A parabola has its vertex at the origin and passes through the point (4, 16). What is the equation of the parabola?
["Title: Finding the Equation of a Parabola with Vertex at the Origin Passing Through (4, 16)", "Meta Description:\nDiscover how to find the equation of a parabola with its vertex at the origin and passing through the point (4, 16). Learn the step-by-step method and the derived equation.", "---", "### Introduction", "Parabolas are fundamental curves in algebra and geometry, widely used in physics, engineering, and mathematics. One of the simplest forms of a parabola occurs when its vertex is at the origin, making it ideal for modeling idealized situations. In this article, we’ll explore how to determine the equation of a parabola whose vertex lies at the origin and passes through the point (4, 16). We’ll walk through the formula, substitution, and calculation to reveal the exact equation.", "---", "### Understanding Parabolas with Vertex at the Origin", "A vertical parabola opening upward or downward with its vertex at the origin has a standard equation of the form:", "[\ny = ax^2\n]", "where:", "- ( (0, 0) ) is the vertex,\n- ( a ) is a constant determining the width and direction of the parabola.", "If ( a > 0 ), the parabola opens upward; if ( a < 0 ), it opens downward.", "---", "### Given Information", "We are told the parabola passes through the point ( (4, 16) ). This means when ( x = 4 ), ( y = 16 ). We can substitute these coordinates into the standard equation to solve for ( a ):", "[\n16 = a(4)^2\n]", "[\n16 = a \cdot 16\n]", "---", "### Solving for ( a )", "Divide both sides by 16:", "[\na = \frac{16}{16} = 1\n]", "---", "### Final Equation", "Now that we have ( a = 1 ), substitute back into the standard form:", "[\ny = x^2\n]", "---", "### Summary and Key Points", "- A parabola with vertex at the origin and vertical axis has the form ( y = ax^2 ).\n- Substituting the point ( (4, 16) ) allows solving for ( a ).\n- The resulting equation is ( y = x^2 ), a simple but foundational quadratic function.", "---", "### Why This Matters", "Understanding how to derive a parabola’s equation from its vertex and a point helps in modeling real-world phenomena like projectile motion, satellite dishes, and architectural designs. The standard form ( y = ax^2 ) serves as a building block for more complex applications.", "---", "Conclusion:\nThe parabola with its vertex at the origin passing through (4, 16) has the equation:", "[\n\boxed{y = x^2}\n]", "---", "Keywords: parabola equation, vertex at origin, quadratic function, y = ax², parabola through point (4,16), Algebra 2, conic sections."]
