(A) Transform the general equation into standard form, if the given equation of a parabola is y² + 4x 14y= -53.

(B) From the equation given above at number 1 (A), give the following

coordinates:

a. Vertex

b. Focus

c. Directrix

d. Axis of symmetry

### (A) Transforming the general equation into standard form

We are given the equation of a parabola:

\[
y^2 + 4x - 14y = -53
\]

**Step 1: Rearrange terms to complete the square**

We aim to transform the quadratic in \( y \) into a perfect square. First, move all \( y \)-related terms to one side:

\[
y^2 - 14y = -4x - 53
\]

**Step 2: Complete the square**

To complete the square, take half of the coefficient of \( y \), square it, and add that value to both sides. The coefficient of \( y \) is -14, so:

\[
\left(\frac{-14}{2}\right)^2 = 49
\]

Add 49 to both sides:

\[
y^2 - 14y + 49 = -4x - 53 + 49
\]

Simplify:

\[
(y - 7)^2 = -4x - 4
\]

**Step 3: Simplify and rewrite the equation**

Now, factor the right side:

\[
(y - 7)^2 = -4(x + 1)
\]

This is the standard form of a parabola equation:

\[
(y - k)^2 = 4p(x - h)
\]

where the vertex is \((h, k)\) and \(4p\) determines the focal length.

### (B) Finding the Coordinates

From the equation:

\[
(y - 7)^2 = -4(x + 1)
\]

we can extract the following information:

- The equation is in the form \((y - k)^2 = 4p(x - h)\), where:
  - Vertex: \((h, k) = (-1, 7)\)
  - \(4p = -4\), so \(p = -1\)

#### a. Vertex
The vertex is at \((-1, 7)\).

#### b. Focus
The focus is located \(p\) units away from the vertex along the axis of symmetry. Since \(p = -1\), the focus is 1 unit to the left of the vertex (because it's a horizontal parabola opening to the left).

\[
\text{Focus} = (-1 + (-1), 7) = (-2, 7)
\]

#### c. Directrix
The directrix is a vertical line \(p\) units to the right of the vertex. Since \(p = -1\), the directrix is:

\[
x = -1 - (-1) = 0
\]

So the directrix is the

Transform the general equation into standard form, if the given equation of a parabola is y² + 4x - 14y = -53. From the equation, give the following coordinates: a. Vertex b. Focus c. Directrix d. Axis of symmetry

This guide walks through transforming the equation y² + 4x - 14y = -53 into standard form and finding key features such as the vertex, focus, directrix, and axis of symmetry. Ideal for high school students studying conic sections and parabola properties.

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