Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Discover detailed solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Given the focus of the parabola [tex]\((0, -2)\)[/tex] and the directrix [tex]\(y = 0\)[/tex]:
1. Finding [tex]\(p\)[/tex]:
- The distance [tex]\(2p\)[/tex] between the focus and the directrix is the distance from [tex]\((0, -2)\)[/tex] to the line [tex]\(y = 0\)[/tex].
- Thus, [tex]\(2p = 2\)[/tex], so [tex]\(p = 1\)[/tex].
2. Finding the vertex:
- The vertex is halfway between the focus and the directrix.
- The midpoint of the y-coordinates [tex]\(-2\)[/tex] and [tex]\(0\)[/tex] is [tex]\(\frac{-2 + 0}{2} = -1\)[/tex].
- Therefore, the vertex is [tex]\((0, -1)\)[/tex].
3. Equation in vertex form:
- The vertex form of the parabola is [tex]\(y = \frac{1}{4p}(x-h)^2 + k\)[/tex].
- Here, [tex]\(p = 1\)[/tex], [tex]\(h = 0\)[/tex], and [tex]\(k = -1\)[/tex].
- Therefore, the equation is [tex]\(y = \frac{1}{4 \cdot 1}(x - 0)^2 - 1 = \frac{1}{4}(x^2) - 1 = 0.25x^2 - 1\)[/tex].
Filling in the blanks:
- The value of [tex]\(p\)[/tex] is [tex]\(1\)[/tex].
- The vertex of the parabola is the point [tex]\((0, -1)\)[/tex].
- The equation of the parabola in vertex form is [tex]\(y = 0.25x^2 - 1\)[/tex].
1. Finding [tex]\(p\)[/tex]:
- The distance [tex]\(2p\)[/tex] between the focus and the directrix is the distance from [tex]\((0, -2)\)[/tex] to the line [tex]\(y = 0\)[/tex].
- Thus, [tex]\(2p = 2\)[/tex], so [tex]\(p = 1\)[/tex].
2. Finding the vertex:
- The vertex is halfway between the focus and the directrix.
- The midpoint of the y-coordinates [tex]\(-2\)[/tex] and [tex]\(0\)[/tex] is [tex]\(\frac{-2 + 0}{2} = -1\)[/tex].
- Therefore, the vertex is [tex]\((0, -1)\)[/tex].
3. Equation in vertex form:
- The vertex form of the parabola is [tex]\(y = \frac{1}{4p}(x-h)^2 + k\)[/tex].
- Here, [tex]\(p = 1\)[/tex], [tex]\(h = 0\)[/tex], and [tex]\(k = -1\)[/tex].
- Therefore, the equation is [tex]\(y = \frac{1}{4 \cdot 1}(x - 0)^2 - 1 = \frac{1}{4}(x^2) - 1 = 0.25x^2 - 1\)[/tex].
Filling in the blanks:
- The value of [tex]\(p\)[/tex] is [tex]\(1\)[/tex].
- The vertex of the parabola is the point [tex]\((0, -1)\)[/tex].
- The equation of the parabola in vertex form is [tex]\(y = 0.25x^2 - 1\)[/tex].
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.