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Wize High School Grade 12 Pre-Calculus Textbook > Conics [BC/AB]

Parabolas

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Parabolas

A parabola is the collection of points equidistant from a point called the focus and the line called the directrix.



Assume:
  • the focus is (0,p)(0,p)
  • the directrix is the line y=−py=-p
  • there is an arbitrary point on the parabola (x,y)(x,y)

Then:

distance to focus=distance to directrix(x−0)2+(y−p)2=(y+p)2x2+y2−2yp+p2=y2+2yp+p2x2=4yp\begin{array}{rcl} \text{distance to focus}&=&\text{distance to directrix}\\\\ \sqrt{(x-0)^2+(y-p)^2}&=&\sqrt{(y+p)^2}\\\\ x^2+y^2-2yp+p^2&=&y^2+2yp+p^2\\\\ x^2&=&4yp \end{array}


Therefore, the standard form of the parabola is x2=4yp\boxed{x^2=4yp} (opens up) or y2=4xp\boxed{y^2=4xp} (open right).

Wize Tip
The transformed form of a parabola moves the vertex from the origin to a point (h, k) and can be expressed as:

(x−h)2=4p(y−k)(x-h)^2=4p(y-k) or

(y−k)2=4p(x−h)(y-k)^2=4p(x-h)

Vertex: (h,k)(h,k)
Focal Length: p
Focus:
  • a point that is a distance of 'p' units away from the vertex
  • the point that the parabola curves around
Directrix:
  • If horizontal: y=k+py=k+p
  • If vertical: x=h+px=h+p
  • the parabola curves away from the directrix
Focal Width: ∣4p∣|4p|

Example: Parabolas

Write an equation for a parabola that opens left, has a vertex (0,2)(0,2) and passes through (−6,−4).(-6,-4).

Standard form opening to the left:

(y−k)2=−4p(x−h)(y−2)2=−4p(x−0)(−4−2)2=−4p(−6−0)36=24p∴ p=32\begin{array}{rcl} (y-k)^2&=&-4p(x-h)\\\\ (y-2)^2&=&-4p(x-0)\\\\ (-4-2)^2&=&-4p(-6-0)\\\\ 36&=&24p\\\\ \therefore~p&=&\dfrac{3}{2} \end{array}

The equation of the parabola is then:

(y−2)2=−4⋅32(x−0)(y−2)2=−6x\begin{array}{rcl} (y-2)^2&=&-4\cdot\dfrac{3}{2}(x-0)\\\\ (y-2)^2&=&-6x \end{array}

Practice: Parabolas

What is the vertex, focal length, focus, directrix, and focal width of 2x2+16x+y=02x^2+16x+y=0?
A.
Vertex
B.
Focal Length
C.
Focus
D.
Directrix
E.
Focal Width
(−4,32)(-4,32)
18\dfrac{1}{8}
(−4,2558)\Bigg(-4, \dfrac{255}{8}\Bigg)
y=2578y=\dfrac{257}{8}
12\dfrac{1}{2}

Practice: Parabolas

What is the equation of a parabola with focus at (2,3)(2,3) and directrix at y=5y=5?

Practice: Parabolas

What is the equation of a parabola that opens to the right with focal width from (6,−7)(6,-7) to (6,12)(6,12)?
Extra Practice
Conics
Write an equation for a parabola that opens left, has a vertex (3,5)(3,5) and passes through (−2,7).(-2,7).
Parabolas
Practice: Parabolas
What is the equation of a parabola that opens to the left with focal width from (−3,4)(-3,4) to (−3,−4)(-3,-4)?
Parabolas
Practice: Parabolas
What is the equation of a parabola that opens up with a focal width from (−16,3)(-16,3) to (7,3)(7,3)?
Parabolas
Practice: Parabolas
What is the equation of a parabola with focus at (5,2)(5,2) and directrix at x=−4x=-4?
Parabolas
Practice: Parabolas
What is the equation of a parabola with focus at (−1,−4)(-1,-4) and directrix at y=1y=1?
Parabolas
Practice: Parabolas
What is the vertex, focal length, focus, directrix, and focal width of 3x2+27x+y=03x^2+27x+y=0?
Parabolas
Practice: Parabolas
What is the vertex, focal length, focus, directrix, and focal width of 5x2+10x+y=05x^2+10x+y=0?