https://file.aiquickdraw.com/mathbot/file/1727102206304ib7qm2wj.jpg

The equation in the image is:

\[
y^2 + 8x - 18y + 18 = 0
\]

To graph this as a parabola, we need to rewrite it in a standard form for conic sections. Let's manipulate the equation step by step:

### Step 1: Group terms involving \( y \).
\[
y^2 - 18y + 8x + 18 = 0
\]

### Step 2: Move \( 8x + 18 \) to the other side:
\[
y^2 - 18y = -8x - 18
\]

### Step 3: Complete the square for \( y \)-terms.
- Take half of the coefficient of \( y \) (which is \( -18 \)), square it: 
\[
\left(\frac{-18}{2}\right)^2 = 81
\]
- Add 81 to both sides:
\[
y^2 - 18y + 81 = -8x - 18 + 81
\]
\[
(y - 9)^2 = -8x + 63
\]

### Step 4: Simplify:
\[
(y - 9)^2 = -8(x - \frac{63}{8})
\]

This is now in the form \( (y - k)^2 = 4p(x - h) \), which is the equation of a parabola that opens horizontally (since the \( y \)-variable is squared). Here:
- \( h = \frac{63}{8} \)
- \( k = 9 \)
- \( 4p = -8 \), so \( p = -2 \) (this indicates that the parabola opens to the left).

### Step 5: Plotting the graph
- The vertex is \( \left(\frac{63}{8}, 9\right) \), which is approximately \( (7.875, 9) \).
- Since \( p = -2 \), the parabola opens to the left.

Would you like me to provide more details or plot this parabola for you?

### Relative Questions:
1. How does completing the square help in rewriting equations of conic sections?
2. What is the significance of \( p \) in the standard form of a parabola?
3. How does the direction of the parabola's opening relate to the signs of coefficients in the equation?
4. How can you determine the vertex of a parabola from its equation?
5. How would you convert the equation if it opened vertically instead of horizontally?

### Tip:
Completing the square is a crucial technique for transforming quadratic equations into recognizable forms such as parabolas, circles, or ellipses.

Graph this equation with a parabola: y^2 + 8x - 18y + 18 = 0

Learn how to graph the quadratic equation y^2 + 8x - 18y + 18 = 0 by completing the square and converting it into the standard form of a parabola. This step-by-step solution covers conic sections and is suitable for high school students.

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