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The problem states that we need to **determine the equation of a parabola** with the following conditions:

- The parabola has a **vertical axis of symmetry**.
- The points \((4, -1)\), \((-2, 5)\), and \((0, 3)\) lie on the parabola.

Since the axis of symmetry is vertical, the equation of the parabola will be in the form:

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

### Step 1: Set up the system of equations
We will substitute each of the given points into the general equation to create a system of three equations.

1. For \((4, -1)\), substitute into \(y = ax^2 + bx + c\):

   \[
   -1 = a(4)^2 + b(4) + c \quad \Rightarrow \quad -1 = 16a + 4b + c
   \]

2. For \((-2, 5)\):

   \[
   5 = a(-2)^2 + b(-2) + c \quad \Rightarrow \quad 5 = 4a - 2b + c
   \]

3. For \((0, 3)\):

   \[
   3 = a(0)^2 + b(0) + c \quad \Rightarrow \quad 3 = c
   \]

### Step 2: Solve the system of equations
Now we have the following system of equations:

\[
\begin{aligned}
1. & \quad -1 = 16a + 4b + c \\
2. & \quad 5 = 4a - 2b + c \\
3. & \quad 3 = c
\end{aligned}
\]

Substitute \(c = 3\) into the first two equations:

\[
\begin{aligned}
1. & \quad -1 = 16a + 4b + 3 \quad \Rightarrow \quad -4 = 16a + 4b \\
2. & \quad 5 = 4a - 2b + 3 \quad \Rightarrow \quad 2 = 4a - 2b
\end{aligned}
\]

### Step 3: Solve for \(a\) and \(b\)
We now have two equations:

1. \( -4 = 16a + 4b \)
2. \( 2 = 4a - 2b \)

Solve the second equation for \(b\):

\[
2 = 4a - 2b \quad \Rightarrow \quad b = 2a - 1
\]

Substitute \(b = 2a - 1\) into the first equation:

\[
-4 = 16a + 4(2a - 1) \quad \Rightarrow \quad -4 = 16a + 8a - 4 \quad \Rightarrow \quad 0 = 24a \quad \Rightarrow \quad a = 0
\]

Now substitute \(a = 0\) into \(b = 2a - 1\):

\[
b = 2(0) - 1 = -1
\]

### Step 4: Write the final equation
Since \(a = 0\), \(b = -1\), and \(c = 3\), the equation of the parabola is:

\[
y = -x + 3
\]

### Final Answer:
The equation of the parabola is:

\[
y = -x + 3
\]

Would you like further clarification on any of the steps? Here are 5 related questions to deepen your understanding:

1. How can we identify whether the parabola opens upwards or downwards?
2. What if the axis of symmetry wasn't vertical? How would that change the approach?
3. Can we use other forms of the parabola equation, like vertex form, for this problem?
4. What does the slope of \(-1\) in the equation \(y = -x + 3\) tell us about the parabola's shape?
5. How would adding more points change the complexity of solving this problem?

**Tip:** Whenever you're dealing with systems of equations, double-check your substitutions to avoid small calculation errors that can affect the final result.

Determine the equation of a parabola with the following conditions: Vertical axis of symmetry and points (4, -1), (-2, 5), and (0, 3) on the graph.

Learn how to find the equation of a parabola with a vertical axis of symmetry through the points (4, -1), (-2, 5), and (0, 3). This problem covers key concepts in quadratic equations and solving systems of equations. Suitable for students in Grades 9-12.

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