The given function is \(f(x) = 2x^2 - 12x + 18\), which is a quadratic function in standard form \(ax^2 + bx + c\), with \(a = 2\), \(b = -12\), and \(c = 18\).

["# Understanding the Quadratic Function ( f(x) = 2x^2 - 12x + 18 )", "Quadratic functions are fundamental in algebra and appear frequently across science, engineering, economics, and data analysis. One key example is the function:", "[ f(x) = 2x^2 - 12x + 18 ]", "This standard quadratic function is in the form ( f(x) = ax^2 + bx + c ), where ( a = 2 ), ( b = -12 ), and ( c = 18 ). In this article, we explore its properties, graph, vertex, roots, and practical applications to deepen your understanding of quadratic behavior.", "---", "## What Makes ( f(x) = 2x^2 - 12x + 18 ) a Quadratic Function?", "A quadratic function is defined by a polynomial of degree 2 — meaning the highest power of ( x ) is 2. The presence of ( x^2 ) confirms ( f(x) ) is quadratic. The general form is ( f(x) = ax^2 + bx + c ), where:", "- ( a ), ( b ), and ( c ) are constants\n- ( a <br/>\neq 0 )\n- The graph forms a parabola", "For our function, ( a = 2 > 0 ), so the parabola opens upward — resulting in a minimum point.", "---", "## Analyzing the Coefficients", "The coefficients determine the shape and position of the parabola:", "- Leading coefficient ( a = 2 ): Since ( a > 0 ), the parabola opens upward. The positive value makes the curve wider or narrower depending on its magnitude. Here, ( a = 2 ) causes a moderate spread.", "- Linear coefficient ( b = -12 ): This influences the horizontal placement of the vertex. It relates to the axis of symmetry.", "- Constant term ( c = 18 ): This is the ( y )-intercept of the graph — where the curve intersects the ( y )-axis when ( x = 0 ).", "---", "## Vertex: The Center of the Parabola", "The vertex of a quadratic function in standard form is found using the formula:", "[ x_{\ ext vertex} = -\frac{b}{2a} ]", "Substituting ( a = 2 ) and ( b = -12 ):", "[ x_{\ ext vertex} = -\frac{-12}{2 \cdot 2} = \frac{12}{4} = 3 ]", "Now, substitute ( x = 3 ) back into the function to find ( y_{\ ext vertex}} ):", "[\nf(3) = 2(3)^2 - 12(3) + 18 = 2(9) - 36 + 18 = 18 - 36 + 18 = 0\n]", "So, the vertex is at ( (3, 0) ).", "### Interpretation:\nThe vertex lies on the ( x )-axis at ( x = 3 ), indicating the parabola touches (but does not cross) the horizontal axis. This suggests a double root (a repeated root), which we’ll explore next.", "---", "## Roots of the Function: Where Does ( f(x) = 0 )?", "To find the ( x )-values where the graph crosses the ( x )-axis, solve:", "[ 2x^2 - 12x + 18 = 0 ]", "First, simplify by dividing all terms by 2:", "[ x^2 - 6x + 9 = 0 ]", "This factors neatly:", "[ (x - 3)^2 = 0 ]", "Thus, the quadratic equation has one real solution:", "[ x_{\ ext{roots}} = 3 \quad \ ext{(multiplicity 2)} ]", "These repeated roots mean the parabola touches the ( x )-axis at ( (3, 0) ) without crossing, forming a tangent to the axis.", "---", "## Symmetry and Axis of the Parabola", "The axis of symmetry passes through the vertex, so its equation is:", "[ x = 3 ]", "This line divides the parabola into two mirror-image halves, essential for graph accuracy and optimization problems.", "---", "## Graph of ( f(x) = 2x^2 - 12x + 18 )", "Plotting key points enhances understanding:", "- Vertex: ( (3, 0) )\n- ( y )-intercept: ( f(0) = 18 ) → point ( (0, 18) )\n- ( x )-intercept: ( (3, 0) ) (only one point, touches but does not cross)", "The parabola opens upward with a gentle curve due to ( a = 2 ) being moderately large.", "", "(Note: Replace placeholder image with actual graph visualization tools for accurate representation.)", "---", "## Applications of Quadratic Functions", "### 1. Optimization Problems\nBecause ( f(x) ) has a minimum at ( (3, 0) ), the function achieves its lowest value there. In real-world modeling — such as profit or cost optimization — understanding minima helps determine greatest efficiency.", "### 2. Projectile Motion\nQuadratic functions model vertical displacement under gravity. While our ( f(x) ) lacks vertical stretch typical in projectile models, its structure inspires equations like ( h(t) = -4.9t^2 + v_0 t + h_0 ).", "### 3. Business and Economics\nProfits, revenue, and cost functions often use quadratics. The double root here might represent a scenario where no quantity yields profit — a cautionary insight into break-even analysis.", "---", "## Conclusion", "The function ( f(x) = 2x^2 - 12x + 18 ) exemplifies core quadratic behavior: a U-shaped parabola touching the ( x )-axis at ( x = 3 ), with a minimum value of 0. Recognizing vertex location, roots, and symmetry empowers solving equations, optimizing outputs, and modeling real-life phenomena. Mastering such functions strengthens algebraic intuition and analytical problem-solving across STEM disciplines.", "---", "## Key Takeaways", "- Standard form: ( ax^2 + bx + c )\n- Vertex formula: ( x = -\frac{b}{2a} )\n- Discriminant analysis helps identify root nature: ( b^2 - 4ac )\n- Double roots imply a tangent at the intercept\n- Applications span optimization, physics, economics, and data science", "Master ( f(x) = 2x^2 - 12x + 18 ) today to unlock deeper insight into quadratics and their powerful uses in mathematics and science.", "---", "Keywords: ( f(x) = 2x^2 - 12x + 18 ), quadratic function, vertex, discriminant, parabola, roots, optimization, standard form, vertex formula, algebraic functions.\nMeta Description: Explore the quadratic function ( f(x) = 2x^2 - 12x + 18 )—its graph, vertex, roots, and real-world applications in algebra and STEM fields."]