f(y) = 3(y - 2)^2 + 12(y - 2) + 7

["Understanding the Quadratic Function f(y) = 3(y - 2)² + 12(y - 2) + 7", "Quadratic functions are essential in algebra and calculus, forming the foundation for modeling real-world phenomena in physics, economics, and engineering. One such quadratic expression is:", "[\nf(y) = 3(y - 2)^2 + 12(y - 2) + 7\n]", "This article explores how to simplify, analyze, and interpret this quadratic function, offering tools you can use to graph, solve, and apply it effectively.", "---", "### Step 1: Simplify the Quadratic Expression", "We begin by simplifying ( f(y) ) to a standard quadratic form. Let’s set ( u = y - 2 ) for convenience:", "[\nf(y) = 3u^2 + 12u + 7\n]", "Now complete the square on the quadratic expression in ( u ):", "1. Factor out the coefficient of ( u^2 ):", "[\nf(y) = 3(u^2 + 4u) + 7\n]", "2. Complete the square inside the parentheses:", "[\nu^2 + 4u = (u + 2)^2 - 4\n]", "3. Substitute back:", "[\nf(y) = 3\left[(u + 2)^2 - 4\right] + 7 = 3(u + 2)^2 - 12 + 7 = 3(u + 2)^2 - 5\n]", "Now revert to original variable ( y ):", "[\nu = y - 2 \Rightarrow u + 2 = y\n]", "Thus:", "[\nf(y) = 3y^2 - 5\n]", "So, surprisingly, the function simplifies elegantly to:", "[\nf(y) = 3y^2 - 5\n]", "---", "### Step 2: Analyze the Quadratic Function", "Even though the simplified form is ( f(y) = 3y^2 - 5 ), it arose from a shifted and transformed version involving ( (y - 2) ). This design reveals key properties:", "- Parabola Shape: Since the coefficient of ( y^2 ) is positive, the parabola opens upward, indicating a minimum point at the vertex.\n- Vertex Location: From original substitution, vertex occurs when ( y - 2 = 0 \Rightarrow y = 2 ). At ( y = 2 ),\n [\n f(2) = 3(2 - 2)^2 + 12(2 - 2) + 7 = 0 + 0 + 7 = 7\n ]\n So the vertex is at ( (2, 7) ).", "- Axis of Symmetry: The axis is ( y = 2 ), the vertical line through the vertex.", "- Minimum Value: The minimum value of ( f(y) ) is ( 7 ), achieved when ( y = 2 ).", "---", "### Step 3: Why This Simplified Form Matters", "While ( f(y) = 3y^2 - 5 ) appears simpler, the original form shows:", "- The function is centered around ( y = 2 ), reflecting a horizontal shift from the basic parabola ( y = x^2 ).\n- Vertex form ( f(y) = 3(y - 2)^2 + 3 ) (after last step above) matches ( 3(y - 2)^2 + 3 ), but original provided inclusion of linear term shows shift integration.", "This form is useful in contexts where transformations are important, such as in optimization or calculus lessons involving counterexamples or applied modeling.", "---", "### Step 4: Graphing the Function", "To sketch the graph:", "- Draw the standard parabola ( f(y) = 3y^2 - 5 ) opening upward.\n- Vertex at ( (2, 7) ).\n- Symmetric about the vertical line ( y = 2 ).\n- As ( y \ o \pm\infty ), ( f(y) \ o \infty ).", "---", "### Step 5: Real-World Applications", "Quadratic functions like this model many phenomena:", "- Physics: Projectile motion under constant gravity (ignoring air resistance). The vertex represents the peak height.\n- Economics: Profit maximization where revenue minus cost forms a quadratic relationship.\n- Engineering: Design optimization, such as minimizing material usage with cost functions involving squared terms.", "---", "### Step 6: Solving Equations and Inequalities", "Using ( f(y) = 3y^2 - 5 ), solving ( f(y) = 0 ) gives:", "[\n3y^2 - 5 = 0 \Rightarrow y^2 = \frac{5}{3} \Rightarrow y = \pm \sqrt{\frac{5}{3}}\n]", "For inequalities:", "- ( f(y) > 0 ) when ( y < -\sqrt{\frac{5}{3}} ) or ( y > \sqrt{\frac{5}{3}} )\n- ( f(y) < 0 ) when ( -\sqrt{\frac{5}{3}} < y < \sqrt{\frac{5}{3}} )", "---", "### Conclusion", "The function\n[\nf(y) = 3(y - 2)^2 + 12(y - 2) + 7\n]\nsimplifies beautifully to\n[\nf(y) = 3y^2 - 5\n]\noffering clear insight into a shifted parabola with vertex at ( (2, 7) ). Recognizing such transformations helps deepen understanding of quadratic behavior and enables efficient problem-solving across mathematics and applied sciences.", "---", "### SEO Keywords:\n- Simplify quadratic function\n- f(y) = 3(y – 2)^2 + 12(y – 2) + 7\n- Graph quadratic vertex form\n- Analyze parabola vertex and axis\n- Solve f(y) = 0 and inequalities\n- Quadratic functions real-world applications\n- Complete the square quadratic equation", "---", "Ready to master quadratics? Practice transforming, simplifying, and graphing functions like this to unlock faster problem-solving across algebra and calculus."]