Graph y^2=-32x

Problem

y2=−32*x

Solution

1. Identify the type of conic section. The equation y2=−32*x is in the standard form of a parabola that opens horizontally, y2=4*p*x

2. Determine the value of p by setting 4*p equal to the coefficient of x

4*p=−32

p=−8

3. Locate the vertex. Since there are no h or k offsets in the equation, the vertex is at the origin.

Vertex=(0,0)

4. Find the focus. For a horizontal parabola, the focus is at (p,0)

Focus=(−8,0)

5. Identify the directrix. The directrix is the vertical line x=−p

x=8

6. Calculate the endpoints of the latus rectum to determine the width. The length of the latus rectum is |4*p|=32 The endpoints are (p,2*p) and (p,−2*p)

Endpoints=(−8,16),(−8,−16)

7. Sketch the graph. Plot the vertex (0,0) the focus (−8,0) and the endpoints (−8,16) and (−8,−16) Draw a smooth curve opening to the left through these points and include the vertical line x=8 as the directrix.

Final Answer

y2=−32*x⇒Parabola opening left with vertex *(0,0), focus *(−8,0), and directrix *x=8

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