Final answer:
To find the quadratic function with a vertex and a point, we can use the vertex form of a quadratic equation. By substituting the given coordinates into the equation and solving for the unknowns, we can find the values of a, b, and c. The resulting quadratic function is then used to find the desired equation.
Step-by-step explanation:
To find the quadratic function with a vertex (2,7) passing through point (5,25), we can use the general form of a quadratic function, which is f(x) = ax^2 + bx + c. We can plug in the vertex coordinates into this equation to find the values of a, b, and c. Using the vertex form of a quadratic equation, we have f(x) = a(x - h)^2 + k. Substituting the vertex coordinates, we get f(x) = a(x - 2)^2 + 7.
Next, we can plug in the coordinates of the point (5,25) into this equation and solve for a. Substituting x = 5 and y = 25, we get 25 = a(5 - 2)^2 + 7. Simplifying this equation gives us 25 = 9a + 7. Solving for a, we subtract 7 from both sides and divide by 9, giving us a = 2.
Finally, we can substitute the value of a into the equation f(x) = 2(x - 2)^2 + 7 to find the quadratic function that satisfies the given conditions.