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To determine which equations correctly represent the system for finding the parabolic equation passing through the given points \((3, 5)\), \((-2, 1)\), and \((0, 4)\), we need to use the general form of a parabola:

\[
y = ax^2 + bx + c
\]

Substitute the points into this equation to get three separate equations:

1. Using \((3, 5)\):
   \[
   5 = 9a + 3b + c
   \]

2. Using \((-2, 1)\):
   \[
   1 = 4a - 2b + c
   \]

3. Using \((0, 4)\):
   \[
   4 = c
   \]

Now, let's compare these with the given options:

1. \(5 = 9x^2 + 5x + 5\): 
   - This represents a specific parabola equation, not an equation for the system. **NO**

2. \(1 = 4a - 2b + c\):
   - Matches the equation derived from point \((-2, 1)\). **YES**

3. \(5 = 3a + 3b + c\):
   - This does not match any of the derived equations. **NO**

4. \(4 = a + b + c\):
   - This equation doesn't correspond to any of the derived equations. **NO**

5. \(1 = 4x^2 - 2x + c\):
   - This form is not correct for the system as it involves \(x\). **NO**

6. \(5 = 9a + 3b + c\):
   - Matches the equation derived from point \((3, 5)\). **YES**

### Summary of Answers:
1. NO
2. YES
3. NO
4. NO
5. NO
6. YES

Would you like more details or have any questions? 

Here are some related questions to consider:
1. How can you solve a system of equations to find \(a\), \(b\), and \(c\)?
2. What is the vertex form of a parabolic equation, and how does it relate to the standard form?
3. How can you verify if a parabola passes through a specific point?
4. What methods are there for solving quadratic systems?
5. How do changes in the coefficients \(a\), \(b\), and \(c\) affect the shape of a parabola?

**Tip:** When solving a system of equations derived from multiple points, substituting each point into the general equation is a systematic way to ensure the resulting system is accurate.

Joe is trying to set up a system to find the equation of a parabola that passes through the points (3,5), (-2,1), and (0,4). Which are correct equations for the system? Enter YES or NO for each equation.

Learn how to set up and solve a system of quadratic equations for a parabola passing through the points (3,5), (-2,1), and (0,4). This problem explains how to form and check equations using substitution.

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