f(y) = 2(y - 3)^2 + 10(y - 3) + 11 = 2(y^2 - 6y + 9) + 10y - 30 + 11 = 2y^2 - 12y + 18 + 10y - 19 = 2y^2 - 2y - 1.

["# Solving the Quadratic Equation: f(y) = 2(y - 3)² + 10(y - 3) + 11", "Mastering quadratic equations is essential for students, engineers, and developers alike, as quadratics appear frequently in physics, economics, computer graphics, and more. In this article, we’ll thoroughly solve the quadratic expression:\nf(y) = 2(y - 3)² + 10(y - 3) + 11, simplify it into standard form, analyze its vertex, and explore the significance of this elegant transformation.", "---", "## Understanding the Given Quadratic", "Begin with the original function:\nf(y) = 2(y - 3)² + 10(y - 3) + 11", "At first glance, this resembles a completed square form, which often simplifies further into the standard quadratic format:\nf(y) = a(y - h)² + k\nFrom here, you can identify key features such as the vertex and axis of symmetry.", "Let’s expand and simplify step-by-step to reveal the full story.", "---", "## Step-by-Step Simplification", "We begin with:\nf(y) = 2(y - 3)² + 10(y - 3) + 11", "### 1. Expand (y - 3)²\nUse the identity:\n(y - 3)² = y² - 6y + 9", "Substitute into the expression:\nf(y) = 2(y² - 6y + 9) + 10(y - 3) + 11", "### 2. Distribute the constants\n= 2y² - 12y + 18 + 10y - 30 + 11", "### 3. Combine like terms\nGroup the quadratic, linear, and constant terms:\n- Quadratic: 2y²\n- Linear: -12y + 10y = -2y\n- Constant: 18 - 30 + 11 = -1", "Thus:\nf(y) = 2y² - 2y - 1", "---", "## Algebraic Form vs. Standard Form", "While f(y) = 2y² - 2y - 1 is fully expanded, the vertex form f(y) = 2(y - 3)² - 2y - 1 + 11 (already simplified to 2(y - 3)² + 10(y - 3) + 11) helps reveal key properties.", "### Vertex Form Overview:\nf(y) = 2(y - 3)² + 10(y - 3) + 11\nThis matches the vertex form f(y) = a(y - h)² + k, where:\n- a = 2 ( opens the parabola upward since a > 0)\n- h = 3 ( vertex’s x-coordinate)\n- k must be calculated by evaluating the full expression at y = 3, or by simplifying to standard form and expanding:\nf(y) = 2y² - 2y - 1, plug in y = 3:\nf(3) = 2(3)² - 2(3) - 1 = 18 - 6 - 1 = 11, confirming k = 11", "So the vertex is at (3, 11).", "---", "## Analyzing the Parabola", "With a = 2, the parabola:\n- Opens upward\n- Has a vertex at (3, 11)\n- Is symmetric about the vertical line y = 3\n- Has a minimum value of f(y) = 11 at y = 3", "The vertex form immediately gives us the minimum point and behavior.", "---", "## Graphical Interpretation", "Plotting f(y) = 2y² - 2y - 1 reveals a U-shaped parabola crossing the y-axis at:", "Set f(y) = 0 and solve:\n2y² - 2y - 1 = 0\nUse quadratic formula:\ny = [2 ± √(4 + 8)] / 4 = [2 ± √12]/4 = [2 ± 2√3]/4 = (1 ± √3)/2", "So the parabola intersects the x-axis at y = (1 + √3)/2 and y = (1 - √3)/2 — key roots for graphing and analysis.", "---", "## Practical Applications", "Why simplify and convert quadratics like this?", "- Optimization problems — Finding maxima/minima, e.g., profit, cost minimization.\n- Motion modeling — Projectile paths modeled with quadratic functions.\n- Algorithm design — Including quadratic cost functions in computational models.\n- Equation solving — Vertex form clarifies symmetry and critical points, simplifying analysis.", "---", "## Summary", "- Starting from 2(y - 3)² + 10(y - 3) + 11, algebraic expansion and simplification yield 2y² - 2y - 1 in standard form.\n- Vertex form f(y) = 2(y - 3)² + 10(y - 3) + 11 reveals key properties such as vertex (3, 11), upward-opening nature, and axis of symmetry y = 3.\n- This transformation enhances understanding of the function’s behavior and is invaluable in applied math and computing.", "Whether you’re a student mastering quadratics, a programmer modeling real-world systems, or a math enthusiast exploring symmetry, recognizing how to simplify and interpret quadratics is a powerful tool.", "---", "## Further Reading & Exploration", "- Understanding Quadratic Equations and Their Graphs\n- Vertex Form and Completing the Square\n- Real-World Applications of Quadratic Functions", "Start solving quadratics with confidence — and remember, every expanded form tells a story!"]