How to find the focus of a parabola in standard form?

The focus of a parabola is a fixed point that defines the curve: every point on the parabola is equidistant from the focus and a fixed line called the directrix. To find the focus of a parabola in standard form, you need to identify its orientation, vertex, and the value of (p), which represents the focal length.

Understanding Parabola Standard Forms

Parabolas can open horizontally (right or left) or vertically (up or down). The standard form of the equation reveals these characteristics and allows you to pinpoint the focus.

Here's a breakdown of the standard forms and how they relate to the vertex and focus:

• ((h,k)): This represents the coordinates of the vertex of the parabola. The vertex is the turning point of the parabola.
  • If the equation has ((x-h)), then (h) is the x-coordinate of the vertex.
  • If the equation has ((y-k)), then (k) is the y-coordinate of the vertex.
  • If you see ((x+h)) or ((y+k)), remember that this is equivalent to ((x-(-h))) or ((y-(-k))), meaning the respective coordinate is negative. For instance, in ((y+2)^2), (k) would be (-2).
• (p): This is the focal length, which is the directed distance from the vertex to the focus.
  • The term (4p) is the coefficient of the non-squared term in the standard form.
  • The sign of (p) indicates the direction the parabola opens:
    • For ((y-k)^2 = 4p(x-h)): If (p > 0), it opens right. If (p < 0), it opens left.
    • For ((x-h)^2 = 4p(y-k)): If (p > 0), it opens up. If (p < 0), it opens down.

Step-by-Step Guide to Finding the Focus

Follow these steps to locate the focus of any parabola in standard form:

1. Identify the Standard Form: Determine whether the parabola's equation is in the form ((y-k)^2 = 4p(x-h)) (horizontal) or ((x-h)^2 = 4p(y-k)) (vertical). This tells you the parabola's orientation.
2. Determine the Vertex ((h,k)):
   • Identify the values of (h) and (k) directly from the standard form. Remember to account for signs (e.g., (x+3) means (h=-3)).
3. Find the Value of (p):
   • Equate the coefficient of the non-squared term to (4p).
   • Solve for (p). For example, if you have (12(x-h)), then (4p = 12), so (p = 3).
4. Apply the Focus Formula:
   • If the parabola is horizontal (((y-k)^2 = 4p(x-h))), the focus is at ((h+p, k)).
   • If the parabola is vertical (((x-h)^2 = 4p(y-k))), the focus is at ((h, k+p)).

Examples of Finding the Focus

Let's walk through a couple of examples.

Example 1: Horizontal Parabola

Question: Find the focus of the parabola given by the equation ((y-3)^2 = 8(x+1)).

Solution:

1. Standard Form: The equation is in the form ((y-k)^2 = 4p(x-h)), indicating a horizontal parabola.
2. Vertex ((h,k)):
   • From ((y-3)^2), we have (k=3).
   • From ((x+1)), which is ((x-(-1))), we have (h=-1).
   • Therefore, the vertex is ((-1, 3)).
3. Value of (p):
   • We have (4p = 8).
   • Dividing by 4, we get (p = 2). Since (p > 0), the parabola opens to the right.
4. Focus Formula: For a horizontal parabola, the focus is ((h+p, k)).
   • Substitute the values: Focus = ((-1+2, 3))
   • Focus = ((1, 3))

Example 2: Vertical Parabola (Requires Completing the Square)

Question: Find the focus of the parabola given by the equation (x^2 - 4x - 12y - 20 = 0).

Solution:

1. Convert to Standard Form: This equation is not yet in standard form. We need to complete the square for the (x) terms.
   • Move the (y) and constant terms to the other side:
     (x^2 - 4x = 12y + 20)
   • Complete the square for (x^2 - 4x). Take half of the coefficient of (x) ((-4/2 = -2)) and square it (((-2)^2 = 4)). Add 4 to both sides:
     (x^2 - 4x + 4 = 12y + 20 + 4)
   • Factor the left side and simplify the right side:
     ((x-2)^2 = 12y + 24)
   • Factor out the coefficient of (y) on the right side to match the standard form (4p(y-k)):
     ((x-2)^2 = 12(y+2))
   • Now the equation is in the form ((x-h)^2 = 4p(y-k)), indicating a vertical parabola.
2. Vertex ((h,k)):
   • From ((x-2)^2), we have (h=2).
   • From ((y+2)), which is ((y-(-2))), we have (k=-2).
   • Therefore, the vertex is ((2, -2)).
3. Value of (p):
   • We have (4p = 12).
   • Dividing by 4, we get (p = 3). Since (p > 0), the parabola opens upwards.
4. Focus Formula: For a vertical parabola, the focus is ((h, k+p)).
   • Substitute the values: Focus = ((2, -2+3))
   • Focus = ((2, 1))

Key Insights and Tips

• The value of (p) not only indicates the distance from the vertex to the focus but also the distance from the vertex to the directrix. The directrix is a line perpendicular to the axis of symmetry.
• The focus always lies inside the curve of the parabola.
• Understanding the role of (h), (k), and (p) is fundamental to analyzing parabolas and their properties beyond just finding the focus.