The vertex \( x \)-coordinate is given by \( x = -rac{b}{2a} = -rac{-4}{4} = 1 \).

["# Finding the Vertex ( x )-Coordinate: A Complete Guide Using the Quadratic Formula", "Understanding the vertex of a quadratic function is essential in algebra, especially when analyzing parabolas, optimization problems, and graph behavior. One key component is the ( x )-coordinate of the vertex, calculated using the formula ( x = -\frac{b}{2a} ), where ( ax^2 + bx + c = 0 ) represents a quadratic equation. In this article, we’ll explore how this formula works, walk through a specific example, and highlight its importance in graphing and maximizing/minimizing quadratic functions.", "## The General Quadratic Form and Vertex Formula", "The standard form of a quadratic equation is:", "[\nf(x) = ax^2 + bx + c\n]", "The vertex of this parabola represents either its maximum (if ( a < 0 )) or minimum point (if ( a > 0 )). The ( x )-coordinate of the vertex is found using:", "[\nx = -\frac{b}{2a}\n]", "This formula comes from completing the square or using calculus to find the axis of symmetry, which passes through the vertex.", "Note: A quick note on the signs: even if ( b ) or ( a ) are negative, the formula remains consistent—what matters is the ratio ( -\frac{b}{2a} ).", "## Step-by-Step Example: Calculating the Vertex ( x )-Coordinate", "### Given:\nSuppose we have the quadratic function:", "[\nf(x) = -4x^2 - 4x + 6\n]", "Here, ( a = -4 ), ( b = -4 ), and ( c = 6 ).", "### Step 1: Identify Coefficients\nExtract the coefficients ( a ), ( b ), and ( c ) from the equation:\n- ( a = -4 )\n- ( b = -4 )", "### Step 2: Apply the Vertex Formula\nUse ( x = -\frac{b}{2a} ):", "[\nx = -\frac{-4}{2 \ imes (-4)}\n]", "Simplify numerator and denominator:", "[\nx = \frac{4}{-8} = -\frac{1}{2} = -0.5\n]", "So, the ( x )-coordinate of the vertex is ( x = -0.5 ), or ( x = -\frac{1}{2} ).", "Wait! The original problem states ( x = -\frac{b}{2a} = -\frac{-4}{4} = 1 ). That’s a discrepancy. Let’s recheck values carefully.", "Ah — notice a sign error in our calculation. Since ( b = -4 ), plugging in correctly:", "[\nx = -\frac{-4}{2 \ imes (-4)} = \frac{4}{-8} = -0.5\n]", "But the problem statement says:\n[\nx = -\frac{b}{2a} = -\frac{-4}{4} = 1\n]", "This suggests a possible typo in the problem (e.g., ( a = 4 ), not ( -4 )), or a different interpretation. Let’s test with ( a = 4 ), ( b = -4 ) just to verify consistency.", "Try ( a = 4 ), ( b = -4 ):", "[\nx = -\frac{-4}{2 \ imes 4} = \frac{4}{8} = 0.5\n]", "Still not 1.", "Now reverse: suppose ( a = -4 ), ( b = 4 ):", "[\nx = -\frac{4}{2 \ imes (-4)} = \frac{4}{-8} = -0.5\n]", "No match.", "But wait — perhaps the original coefficient is ( b = -8 )? Try ( a = 4 ), ( b = -8 ):", "[\nx = -\frac{-8}{2 \ imes 4} = \frac{8}{8} = 1\n]", "Yes! That works. So likely, in the problem, ( b ) was intended as ( -8 ), not ( -4 ), to yield ( x = 1 ).", "Conclusion: The formula ( x = -\frac{b}{2a} ) is consistent—given ( x = 1 ), carefully solve:", "If ( -\frac{b}{2a} = 1 ), then ( -b = 2a \Rightarrow b = -2a ).", "So unless ( a = -2 ), ( b = 4 ), but ( -\frac{4}{2 \ imes -2} = -\frac{4}{-4} = 1 ).", "Thus, the correct interpretation is likely:", "Let ( a = -2 ), ( b = 4 ), then:", "[\nx = -\frac{4}{2 \ imes (-2)} = -\frac{4}{-4} = 1\n]", "So the complete function might be ( f(x) = -2x^2 + 4x + c ).", "Either way, the key point is that the formula ( x = -\frac{b}{2a} ) is consistent—use it reliably.", "## Why This Formula Matters", "### 1. Axis of Symmetry\nThe ( x )-coordinate of the vertex lies at the axis of symmetry: ( x = -\frac{b}{2a} ). This vertical line passes through both turning points of symmetry.", "### 2. Maximizing/Minimizing Quadratic Functions\nSince the parabola opens upward when ( a > 0 ) (minimum point) and downward when ( a < 0 ) (maximum), knowing ( x ) lets you find the vertex and thus the extremum.", "### 3. Graphing Parabolas\nPlot the vertex using ( \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) ), then sketch the symmetric branches.", "## Final Example with Corrected Values", "Let’s use a working example:", "Given ( f(x) = -2x^2 + 8x - 5 ), find the ( x )-coordinate of the vertex.", "Here, ( a = -2 ), ( b = 8 ).", "[\nx = -\frac{8}{2 \ imes (-2)} = -\frac{8}{-4} = 2\n]", "So, the vertex occurs at ( x = 2 ). The symmetry axis is ( x = 2 ), identity function value is ( f(2) = -2(4) + 8(2) - 5 = -8 + 16 - 5 = 3 ), so vertex is ( (2, 3) ).", "## Practicing with Variations", "Try solving other forms:\n- From standard form: Identify ( a ), ( b ), plug in directly: ( x = -\frac{b}{2a} ).\n- From expanded form: Expand ( (x - p)^2 ) style and identify ( p ) as vertex.\n- Graph insight: The vertex lies halfway between roots (if real) via ( \frac{x_1 + x_2}{2} ), directly linked to ( -\frac{b}{2a} ).", "## Summary", "- The vertex ( x )-coordinate formula is ( x = -\frac{b}{2a} ).\n- Always plug in coefficients correctly—signs matter.\n- This formula defines the parabola’s axis of symmetry and extremum location.\n- Mastering it boosts fluency in quadratic analysis for geometry, optimization, and calculus prep.", "Whether you’re graphing algebraically or solving word problems in physics and economics, knowing how to find the vertex coordinates ensures confidence and precision.", "---", "Keywords: vertex ( x )-coordinate, quadratic formula, ( x = -\frac{b}{2a} ), parabola axis of symmetry, graphing quadratics, vertex formula explained, algebra tutorial."]