Thus, \( f(x) = 3x^2 - 12x + 9 \), so \( a = 3 \), \( b = -12 \), \( c = 9 \).

["# Understanding the Quadratic Function: ( f(x) = 3x^2 - 12x + 9 )", "When analyzing quadratic functions, understanding the coefficients and their roles is essential for graphing, optimizing, and interpreting the behavior of the function. Consider the standard quadratic formula:\n[\nf(x) = ax^2 + bx + c\n]\nFor the function ( f(x) = 3x^2 - 12x + 9 ), the coefficients are clearly defined as:\n- ( a = 3 )\n- ( b = -12 )\n- ( c = 9 )", "## Why These Coefficients Matter", "The coefficient ( a ) determines the direction and narrowness of the parabola. Since ( a = 3 ), which is positive, the parabola opens upwards, indicating that the function has a minimum point (vertex). The relatively large value of ( a ) means the parabola is narrower compared to, say, ( f(x) = x^2 ).", "The coefficient ( b = -12 ) influences the horizontal position of the vertex and the axis of symmetry. Using the vertex formula, the axis of symmetry is given by:\n[\nx = -\frac{b}{2a} = -\frac{-12}{2 \cdot 3} = \frac{12}{6} = 2\n]\nThis tells us the parabola peaks (or is minimized) at ( x = 2 ), the vertex’s ( x )-coordinate.", "The constant term ( c = 9 ) is the y-intercept of the function—where the graph crosses the ( y )-axis when ( x = 0 ). For this function, substituting ( x = 0 ) gives ( f(0) = 9 ), confirming the y-intercept is at ( (0, 9) ).", "## Analyzing the Function’s Behavior", "Let’s explore key features derived from ( a = 3 ), ( b = -12 ), ( c = 9 ):", "- Vertex Form & Minimum Value:\n To find the vertex’s ( y )-coordinate, substitute ( x = 2 ) into ( f(x) ):\n [\n f(2) = 3(2)^2 - 12(2) + 9 = 3(4) - 24 + 9 = 12 - 24 + 9 = -3\n ]\n The vertex is at ( (2, -3) ), confirming the minimum value of the function.", "- Roots (Zeros) and Natural Factorization:\n To solve ( 3x^2 - 12x + 9 = 0 ), start by factoring out the common coefficient:\n [\n 3(x^2 - 4x + 3) = 0 \Rightarrow x^2 - 4x + 3 = 0\n ]\n Factoring:\n [\n (x - 1)(x - 3) = 0\n ]\n So, the roots are ( x = 1 ) and ( x = 3 ). These are the ( x )-intercepts of the graph.", "- Graph Shape and Meetings with Axes:\n - Parabola opens upward (minimum at ( (2, -3) )).\n - Crosses the ( y )-axis at ( (0, 9) ).\n - Crosses the ( x )-axis at ( (1, 0) ) and ( (3, 0) ).\n - Symmetric about the line ( x = 2 ).", "## Real-World Applications", "Quadratic functions like ( f(x) = 3x^2 - 12x + 9 ) appear in physics (e.g., projectile motion), economics (profit maximization), and engineering. The coefficients guide crucial decisions—like timing in projectile launches (vertex timing) or break-even analysis where ( f(x) = 0 ) indicates cost-revenue balance.", "## Conclusion", "Breaking down ( f(x) = 3x^2 - 12x + 9 ) into its components ( a = 3 ), ( b = -12 ), and ( c = 9 ) reveals vital insights about its graph and behavior. Recognizing how each coefficient shapes the function empowers deeper analysis, accurate graphing, and practical application. Whether optimizing a business model or predicting motion paths, mastering quadratics starts here.", "---", "Keywords: quadratic function, ( f(x) = 3x^2 - 12x + 9 ), coefficients ( a, b, c ), vertex, axis of symmetry, roots, parabola, graphing quadratic, application of quadratic equations.\nMeta Description: Learn how ( a = 3 ), ( b = -12 ), and ( c = 9 ) define the quadratic function ( f(x) = 3x^2 - 12x + 9 )—its vertex, roots, and shape. Ideal for math students and educators."]