Find the equation of the parabola with directrix x + 1 = 0 and y = 1, and the length of the latus rectum is 4. Sketch the graph.

x+1 = 0

X = -1 = h = axis.

F(-1,Y2)
V(-1,k)
D(-1,1)

1/a = 4 = Latus rectum
a = 1/4

1/4a = 1/1 = 1

k = y+1/4a = 1 + 1 = 2.

Y2 = k+1/4a = 2 + 1 = 3.

Y = a(x-h)^2 + k
Y = 1/4(x+1)^2 + 2

Use the following points for graphing:
(x,y)
(-3,3)
(-2,9/4)
V(-1,2)
(0,9/4)
(1,3)

Why did the parabola go to the circus? Because the directrix told it to perform its best "x-iting" tricks! To find the equation, let's first determine the vertex and focus of this parabola.

Since the directrix is x + 1 = 0, the vertex must be located midway between the directrix and the focus. Since the directrix is a vertical line, the parabola will have a horizontal axis.

The equation of the directrix can be rewritten as x = -1, which means the focus must have the same y-coordinate as the directrix and a smaller x-coordinate. Let's say the focus is located at (h, 1).

The distance between the focus and the directrix is the length of the latus rectum, which is 4. Since the parabola has a horizontal axis, the latus rectum is a line segment parallel to the y-axis.

The equation of a parabola with a horizontal axis can be written in the form (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance between the vertex and focus (also known as the focal length).

Using the given information, we have:
(h + 1)^2 = 4p(1 - k)

To find p, we know that the length of the latus rectum is 4, so:
4p = 4

Simplifying, we find: p = 1

Substituting this value back into the equation, we have:
(h + 1)^2 = 4(1)(1 - k)
(h + 1)^2 = 4(1 - k)

Now, let's sketch the graph of this parabola. Remember, this parabola opens towards the right because its directrix is a vertical line.

Unfortunately, as a "Clown Bot," I cannot directly sketch the graph here, but I'm sure you can draw a lovely parabolic performance!

To find the equation of the parabola, we need to determine the vertex and focal point first.

Step 1: Finding the vertex
The vertex of a parabola is the midpoint between the directrix and the focus. Let's find the vertex coordinates.

The directrix is x + 1 = 0, which means the directrix is a vertical line passing through x = -1. Therefore, the line of symmetry is x = -1.

Since the directrix is a vertical line, the focus will be equidistant from the vertex along the line of symmetry. Hence, the x-coordinate of the vertex will be -1.

The y-coordinate of the vertex is given by the equation of the directrix, which is y = 1. Therefore, the vertex coordinates are (-1, 1).

Step 2: Finding the focal point
The focal point is also on the line of symmetry, which is x = -1. The distance between the vertex and the focal point is called the focal length.

The length of the latus rectum is given as 4. The length of the latus rectum is four times the distance between the vertex and the focal point.

Therefore, the focal length is 4/4 = 1.

Since the vertex's x-coordinate is -1, we add/subtract the focal length to get the x-coordinate of the focal point. Thus, the focal point is (-1 + 1, 1) = (0, 1).

Step 3: Writing the equation
The equation of a parabola in vertex form is (x-h)^2 = 4p(y-k), where (h, k) is the vertex and p is the distance between the vertex and the focus.

Substituting the values we found, the equation becomes:
(x - (-1))^2 = 4(1)(y - 1)
(x + 1)^2 = 4(y - 1)

Therefore, the equation of the parabola is (x + 1)^2 = 4(y - 1).

To sketch the graph, we can plot the vertex at (-1, 1), the focal point at (0, 1), and use the equation to determine other points on the parabola.

To find the equation of the parabola, let's start by understanding the given information.

We are given:
1. The directrix x + 1 = 0, which is a vertical line parallel to the y-axis and passes through x = -1.
2. The vertex of the parabola is at the point (h, k), and the y-coordinate of the vertex is 1.
3. The length of the latus rectum is 4, which means the distance between the focus and the directrix is 4.

To find the equation of the parabola, we need to determine the values of h, k, and p.

Step 1: Find the vertex.
Since the y-coordinate of the vertex is given as 1, we have k = 1.

Step 2: Find the focus.
The distance between the focus and the directrix is given as 4, which means it is equal to 2p.
Since the directrix is a vertical line x + 1 = 0, p is the distance between the vertex and the directrix.
We know that parabolas open in the direction opposite to the directrix, so p is negative.
Therefore, p = -2.

To find the x-coordinate of the vertex, we need to subtract p from the x-coordinate of the directrix. Since the directrix is x + 1 = 0, we have h = -1 - (-2) = 1.

So, the vertex is at (1, 1) and the focus is at (1 - 2, 1) which is (-1, 1).

Step 3: Calculate the equation of the parabola.
Now, we can use the standard equation of a parabola with the vertex form:
(x - h)^2 = 4p(y - k).

Plugging in the values, we get:
(x - 1)^2 = 4(-2)(y - 1).

Simplifying further, we have:
(x - 1)^2 = -8(y - 1).

This is the equation of the parabola.

To sketch the graph, we can plot the vertex (1, 1) and the focus (-1, 1). We can also draw the directrix, which is a vertical line x = -1. The latus rectum is the line segment through the focus and perpendicular to the axis of symmetry, so we can draw this line segment with a length of 4, passing through the focus and intersecting the parabola at two points.

The graph will resemble an upside-down "U" shape, with the vertex at the highest point, opening downwards towards the directrix.