Question

Use the given information to write an equation for each parabola below: The parabola has focus (3, 3) and vertex (1, 3)

Answer 1

The answer to your question is:

Step-by-step explanation:

Data

Focus = (3, 3)

Vertex = (1, 3 )

Horizontal parabola that opens to the right

Process

p = 2 distance to the vertex to the focus

Formula

(y - k)² = 4p (x - h)

( y - 3)² = 4(2)(x - 1)

y² - 6y + 9 = 8x - 8

y² - 8x - 6y + 17 = 0

Answer 2

The equation of the parabola is
`\[ (y - 3)^2 = 8(x - 1) \]`.

Step-by-step explanation:

To write the equation of a parabola, we can use the definition that a parabola is the set of all points that are equidistant from the focus and the directrix. Let's follow the steps to find this equation:

Step 1: Determine the axis of symmetry
The vertex of the parabola is at (1, 3), and the focus is at (3, 3). We see that the y-coordinates of the focus and the vertex are the same, which means the axis of symmetry is vertical and given by the line x = 1, the x-coordinate of the vertex.

Step 2: Determine the directrix line
Since the focus is at (3, 3) and the vertex is at (1, 3), the parabola opens rightward because the focus is to the right of the vertex. The directrix will be a vertical line that is equidistant from the vertex but on the opposite side of the focus. The distance between the vertex and the focus along the x-axis is 2 units (from x = 1 to x = 3). Therefore, the directrix is a vertical line 2 units to the left of the vertex, which is the line x = -1.

Step 3: Determine the focal length "p"
The focal length "p" is the distance between the vertex and the focus. In this case, p = 2 (since the focus (3, 3) is 2 units to the right of the vertex (1, 3)).

Step 4: Write the standard form of the parabola's equation
For a parabola with a vertical axis of symmetry that opens rightward, vertex to the focus (or to the directrix).

Plugging in the values, we have vertex (h, k) = (1, 3) and p = 2. So we get:

`\[ (y - 3)^2 = 4(2)(x - 1) \]`

`\[ (y - 3)^2 = 8(x - 1) \]`

Step 5: Simplify, if necessary
Our equation looks simplified already. So, we can provide the equation of the parabola in standard form as:

`\[ (y - 3)^2 = 8(x - 1) \]`

This is the equation of the parabola with focus at (3, 3) and vertex at (1, 3) that opens rightward.