This is a quadratic function in vertex form. The vertex occurs at \( x = -\frac{b}{2a} = -\frac{20}{2(-2)} = 5 \).

["# Understanding Quadratic Functions in Vertex Form: A Deep Dive", "A quadratic function is one of the most fundamental and widely used polynomial functions in mathematics. Its general form can vary, but expressing it in vertex form provides key insights into its graph and behavior, especially by clearly revealing the vertex of the parabola. Today, let’s explore a classic example of a quadratic function in vertex form:", "[\nf(x) = a(x - h)^2 + k\n]\nwhere ((h, k)) is the vertex, and (a) determines the parabola’s width and direction.", "---", "### What Does Vertex Form Mean?", "Vertex form explicitly displays the vertex ((h, k)), making it easier to identify the graph’s peak (maximum or minimum) and its orientation. For horizontal and vertical stretches, reflections, and shifts, the vertex serves as the anchor point.", "In our example:\n[\nf(x) = -2(x - 5)^2 + k\n]\nWe already know the vertex occurs at (x = -\frac{b}{2a} = 5), confirming the vertex’s (x)-coordinate. To find the full vertex, substitute (x = 5):", "[\nf(5) = -2(5 - 5)^2 + k = k\n]", "So the vertex is exactly ((5, k)), a clear marker of where the parabola reaches its peak (since (a = -2 < 0), the parabola opens downward and (k) is the maximum value).", "---", "### Deriving the Vertex: The Role of the Formula", "Remember the vertex formula (x = -\frac{b}{2a})? It’s derived from completing the square—a core algebraic technique. When transforming a standard quadratic (ax^2 + bx + c) into vertex form, completing the square isolates ((x - h)^2), isolating (x = h) as the vertex’s (x)-coordinate.", "In our case:\n[\nf(x) = -2(x - 5)^2 + k\n]", "Expanding confirms the original form (if desired):\n[\nf(x) = -2(x^2 - 10x + 25) + k = -2x^2 + 20x - 50 + k\n]", "But vertex form skips expansion, prioritizing direct insight into the parabola’s transformation.", "---", "### Key Characteristics Explained", "- Vertex: At ((5, k)), the minimum or maximum point. Since (a = -2), (k) is the highest point on the graph.\n- Axis of Symmetry: The vertical line (x = 5) divides the parabola symmetrically.\n- Opening Direction: Negative (a) means opening downward.\n- Stretch/Fold: The coefficient (-2) compresses the parabola vertically by a factor of 2 and reflects it across the (x)-axis.", "---", "### Practical Applications", "The vertex form’s clarity empowers solving real-world problems:", "- Projectile Motion: The vertex represents the peak height and time. If (f(t)) models height at time (t), the vertex gives maximum altitude.\n- Profit Maximization: In economics, (f(x) = -2x^2 + 20x + k) models revenue, with the vertex at (x = 5) showing optimal output.", "---", "### Final Thoughts", "Mastering quadratic functions in vertex form transforms abstract equations into visual, actionable insights. Knowing the vertex (x = -\frac{b}{2a}) anchors your understanding, and explicit manipulations uncover how each parameter shapes the graph. Whether in calculus, physics, or business, the vertex form remains an essential tool for analysis and prediction.", "Next time you see a quadratic function written as (f(x) = a(x - 5)^2 + k), you’ll instantly recognize the vertex, direction, and stretch—and use that knowledge powerfully.", "---", "Keywords: quadratic function, vertex form, vertex coordinates, parabola, axis of symmetry, vertex formula, (x = -\frac{b}{2a}), (a(x - h)^2 + k), graphing quadratics, real-world applications.", "for more math tips and visual guides, explore our full library of top-tier algebra resources!"]