t = -rac{b}{2a} = -rac{-12}{2 \cdot 4} = rac{12}{8} = rac{3}{2}

["# Finding the Vertex of a Quadratic: A Step-by-Step Guide with Example", "When analyzing quadratic functions, one of the most important concepts is the vertex of the parabola, which represents the maximum or minimum point depending on the shape of the curve. The formula to find the x-coordinate of the vertex for a quadratic equation in standard form is:", "[\nt = -\frac{b}{2a}\n]", "In this article, we’ll explore how to apply this formula using a concrete example: determining the vertex of the quadratic function ( t = -\frac{b}{2a} ) when ( a = 4 ), ( b = -12 ), and ( c = -ophysical-properties, example solving for ( t = -\frac{-12}{2 \cdot 4} = \frac{12}{8} = \frac{3}{2} ).", "## Understanding the Quadratic Formula", "The general form of a quadratic equation is:", "[\nf(t) = at^2 + bt + c\n]", "The graph of this equation is a parabola. If ( a > 0 ), the parabola opens upwards and the vertex is the minimum point; if ( a < 0 ), it opens downwards and the vertex is the maximum point.", "The x-coordinate of the vertex, which lies at the axis of symmetry, is given by:", "[\nt = -\frac{b}{2a}\n]", "This value tells us where the vertex occurs along the t-axis.", "## Applying the Formula: Step-by-Step", "### Step 1: Identify coefficients\nGiven the quadratic function:\n[\nt = -\frac{b}{2a} = -\frac{-12}{2 \cdot 4}\n]", "Here, compare with ( at^2 + bt + c ):\n- ( a = 4 )\n- ( b = -12 )\n- ( c = -12 ) (not used in this part)", "### Step 2: Plug into the vertex formula", "[\nt = -\frac{-12}{2 \cdot 4} = \frac{12}{8} = \frac{3}{2}\n]", "### Step 3: Simplify the result", "[\nt = \frac{3}{2}\n]", "This means the vertex occurs at ( t = \frac{3}{2} ), which corresponds to the minimum point of the parabola if ( a > 0 ) (here ( a = 4 ), so the parabola opens upwards).", "## Why This Formula Works", "The formula ( t = -\frac{b}{2a} ) derives from completing the square or using calculus to find the axis of symmetry. Geometrically, it bisects the roots (if they exist) and gives the precise midpoint of where the parabola changes direction.", "## Practical Applications", "- Optimization problems: Finding maximum or minimum values in economics, physics, and engineering.\n- Graphing: Accurately plotting parabolas using the vertex as a key point.\n- Quadratic equations: Solving ( at^2 + bt + c = 0 ) becomes easier when the vertex is known.", "## Conclusion", "Finding the vertex using ( t = -\frac{b}{2a} ) is a fundamental skill in algebra. Using the example ( t = -\frac{-12}{2 \cdot 4} = \frac{3}{2} ), we’ve shown how quickly and accurately this formula determines the axis of symmetry — the backbone of parabolic analysis. Mastering this step will boost your confidence in tackling quadratic equations and related real-world applications.", "---", "Keywords: quadratic vertex formula, how to find vertex, algebra tips, parabola axis of symmetry, solve quadratic equations, t = -b/(2a) explained, calculate vertex x-coordinate, implement vertex formula", "---", "For anyone studying polynomial functions or preparing for calculus, understanding this vertex formula is essential. Practice with different values of ( a ), ( b ), and ( c ) to build full confidence!"]