Sketch the graph and find following info:
Vertex:
Focus:
Directrix:
Axis of Symmetry:
Ends of Latus Rectum:
(x-2)^2=4(y-3)
I think the vertex is (2,3) is this correct, and how do I proceed on?
Thank you!
(x-2)^2 = 4(y-3).
(x-2)^2 = 4y - 12,
(x-2)^2 + 12 = 4y,
Divide both sides by 4:
y = 1/4(x-2)^2 +3, Vertex Form.
V(h,k) = (2,3).
a = 1/4.
4a = 1.
F(2,Y2). Y2 = k + 4a = 3 + 1 = 4.
V(2,3)
D(2,Y1). Y1 = k-4a = 3 - 1 = 2.
Axis: x = 2.
Use the following points for graphing:
(0,4), (1,1/4), V(2,3), (3,1/4), (4,4).
Yes, you are correct. The vertex of the given equation, (x-2)^2=4(y-3), is indeed (2, 3). To further find the focus, directrix, axis of symmetry, and ends of the latus rectum, we can use the standard form equation for a parabola with vertical axis:
(x - h)^2 = 4p (y - k)
where (h, k) represents the vertex, p represents the distance between the vertex and the focus/directrix, and p also represents the distance between the vertex and the endpoints of the latus rectum.
Comparing this standard form with our given equation (x-2)^2=4(y-3), we can see that the vertex is (h, k) = (2, 3) and p = 1.
1. Vertex: The vertex is already given as (2, 3).
2. Focus: To find the focus, we can use the formula (h, k + p). Since h = 2 and k = 3, the focus is at (2, 3 + 1), which simplifies to (2, 4).
3. Directrix: The directrix is a horizontal line that is p units below the vertex. Since p = 1, the directrix is a horizontal line y = 3 - 1, which simplifies to y = 2.
4. Axis of Symmetry: The axis of symmetry is always a vertical line passing through the vertex. Here, the axis of symmetry is the line x = 2.
5. Ends of Latus Rectum: The latus rectum is the line segment passing through the focus and perpendicular to the axis of symmetry. It has a length of 4p. In this case, the latus rectum has a length of 4 * 1 = 4 units. Since the axis of symmetry is x = 2, the latus rectum will be parallel to the y-axis. The endpoints of the latus rectum will be (2, 4 + 2) and (2, 4 - 2), which simplify to (2, 6) and (2, 2).
So, to summarize:
- Vertex: (2, 3)
- Focus: (2, 4)
- Directrix: y = 2
- Axis of Symmetry: x = 2
- Ends of Latus Rectum: (2, 6) and (2, 2)
To sketch the graph, plot points for the vertex, focus, and endpoints of the latus rectum on a coordinate plane, and use those points to draw a smooth curve passing through them.