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For the equation of the parabola given in the form y2 = 4px,

(a) Identify the vertex, value of p, focus, and focal diameter of the parabola.
(b) Identify the endpoints of the latus rectum.
(c) Graph the parabola.
(d) Write equations for the directrix and axis of symmetry.
Express numbers in exact, simplest form.
4y2=16x

User Ty Smith
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(a) Vertex: The parabola is in the form y2 =4px, where the vertex is at the origin (0,0).

Value of p: Comparing with the standard form, we have p=4.

Focus: The focus is at (p,0)=(4,0).

Focal Diameter: Focal diameter is the absolute value of 4p, so it is ∣4×4∣=16.

(b) Endpoints of Latus Rectum: Latus rectum is perpendicular to the axis of the parabola and passes through the focus. Its length is 4p, so the endpoints are (−2p,p) and (2p,p). In this case, the endpoints are (−8,4) and (8,4).

(d) Equation for Directrix: Since the parabola opens to the right, the equation for the directrix is x=−p=−4.

Axis of Symmetry: The axis of symmetry is the line x=0, passing through the vertex.

The given parabola is 4y^2 =16x.

(a) Vertex: For a parabola in the form y^2 =4px, the vertex is at the origin (0,0).

Value of p: In the given equation 4y^2 =16x, comparing with the standard form y^2 =4px, we find p=4.

Focus: The focus of the parabola is at the point (p,0)=(4,0). This is because the parabola opens to the right, and the focus is located at a distance p from the vertex along the axis of symmetry.

Focal Diameter: The focal diameter is the absolute value of 4p, which is ∣4×4∣=16.

(b) Endpoints of Latus Rectum: The latus rectum is a line segment perpendicular to the axis of the parabola and passes through the focus. Its length is 4p, so the endpoints are given by (−2p,p) and (2p,p). Substituting p=4, we get (−8,4) and (8,4).

(c) Graph: The graph of the parabola is a U-shaped curve that opens to the right. The vertex is at the origin, and the focus is to the right of the vertex.

(d) Equation for Directrix: Since the parabola opens to the right, the equation for the directrix is x=−p=−4.

Axis of Symmetry: The axis of symmetry is the vertical line passing through the vertex, which is x=0.

User WernerW
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