An equation of a parabola is given.
y = −4x2
(a)
Find the focus, directrix, and focal diameter of the parabola.
focus	
(x, y)
= 

  

directrix			
 
focal diameter			


The equation of the parabola given is \( y = -4x^2 \). This can be rewritten in the standard form for a parabola that opens downwards, which is \( y = a(x - h)^2 + k \). Here, the vertex is at the origin \((0, 0)\), so we have \( h = 0 \) and \( k = 0 \).

### Parameters of the Parabola:
1. **Vertex:** \((h, k) = (0, 0)\)
2. **Coefficient \( a \):** In this case, \( a = -4 \).

### Calculating Focus, Directrix, and Focal Diameter:
- The formula for the focus of a parabola in this form is given by:
  \[
  \text{Focus} = (h, k + \frac{1}{4p})
  \]
  where \( p \) is the distance from the vertex to the focus. For a parabola of the form \( y = -4x^2 \), \( p = -\frac{1}{4}a = -\frac{1}{4} \cdot (-4) = 1 \). Thus:
  - **Focus:** \((0, 0 + 1) = (0, 1)\)

- The directrix is given by:
  \[
  \text{Directrix} = y = k - p = 0 - 1 = -1
  \]

- The focal diameter (the width of the parabola at the focus) is given by \( |4p| \):
  \[
  \text{Focal Diameter} = |4 \cdot 1| = 4
  \]

### Summary of Results:
- **Focus:** \((0, 1)\)
- **Directrix:** \( y = -1 \)
- **Focal Diameter:** \( 4 \)

If you have any questions or need further details, let me know! Here are some related questions you might consider:
1. How can you derive the equation of the directrix from the vertex?
2. What would be the effects on the parabola if \( a \) were positive?
3. How do you graph the parabola based on its focus and directrix?
4. Can you find the intersection points of the parabola with the line \( y = 0 \)?
5. How does changing the value of \( p \) affect the shape of the parabola?

**Tip:** Remember that the distance \( p \) determines how "wide" or "narrow" the parabola appears. The larger the absolute value of \( a \), the narrower the parabola.

An equation of a parabola is given.
y = -4x^2
(a)
Find the focus, directrix, and focal diameter of the parabola.

Explore the solution for finding the focus, directrix, and focal diameter of the parabola y = -4x^2. This detailed explanation covers key concepts in conic sections, including calculating the distance to the focus and the equation of the directrix. Suitable for high school students studying advanced algebra or pre-calculus.

Math Problem Statement

An equation of a parabola is given. y = −4x2 (a) Find the focus, directrix, and focal diameter of the parabola. focus (x, y)

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