Solution: The vertex form of a parabola is $ y = -a(x - h)^2 + k $, where $ (h, k) $ is the vertex. Given the vertex $ (3, 10) $, the equation becomes $ y = -a(x - 3)^2 + 10 $. Substituting the point $ (0, 4) $:

["Solution: Finding the Equation of a Parabola in Vertex Form", "Understanding the behavior of parabolas is essential in algebra, and one of the most efficient forms for analyzing them is the vertex form:\n   ( y = -a(x - h)^2 + k )", "Here, ( (h, k) ) represents the vertex—the turning point of the parabola—while the parameter ( a ) determines the width and direction the parabola opens. Because the coefficient ( a ) is negative in this case (( -a )), the parabola opens downward, indicating a maximum point at the vertex.", "---", "### Given: Vertex and a Point", "We are provided the vertex ( (h, k) = (3, 10) ), so the equation simplifies to:\n[\ny = -a(x - 3)^2 + 10\n]", "To determine the precise value of ( a ), we use a known point on the parabola: ( (0, 4) ). Substituting ( x = 0 ) and ( y = 4 ) into the equation:", "[\n4 = -a(0 - 3)^2 + 10\n]", "Simplify the expression:\n[\n4 = -a(9) + 10\n]\n[\n4 = -9a + 10\n]", "Solve for ( a ):\n[\n-9a = 4 - 10 = -6\n]\n[\na = \frac{6}{9} = \frac{2}{3}\n]", "---", "### Final Equation", "Substituting ( a = \frac{2}{3} ) back into the vertex form gives:\n[\ny = -\frac{2}{3}(x - 3)^2 + 10\n]", "This equation fully describes a downward-opening parabola with vertex at ( (3, 10) ) that passes through the point ( (0, 4) ).", "---", "### Why Use Vertex Form?", "The vertex form is powerful for quickly identifying key features:\n- The vertex is directly visible.\n- The symmetry of the parabola is centered at ( x = 3 ).\n- It allows easy graphing by locating the vertex and using the value of ( a ) to stretch, compress, and flip the basic parabola shape.", "Whether solving real-world optimization problems or graphing complex functions, mastering the vertex form gives clarity and efficiency in working with parabolas.", "---", "Conclusion\nBy leveraging the vertex form with the vertex ( (3, 10) ) and the checking point ( (0, 4) ), we determined that ( a = \frac{2}{3} ), yielding the equation:\n[\ny = -\frac{2}{3}(x - 3)^2 + 10\n]", "This solution demonstrates how substitution and algebraic manipulation solve for unknowns in standard forms—key skills for students and learner of algebra."]