", "---", "### Step 5: Verification", "To ensure accuracy, verify the polynomial with all given points:", "- ( f(0) = -0 - 0 + 4 = 4 ) ✅
\n- ( f(1) = -1 - 2 + 4 = 1 ) ✅
\n- ( f(2) = -4 - 4 + 4 = -4 ) ✅", "All conditions are satisfied.", "---", "### Why This Method Works", "This approach leverages the fact that a quadratic polynomial has three unknown coefficients—requiring three independent equations to uniquely determine it. By substituting the known function values and using substitution or elimination, we efficiently solve for these coefficients. This method applies broadly to polynomial interpolation problems.", "---", "### Practical Applications", "Finding polynomials through values is essential in engineering, physics, computer graphics, and data fitting. Accurate interpolation ensures models reflect real-world behavior at specific data points.", "---", "### Summary", "To find the quadratic polynomial ( f(x) ) such that ( f(0) = 4 ), ( f(1) = 1 ), and ( f(2) = -4 ), express ( f(x) ) in general form, substitute the known values, solve the resulting linear system, and verify the solution. The resulting polynomial—( f(x) = -x^2 - 2x + 4 )—not only fits the data precisely but is optimal due to the uniqueness and simplicity it guarantees.", "---", "Keywords: quadratic polynomial, polynomial interpolation, f(0)=4, f(1)=1, f(2)=-4, algebraic solution, function values, a x² + b x + c, step-by-step algebra."]
