t = -\frac{b}{2a} = -\frac{-12}{2 \cdot 3} = \frac{12}{6} = 2

["Understanding the Vertex Formula: Finding the Maximum or Minimum of a Quadratic Function", "When studying quadratic equations, one of the most essential concepts is determining the vertex of a parabola — the point where the function reaches its maximum or minimum value. This critical point is calculated using the vertex formula:", "[ t = -\frac{b}{2a} ]", "This formula provides a quick and accurate way to locate the axis of symmetry for any quadratic function in standard form:\n[ f(t) = at^2 + bt + c ]", "Let’s break down how this formula works and apply it step-by-step using an example for clarity.", "---", "### The Vertex Formula Explained", "For a quadratic equation of the form:\n[ f(t) = at^2 + bt + c ]\nthe vertex occurs at:\n[ t = -\frac{b}{2a} ]", "This value ( t ) represents the axis of symmetry — the vertical line that divides the parabola into two mirror-image halves. The corresponding ( f(t) ) value gives the function’s minimum or maximum value, depending on the direction the parabola opens.", "- If ( a > 0 ), the parabola opens upward, and the vertex is a minimum point.\n- If ( a < 0 ), the parabola opens downward, and the vertex is a maximum point.", "---", "### Step-by-Step Example: Finding the Vertex", "Consider the quadratic function:\n[ f(t) = 3t^2 - 12t + c ]\n(Note: The constant ( c ) affects the vertical position, but does not affect the location of the vertex.)", "Here, compare to ( at^2 + bt + c ):\n- ( a = 3 )\n- ( b = -12 )", "Plug into the vertex formula:\n[ t = -\frac{b}{2a} = -\frac{-12}{2 \cdot 3} = \frac{12}{6} = 2 ]", "So, the axis of symmetry is at ( t = 2 ). This means the vertex lies on the vertical line ( t = 2 ), and the minimum value of the function occurs at ( t = 2 ) (since ( a = 3 > 0 )).", "For completeness, substitute ( t = 2 ) into the original equation to find the corresponding ( f(t) ):\n[ f(2) = 3(2)^2 - 12(2) + c = 12 - 24 + c = -12 + c ]", "Therefore, the vertex of the parabola is at the point ( (2, -12 + c) ).", "---", "### Why This Formula Matters", "Mastering the vertex formula ( t = -\frac{b}{2a} ) helps solve optimization problems across science, economics, and engineering. It allows quick identification of peak efficiency or optimal performance in real-world models.", "---", "### Conclusion", "The equation ( t = -\frac{b}{2a} ) is the gateway to finding the vertex of any quadratic function efficiently. By identifying the axis of symmetry, you determine where the function achieves its optimal value — a fundamental tool in algebra and beyond.", "Key Takeaway:\nFor ( f(t) = 3t^2 - 12t + c ), the vertex occurs at ( t = 2 ). This elegant formula simplifies vertex calculation and empowers precise analysis of quadratic behavior.", "---", "Keywords: vertex formula, quadratic vertex, find vertex, t = –b/2a, quadratic functions, algebra, optics, optimization, symmetrical parabola, standard form quadratic, a = 3, b = –12", "---", "Discover more about quadratic equations and their real-life applications to strengthen your mathematical toolkit — mastering ( t = -\frac{b}{2a} ) is the first step!"]