The problem asks for the equation of a quadratic graph with a given maximum point at $(2, 5)$.

Step 1: Use the vertex form of a quadratic equation.

The quadratic equation can be written in vertex form as: $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex of the parabola. In this case, the vertex is $(2, 5)$, so we substitute $h = 2$ and $k = 5$: $y = a(x - 2)^2 + 5$ Since the graph has a maximum point, the parabola opens downward, which means $a$ must be negative.

Step 2: Determine the value of $a$.

To find the specific value of $a$, we would need one more point on the graph. However, since no additional point is given, we can leave the equation in the general form with $a$ being negative: $y = a(x - 2)^2 + 5$ If an additional point were provided, you could plug that point into the equation to solve for $a$.

Final equation:

Without more information, the equation in vertex form is: $y = a(x - 2)^2 + 5$ where $a$ is a negative constant.

Would you like to provide another point to determine $a$, or do you have any other questions about this problem?

Here are 5 related questions for further practice:

1. How do you find the equation of a quadratic graph if given the vertex and one more point?
2. What is the significance of the value of $a$ in the vertex form of a quadratic equation?
3. How can you convert a quadratic equation from vertex form to standard form?
4. If the maximum point were at $(3, 7)$, how would the equation change?
5. How does the vertex form of a quadratic equation help in graphing the parabola?

Tip:

Remember that the sign of $a$ determines whether the parabola opens upwards (positive $a$) or downwards (negative $a$).