The axis parallel to the x-axis, and the parabola passes through `(3,3), (6,5)," and "(6,-3)`.

Answer

To find the equation of the parabola that passes through the points (3, 3), (6, 5), and (6, -3) with its axis parallel to the x-axis, we can follow these steps:

### Step 1: Write the standard form of the parabola
The standard form of the equation of a parabola that opens horizontally (with its axis parallel to the x-axis) is given by:
\[ x = Ay^2 + By + C \]

### Step 2: Substitute the points into the equation
We will substitute each of the given points into the equation to create a system of equations.

1. For the point (3, 3):
\[ 3 = A(3^2) + B(3) + C \implies 3 = 9A + 3B + C \quad \text{(Equation 1)} \]

2. For the point (6, 5):
\[ 6 = A(5^2) + B(5) + C \implies 6 = 25A + 5B + C \quad \text{(Equation 2)} \]

3. For the point (6, -3):
\[ 6 = A(-3^2) + B(-3) + C \implies 6 = 9A - 3B + C \quad \text{(Equation 3)} \]

### Step 3: Set up the system of equations
Now we have three equations:
1. \( 9A + 3B + C = 3 \) (Equation 1)
2. \( 25A + 5B + C = 6 \) (Equation 2)
3. \( 9A - 3B + C = 6 \) (Equation 3)

### Step 4: Eliminate \(C\)
We can eliminate \(C\) by subtracting Equation 1 from Equations 2 and 3.

From Equation 2 - Equation 1:
\[ (25A + 5B + C) - (9A + 3B + C) = 6 - 3 \]
\[ 16A + 2B = 3 \quad \text{(Equation 4)} \]

From Equation 3 - Equation 1:
\[ (9A - 3B + C) - (9A + 3B + C) = 6 - 3 \]
\[ -6B = 3 \implies B = -\frac{1}{2} \quad \text{(Equation 5)} \]

### Step 5: Substitute \(B\) back to find \(A\)
Substituting \(B = -\frac{1}{2}\) into Equation 4:
\[ 16A + 2(-\frac{1}{2}) = 3 \]
\[ 16A - 1 = 3 \implies 16A = 4 \implies A = \frac{1}{4} \]

### Step 6: Substitute \(A\) and \(B\) back to find \(C\)
Now we can substitute \(A\) and \(B\) back into one of the original equations to find \(C\). Using Equation 1:
\[ 9(\frac{1}{4}) + 3(-\frac{1}{2}) + C = 3 \]
\[ \frac{9}{4} - \frac{3}{2} + C = 3 \]
Convert \(-\frac{3}{2}\) to quarters:
\[ \frac{9}{4} - \frac{6}{4} + C = 3 \]
\[ \frac{3}{4} + C = 3 \implies C = 3 - \frac{3}{4} = \frac{12}{4} - \frac{3}{4} = \frac{9}{4} \]

### Step 7: Write the final equation of the parabola
Now we have \(A = \frac{1}{4}\), \(B = -\frac{1}{2}\), and \(C = \frac{9}{4}\). Thus, the equation of the parabola is:
\[ x = \frac{1}{4}y^2 - \frac{1}{2}y + \frac{9}{4} \]

### Step 8: Rearranging the equation
To make it more standard:
\[ 4x = y^2 - 2y + 9 \]

### Final Answer
The equation of the parabola is:
\[ 4x = y^2 - 2y + 9 \]

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