The problem asks to find a quadratic function of the form $y = ax^2 + bx + c$ that passes through the points $(0, 3)$, $(5, 78)$, and $(1, 2)$.

We can substitute each of these points into the quadratic equation to create a system of three equations:

1. For $(0, 3)$:
   $3 = a(0)^2 + b(0) + c \implies c = 3$

2. For $(5, 78)$:
   $78 = a(5)^2 + b(5) + c \implies 78 = 25a + 5b + 3$ Simplifying:
   $25a + 5b = 75$

3. For $(1, 2)$:
   $2 = a(1)^2 + b(1) + c \implies 2 = a + b + 3$ Simplifying:
   $a + b = -1$

Now, we have the system of equations:

1. $25a + 5b = 75$
2. $a + b = -1$

We can solve this system to find $a$ and $b$. Let me do that now.The solution to the system of equations gives us $a = 4$ and $b = -5$.

Therefore, the quadratic function is: $y = 4x^2 - 5x + 3$

Would you like more details on this solution or any related questions?

Here are some related questions you might find interesting:

1. How can I verify if a given point lies on a quadratic curve?
2. How do we solve quadratic equations using substitution?
3. What other forms can a quadratic function take?
4. How is the vertex form of a quadratic equation derived from the standard form?
5. What is the significance of the discriminant in quadratic equations?

Tip: Always double-check your solution by substituting the points back into the final equation to ensure they satisfy the quadratic function.