ight] = 4(x - rac{3}{2})^2 - 9 $. For $ y $: $ -

["# Understanding and Graphing the Quadratic Function: $ y = 4\left(x - \frac{3}{2}\right)^2 - 9 $", "The expression $ y = 4\left(x - \frac{3}{2}\right)^2 - 9 $ represents a quadratic function in vertex form. Quadratic functions are foundational in algebra and help model parabolic curves. This article explores the key features of this equation, including its vertex, axis of symmetry, direction of opening, and how it transforms from the basic parabola, enabling better graphing and comprehension.", "---", "## What Is the Vertex Form?", "The standard form of a quadratic function is:\n$$\ny = a(x - h)^2 + k\n$$\nwhere the point $(h, k)$ is the vertex of the parabola, $a$ determines the parabola’s width and direction, and the vertex form makes it easy to identify transformations.", "In your expression:\n$$\ny = 4\left(x - \frac{3}{2}\right)^2 - 9\n$$\nwe identify:\n- $ h = \frac{3}{2} $ (horizontal shift)\n- $ k = -9 $ (vertical shift)\n- $ a = 4 $ (scaling and direction)", "---", "## Key Features of the Function", "### 1. Vertex\nThe vertex of the parabola is at:\n$$\n\left( \frac{3}{2},\ -9 \right)\n$$", "### 2. Direction and Width\nBecause $ a = 4 > 0 $, the parabola opens upwards and is narrower than the standard parabola $ y = x^2 $ — the coefficient 4 compresses the graph vertically by a factor of 4.", "### 3. Axis of Symmetry\nThe axis of symmetry is a vertical line passing through the vertex:\n$$\nx = \frac{3}{2}\n$$", "### 4. Reflection, Stretch, and Shift\n- The positive $ a = 4 $ flips the parabola upside up (though since it’s positive, it opens upward — no reflection).\n- The factor 4 causes a vertical stretch by 4×.\n- The expression $ x - \frac{3}{2} $ shifts the graph left by 1.5 units from the origin.\n- The constant $-9$ shifts the entire graph down 9 units.", "---", "## Graphing the Function Step-by-Step", "To plot $ y = 4\left(x - \frac{3}{2}\right)^2 - 9 $:", "1. Locate the Vertex\nStart at $ \left( \frac{3}{2},\ -9 \right) $, which is the minimum point since $ a > 0 $.", "2. Use the Axis of Symmetry\nPlot the vertical line at $ x = \frac{3}{2} $ — the parabola is symmetric around this line.", "3. Find Key Points\nChoose values around $ x = \frac{3}{2} $ and compute corresponding $ y $-values:\n- At $ x = 1 $:\n$$\ny = 4\left(1 - \frac{3}{2}\right)^2 - 9 = 4\left(-\frac{1}{2}\right)^2 - 9 = 4 \cdot \frac{1}{4} - 9 = 1 - 9 = -8\n$$\n- At $ x = 2 $:\n$$\ny = 4\left(2 - \frac{3}{2}\right)^2 - 9 = 4\left(\frac{1}{2}\right)^2 - 9 = 1 - 9 = -8\n$$\n- At $ x = \frac{3}{2} $: $ y = -9 $ (the vertex)", "Plot points: $ \left(1, -8\right), \left(2, -8\right), \left(\frac{3}{2}, -9\right) $.", "4. Plot and Sketch the Parabola\nDraw a smooth curve passing through these points, opening upwards, with vertex at $ \left(1.5, -9\right) $, elevated symmetrically on both sides.", "---", "## Transformations Summary Table", "| Transformation | Effect on Graph |\n|----------------------|-------------------------------------------------------------|\n| Horizontal shift $+\frac{3}{2}$ | Shifts parabola right by 1.5 units |\n| Vertical shift $-$9 | Shifts entire graph down 9 units |\n| Coefficient $a = 4$ | Stretches vertically (narrower), no flip |\n| Parabola opens up | Standard upward-opening shape |", "---", "## Why This Form Matters", "Understanding the vertex form allows efficient graphing without completing the square or iterative plotting. It emphasizes how transformations modulate the basic shape of $ y = x^2 $. The vertex form is especially useful in application problems—such as modeling projectile motion or optimizing profit functions—where constants shift the graph to fit real-world contexts.", "---", "## Final Thoughts", "The quadratic function $ y = 4\left(x - \frac{3}{2}\right)^2 - 9 $ is a clear example of how shifting, scaling, and transformation modify standard parabolas. Knowing its vertex, direction, and key points empowers students, educators, and developers to interpret and apply quadratic relationships confidently.", "Whether you're graphing for math class, building a model, or analyzing data, mastering vertex form gives you a powerful tool to work with parabolas efficiently.", "---", "Keywords: \nquadratic function, vertex form, graphing quadratic, $ y = 4(x - 3/2)^2 - 9 $, parabola transformation, vertex $ \left(\frac{3}{2}, -9\right) $, axis of symmetry, value of $ a $, parabola opens upwards, algebra life skills", "Meta Description:\nLearn everything about $ y = 4\left(x - \frac{3}{2}\right)^2 - 9 $: vertex, transformations, graphing tips, and applications in quadratic functions. Perfect for students and math enthusiasts!"]