Expand to standard form: \( f(x) = 3(x-2)^2 - 3 = 3(x^2 - 4x + 4) - 3 = 3x^2 - 12x + 12 - 3 \).
["Expanding Quadratic Functions: A Step-by-Step Guide to Standard Form", "When working with quadratic functions in algebra, understanding how to expand expressions into standard form is essential. Standard form, also known as the canonical quadratic form, presents a quadratic equation as:\n[\nf(x) = ax^2 + bx + c\n]\nThis standardized format simplifies graphing, finding the vertex, and identifying key properties like axis of symmetry and y-intercept.", "### Expanding ( f(x) = 3(x - 2)^2 - 3 ) to Standard Form", "Let’s expand the commonly encountered vertex form function:\n[\nf(x) = 3(x - 2)^2 - 3\n]", "Step 1: Expand the squared binomial\nUse the identity ((x - 2)^2 = x^2 - 4x + 4):\n[\nf(x) = 3(x^2 - 4x + 4) - 3\n]", "Step 2: Distribute the coefficient 3\nMultiply each term inside the parentheses by 3:\n[\nf(x) = 3x^2 - 12x + 12 - 3\n]", "Step 3: Combine constant terms\nSimplify the constants:\n[\nf(x) = 3x^2 - 12x + 9\n]", "Now the function is in standard form:\n[\nf(x) = 3x^2 - 12x + 9\n]\nHere, (a = 3), (b = -12), and (c = 9).", "### Why Standard Form Matters", "Representing quadratics in standard form unlocks several advantages:\n- Vertex and Axis of Symmetry: The vertex coordinates (\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)) are directly accessible.\n- Easy Graphing: Plotting the parabola becomes straightforward using the vertex and symmetry.\n- Root Finding: Solving (f(x) = 0) using the quadratic formula or factoring relies on the (ax^2 + bx + c) structure.", "### Conclusion", "Converting expressions like (3(x - 2)^2 - 3) into standard form is a foundational algebra skill. Starting with the vertex form, simple algebraic expansion yields the familiar (ax^2 + bx + c) structure, empowering you to analyze and work with quadratic functions efficiently. Whether you're solving equations or graphing curves, mastering this technique is invaluable.", "Keywords: expand quadratic, standard form, (f(x) = 3(x-2)^2 - 3), algebra, vertex form, standard quadratic form, solve quadratics, expand binomial."]
