The parabola passes through the point (-1,5) and its vertex is at (0,4) with a horizontal axis of symmetry. Get the general form, it's focus, directrix, and endpoints

Given the vertex \((0,4)\) and a horizontal axis of symmetry, the parabola opens horizontally. The equation of such a parabola is generally written in the form:

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

where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus or the directrix.

### Step 1: Equation of the Parabola
The vertex is \((0, 4)\), so we can substitute \(h = 0\) and \(k = 4\):

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

Now, use the given point \((-1, 5)\), which lies on the parabola, to find \(p\):

Substitute \((x, y) = (-1, 5)\) into the equation:

\[
(5 - 4)^2 = 4p(-1)
\]
\[
1 = -4p
\]
\[
p = -\frac{1}{4}
\]

Thus, the equation of the parabola is:

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

### Step 2: Focus and Directrix
- The focus lies at a distance \(p = -\frac{1}{4}\) from the vertex along the axis of symmetry (horizontal axis).
- Since the parabola opens to the left (because \(p\) is negative), the focus will be to the left of the vertex.

The focus is:

\[
\left( h + p, k \right) = \left( 0 - \frac{1}{4}, 4 \right) = \left( -\frac{1}{4}, 4 \right)
\]

The directrix is a vertical line \(p\) units to the right of the vertex, so:

\[
x = h - p = 0 + \frac{1}{4} = \frac{1}{4}
\]

### Step 3: Endpoints of the Latus Rectum
The latus rectum is a line segment that passes through the focus and is perpendicular to the axis of symmetry. The endpoints of the latus rectum are located \(2|p|\) units above and below the focus.

Since \(p = -\frac{1}{4}\), the length of the latus rectum is \(2 \times \frac{1}{4} = \frac{1}{2}\).

Thus, the endpoints of the latus rectum are:

\[
\left( -\frac{1}{4}, 4 + \frac{1}{2} \right) = \left( -\frac{1}{4}, 4.5 \right)
\]
and
\[
\left( -\frac{1}{4}, 4 - \frac{1}{2} \right) = \left( -\frac{1}{4}, 3.5 \right)
\]

### Summary:
- **Equation of the parabola**: \((y - 4)^2 = -x\)
- **Focus**: \(\left( -\frac{1}{4}, 4 \right)\)
- **Directrix**: \(x = \frac{1}{4}\)
- **Endpoints of the latus rectum**: \(\left( -\frac{1}{4}, 4.5 \right)\) and \(\left( -\frac{1}{4}, 3.5 \right)\)

Would you like further details or explanations?

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Here are 5 related questions to expand your knowledge:
1. How do you determine whether a parabola opens horizontally or vertically?
2. What is the significance of the vertex in the equation of a parabola?
3. How can you find the length of the latus rectum for any parabola?
4. What changes in the equation if the parabola opens to the right instead of the left?
5. How do you calculate the distance between the focus and directrix?

**Tip**: When dealing with parabolas, always check the sign of \(p\) to know the direction in which the parabola opens.

The parabola passes through the point (-1,5) and its vertex is at (0,4) with a horizontal axis of symmetry. Get the general form, its focus, directrix, and endpoints.

Solve the parabola with a vertex at (0, 4) and a point (-1, 5). Learn how to determine its equation, focus, directrix, and the endpoints of the latus rectum. This problem covers key geometry concepts in conic sections, suitable for students in Grades 9-11.

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