Find the parabolic equation using the following information, then sketch the graph.  Click here for graph paper.Links to an external site.  Type in the equation for the parabola and compare your graph to the answers at the end of the module.

1) Focus F (0,4)

2)  Directrix x = -6

To find the equation of the parabola given the focus \( F(0, 4) \) and the directrix \( x = -6 \), we can follow these steps:

### Step 1: Identify the standard form of the equation
For a parabola with a vertical axis of symmetry, the standard form of the equation is:

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

Where:
- \( (h, k) \) is the vertex of the parabola.
- \( p \) is the distance from the vertex to the focus (or from the vertex to the directrix).

### Step 2: Determine the vertex
The vertex of the parabola lies midway between the focus and the directrix. Since the directrix is vertical (i.e., \( x = -6 \)), the vertex will be at a point with \( y = k \) where \( k = 4 \) (same as the y-coordinate of the focus).

To find the x-coordinate of the vertex, note that it's halfway between \( x = 0 \) (focus) and \( x = -6 \) (directrix). This gives us:

\[
h = \frac{0 + (-6)}{2} = -3
\]

So, the vertex is at \( (-3, 4) \).

### Step 3: Calculate the value of \( p \)
The distance \( p \) is the horizontal distance from the vertex to the focus:

\[
p = |0 - (-3)| = 3
\]

### Step 4: Write the equation of the parabola
Since the parabola opens horizontally (because the directrix is a vertical line), the equation takes the form:

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

Substituting the known values \( h = -3 \), \( k = 4 \), and \( p = 3 \):

\[
(y - 4)^2 = 12(x + 3)
\]

### Final Equation
The equation of the parabola is:

\[
(y - 4)^2 = 12(x + 3)
\]

### Graph of the Parabola
To sketch the graph:
1. Plot the vertex at \( (-3, 4) \).
2. Plot the focus at \( (0, 4) \).
3. Draw the directrix as the line \( x = -6 \).
4. Sketch the parabola opening to the right.

You can use graph paper to compare your sketch with the given information.

### Next Steps
- Would you like me to generate the graph of this parabola for you?
- Do you have any questions about the solution?

### Questions to Explore Further
1. How does the value of \( p \) affect the shape of the parabola?
2. What happens to the parabola if the directrix changes to \( y = -6 \)?
3. How would the equation change if the parabola opened upwards?
4. Can you find the equation of a parabola with focus \( (0, -4) \) and directrix \( x = 6 \)?
5. How does the equation change if the vertex is at the origin?

### Tip
When dealing with parabolas, always identify the vertex first, as it greatly simplifies finding the equation.

Learn how to find the equation of a parabola given the focus (0, 4) and the directrix x = -6. Understand the standard form of a parabola and its application in geometry. Suitable for high school students studying conic sections.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation