y²=-4(x-1) solve and find the vertex, focus, end point and directrix 

The given equation is:

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

This equation represents a parabola. To solve it and find the vertex, focus, endpoints of the latus rectum, and directrix, let's follow these steps:

### 1. **Standard Form of a Parabola**  
For a parabola that opens horizontally, the standard form is:

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

Where:
- \((h, k)\) is the vertex.
- \(p\) is the distance from the vertex to the focus and also to the directrix.

### 2. **Identify Parameters**
Comparing the given equation \(y^2 = -4(x - 1)\) with the standard form, we can identify:
- \(h = 1\) (shifts the parabola horizontally)
- \(k = 0\) (no vertical shift)
- \(4p = -4\), hence \(p = -1\)

### 3. **Vertex**
The vertex \((h, k)\) is:

\[
\text{Vertex: } (1, 0)
\]

### 4. **Focus**
The focus is located \(p\) units from the vertex in the direction the parabola opens. Since \(p = -1\) and the parabola opens to the left (negative direction):

\[
\text{Focus: } (1 + p, 0) = (0, 0)
\]

### 5. **Directrix**
The directrix is a vertical line \(p\) units away from the vertex in the opposite direction of the focus:

\[
\text{Directrix: } x = 1 - (-1) = 2
\]

### 6. **Endpoints of the Latus Rectum**
The length of the latus rectum is \(|4p| = 4\). The endpoints of the latus rectum are \(2p\) units above and below the focus:

\[
\text{Endpoints: } (0, -2) \text{ and } (0, 2)
\]

### Summary of Results:
- **Vertex**: \((1, 0)\)
- **Focus**: \((0, 0)\)
- **Directrix**: \(x = 2\)
- **Endpoints of the Latus Rectum**: \((0, -2)\) and \((0, 2)\)

Would you like more details or have any questions?  
Here are five related questions for further exploration:

1. How would the equation change if the parabola opened upwards?
2. What is the geometric interpretation of the directrix in relation to the parabola?
3. How does the value of \(p\) affect the width of the parabola?
4. How do you determine the equation of a parabola given the vertex and focus?
5. What is the significance of the latus rectum in a parabola?

**Tip:** The directrix is always perpendicular to the axis of symmetry of the parabola and helps in defining the set of points that form the parabola based on its distance from the focus.

y²=-4(x-1) solve and find the vertex, focus, end point and directrix

This page provides a detailed solution to the equation y² = -4(x - 1), including how to find the vertex, focus, endpoints of the latus rectum, and directrix. Perfect for students in Grades 9-12 studying conic sections.

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