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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.

Find the quadratic function whose graph contains the points (0, 3), (5, 78), and (1, 2).

This problem involves finding a quadratic function of the form y = ax^2 + bx + c that passes through three given points (0, 3), (5, 78), and (1, 2). Learn how to solve this algebraic problem by setting up and solving a system of equations.

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