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Sagot :
Answer:
Equation of the Parabola
Step-by-step explanation:
To determine the equation of the parabola with a vertex at (−2,3)(−2,3) and a focus at (0,3)(0,3), follow these steps:
Identify the Type of Parabola:
Since the focus and the vertex have the same yy-coordinate (y=3y=3), the parabola opens horizontally.
Find the Directrix:
The directrix of a parabola is equidistant from the vertex as the focus but on the opposite side. Given the vertex (−2,3)(−2,3) and focus (0,3)(0,3), the distance between them is 2 units. Therefore, the directrix is at x=−2−2=−4x=−2−2=−4.
Write the Standard Form of the Parabola:
For a parabola opening horizontally, the standard form is (y−k)^2=4p(x−h)(y−k)^2=4p(x−h), where (h,k)(h,k) is the vertex and pp is the distance from the vertex to the focus (or the directrix).
In this case:
The vertex (h,k)=(−2,3)(h,k)=(−2,3)
The distance p=2p=2 (from vertex to focus)
Thus, the equation of the parabola is:
(y−3)^2=4⋅2⋅(x+2)
(y−3)^2=4⋅2⋅(x+2)
(y−3)^2=8(x+2)
(y−3)^2=8(x+2)
Graphing the Parabola
Plot the Vertex: Mark the point (−2,3)(−2,3) on the graph.
Plot the Focus: Mark the point (0,3)(0,3).
Draw the Directrix: Draw a vertical line at x=−4x=−4.
Sketch the Parabola:
The parabola opens to the right since the focus is to the right of the vertex.
The shape should be symmetrical around the line y=3y=3 and should curve towards the right from the vertex.
Mark additional points on the parabola to help in sketching, such as points where x=−1x=−1 or x=−3x=−3 and find corresponding yy values using the equation (y−3)^2=8(x+2)(y−3)^2=8(x+2).
This will give you a complete picture of the parabola with the vertex, focus, and directrix clearly marked.
Answer:
Here is the equation of the parabola:
[tex](y - 3)^2 = 8(x + 2)[/tex]
Step-by-step explanation:
1. Identify the orientation of the parabola:
Since the vertex and focus share the same y-coordinate (3), the parabola opens horizontally. The focus is to the right of the vertex, indicating that the parabola opens to the right.
2. Determine the vertex and the focus-related properties:
[tex]Vertex \: \((h, k)\) = \((-2, 3)\)[/tex]
[tex]Focus \: \((h + p, k)\) = \((0, 3)\)[/tex]
The distance (p) (the distance from the vertex to the focus) is:
[tex]{h + p = 0 \implies -2 + p = 0 \implies p = 2}[/tex]
3. Equation of the parabola:
For a horizontally oriented parabola (opening right), the standard form is:
[tex] (y - k)^2 = 4p(x - h)[/tex]
Substituting (h = -2), (k = 3), and (p = 2):
[tex](y - 3)^2 = 4 \cdot 2 \cdot (x + 2)[/tex]
Simplifying:
[tex](y - 3)^2 = 8(x + 2)[/tex]
4. Graphing the Parabola:
To graph this parabola:
- Plot the vertex at (-2, 3).
- Plot the focus at (0, 3).
- Draw the directrix, which is a vertical line at (x = -4) (since the directrix is the same distance (p) on the opposite side of the vertex from the focus).
- Sketch the parabola ensuring it curves around the focus and opens towards the right.
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