 y ^ 2 + 1/4 * x - 5/4 = - 18y - 81
Do the following to each item.

a. Rewrite the following equations in standard form.

b. Determine the

i. vertex,

ii. directrix,

iii. focus,

iv. axis of symmetry, and

V. the opening of the graph of the parabola.




### Part (a): Rewrite the equation in standard form.

The given equation is:
\[
y^2 + \frac{1}{4}x - \frac{5}{4} = -18y - 81
\]

To rewrite this in standard form, let's first move all terms involving \(x\) and \(y\) to one side of the equation and constant terms to the other side:

\[
y^2 + \frac{1}{4}x - \frac{5}{4} + 18y + 81 = 0
\]

Next, let's complete the square for the \(y^2\) and \(18y\) terms.

1. The quadratic expression in \(y\) is \(y^2 + 18y\).
2. To complete the square, take half of the coefficient of \(y\), square it, and add it to both sides of the equation. The coefficient of \(y\) is 18, so half of 18 is 9, and \(9^2 = 81\).

Thus, we add and subtract 81 to complete the square:

\[
y^2 + 18y + 81 - 81 + \frac{1}{4}x - \frac{5}{4} + 81 = 0
\]

Simplifying:

\[
(y + 9)^2 - 81 + \frac{1}{4}x - \frac{5}{4} + 81 = 0
\]

Now, combine like terms:

\[
(y + 9)^2 + \frac{1}{4}x + \frac{76}{4} = 0
\]

This simplifies further to:

\[
(y + 9)^2 + \frac{1}{4}x + 19 = 0
\]

Subtract 19 from both sides:

\[
(y + 9)^2 = -\frac{1}{4}x - 19
\]

Factor out the negative sign from the right side:

\[
(y + 9)^2 = -\frac{1}{4}(x + 76)
\]

This is the standard form of the parabola. Now, we move on to part (b).

### Part (b): Determine the properties of the parabola

The standard form of a parabola is:
\[
(y - k)^2 = 4p(x - h)
\]
Comparing this with our equation:
\[
(y + 9)^2 = -\frac{1}{4}(x + 76)
\]

From this equation:

- \(h = -76\)
- \(k = -9\)
- \(4p = -\frac{1}{4}\), so \(p = -\frac{1}{16}\)

#### i. Vertex:
The vertex of the parabola is given by \((h, k)\):

\[
\text{Vertex} = (-76, -9)
\]

#### ii. Directrix:
The directrix of the parabola is given by the equation \(x = h - p\):

\[
\text{Directrix} = x = -76 - \left(-\frac{1}{16}\right) = -76 + \frac{1}{16} = -\frac{1216}{16} + \frac{1}{16} = -\frac{1215}{16}
\]

#### iii. Focus:
The focus of the parabola is given by \((h + p, k)\):

\[
\text{Focus} = \left(-76 - \frac{1}{16}, -9\right) = \left(-\frac{1216}{16} - \frac{1}{16}, -9\right) = \left(-\frac{1217}{16}, -9\right)
\]

#### iv. Axis of Symmetry:
The axis of symmetry is the vertical line that passes through the vertex. In this case, it is given by:

\[
\text{Axis of Symmetry} = y = -9
\]

#### v. Opening of the Graph:
Since the coefficient of \(x\) is negative (\(-\frac{1}{4}\)), the parabola opens to the left.

### Summary:
- **Vertex**: \((-76, -9)\)
- **Directrix**: \(x = -\frac{1215}{16}\)
- **Focus**: \(\left(-\frac{1217}{16}, -9\right)\)
- **Axis of Symmetry**: \(y = -9\)
- **Opening**: To the left

---

Would you like a more detailed explanation on any of these steps? Or perhaps a graph of the parabola? Here are some related questions to explore:

1. How do you complete the square for quadratic equations in general?
2. What is the geometric interpretation of the directrix and focus in a parabola?
3. How does the value of \(p\) affect the shape of the parabola?
4. Can you find the equation of the parabola if it opens upwards or downwards?
5. How do we convert the standard form of a parabola into the general form?

**Tip**: For parabolas, the distance between the vertex and the focus (or directrix) determines the "sharpness" or "width" of the curve. Smaller values of \(p\) result in a narrower parabola.

Rewrite the equation y^2 + 1/4 * x - 5/4 = -18y - 81 in standard form. Determine the vertex, directrix, focus, axis of symmetry, and the opening of the graph of the parabola.

Learn how to rewrite the quadratic equation y^2 + 1/4 * x - 5/4 = -18y - 81 into its standard form and find the key features of the parabola. This includes determining the vertex, directrix, focus, axis of symmetry, and the graph's direction of opening. Suitable for Grades 10-12.

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