1. (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

### 1 (A). Transforming the general equation of the parabola into standard form:

The given equation is:
\[
y^2 + 4x - 14y = -53
\]

We will first rearrange the terms and complete the square to convert this into the standard form of a parabola.

1. Rearrange the terms by isolating \(x\):
   \[
   y^2 - 14y = -4x - 53
   \]

2. Complete the square on the left side for the \(y\)-terms. To complete the square, take half of the coefficient of \(y\) (which is \(-14\)), square it, and add it to both sides. Half of \(-14\) is \(-7\), and \((-7)^2 = 49\).

   So, we add 49 to both sides:
   \[
   y^2 - 14y + 49 = -4x - 53 + 49
   \]
   Simplifying:
   \[
   (y - 7)^2 = -4x - 4
   \]
   \[
   (y - 7)^2 = -4(x + 1)
   \]

Thus, the equation in **standard form** is:
\[
(y - 7)^2 = -4(x + 1)
\]

This is the standard form of a parabola that opens left or right, \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex and \(p\) determines the direction and distance of the focus from the vertex.

### 1 (B). Finding the coordinates:

From the equation \((y - 7)^2 = -4(x + 1)\), we can extract the following details:

1. **Vertex**: The vertex \((h, k)\) is given by the point \((-1, 7)\), directly from the form of the equation.
   
   **Vertex**: \((-1, 7)\)

2. **Focus**: The general equation is \((y - k)^2 = 4p(x - h)\). In our equation, \(4p = -4\), so \(p = -1\). Since the parabola opens to the left (because \(p\) is negative), the focus is located 1 unit to the left of the vertex. Thus, the focus is at:
   
   **Focus**: \((-2, 7)\)

3. **Directrix**: The directrix is a vertical line, located \(p\) units to the right of the vertex. Since \(p = -1\), the directrix is 1 unit to the right of the vertex, at:
   
   **Directrix**: \(x = 0\)

4. **Axis of Symmetry**: The axis of symmetry is the horizontal line passing through the vertex, parallel to the \(y\)-axis. The equation of this line is:
   
   **Axis of Symmetry**: \(y = 7\)

---

Would you like further clarification on any of these steps?

### 5 Related Questions:
1. How do you complete the square for quadratic terms?
2. What does the value of \(p\) represent in the standard form of a parabola?
3. How does the equation change if the parabola opens upwards or downwards?
4. What is the general form of the equation for a parabola that opens vertically?
5. How do you derive the equation of the directrix from the standard form of a parabola?

### Tip:
Always remember that in the standard form of a parabola \((y - k)^2 = 4p(x - h)\), the sign of \(p\) tells you the direction in which the parabola opens.

1. (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

This solution transforms the equation y² + 4x - 14y = -53 into its standard form and explains how to find the vertex, focus, directrix, and axis of symmetry. Step-by-step instructions are provided, focusing on completing the square and applying the properties of parabolas. Suitable for students in Grades 9-11.

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