The given profit equation is:

$y = -x^2 + 75x - 636$

This is a quadratic equation of the form $y = ax^2 + bx + c$, where $a = -1$, $b = 75$, and $c = -636$.

Finding the maximum profit:

Since the coefficient of $x^2$ is negative ($a = -1$), the parabola opens downwards, meaning there is a maximum point (vertex).

The formula to find the $x$-coordinate of the vertex of a parabola $y = ax^2 + bx + c$ is:

$x = \frac{-b}{2a}$

Substitute $a = -1$ and $b = 75$:

$x = \frac{-75}{2(-1)} = \frac{75}{2} = 37.5$

Now that we have the $x$-coordinate of the vertex, we can substitute $x = 37.5$ into the original equation to find the maximum profit:

$y = -(37.5)^2 + 75(37.5) - 636$

Let's compute this.The maximum profit the company can make is approximately $770.

Would you like further details or have any questions? Here are some related questions:

1. How do changes in the coefficients of a quadratic equation affect its graph?
2. What is the significance of the vertex in a quadratic function in terms of real-world applications?
3. How do we interpret the maximum or minimum point in a quadratic profit model?
4. Can there be cases where a company doesn't make any profit according to such models?
5. What is the relationship between the vertex and axis of symmetry in parabolas?

Tip: The vertex formula $\frac{-b}{2a}$ is essential for finding maximum or minimum values in quadratic functions!