Answer:
[tex]y = \dfrac{1}{2}(x - 3)^2 + 4[/tex]
Step-by-step explanation:
To write an equation for a parabola with a given vertex and a point it passes through, we can use the vertex form of a parabola's equation:
[tex]\boxed{ \begin{array}{ccc} \text{\underline{Vertex Form of a Parabola's Equation:}} \\\\ y = a(x - h)^2 + k \\\\ \text{Where:} \\ \bullet \ y \ \text{and} \ x \ \text{are the coordinates on the graph} \\ \bullet \ a \ \text{determines the width and direction of the parabola} \\ \bullet \ (h, k) \ \text{are the coordinates of the vertex of the parabola} \end{array}}[/tex]
Here, the vertex is (3,4), so the equation becomes:
[tex]\Longrightarrow y = a(x - 3)^2 + 4[/tex]
We need to find the value of 'a'. To do this, we use the given point that the parabola passes through, (5,6). Substituting x = 5 and y = 6 into the equation gives us:
[tex]\Longrightarrow 6 = a(5 - 3)^2 + 4\\\\\\\\\Longrightarrow 6 = a(2)^2 + 4\\\\\\\\\Longrightarrow 6 = 4a+ 4\\\\\\\\\Longrightarrow 2 = 4a\\\\\\\\\therefore a = \dfrac{1}{2}[/tex]
Thus, the equation of the parabola is:
[tex]\boxed{ y = \dfrac{1}{2}(x - 3)^2 + 4}[/tex]
