Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

4x = -y²

Accepted Answer

Rewrite the given equation $$4x = -y^2$ in a more standard form by isolating x$. This gives x$$ = -\frac{1}{4} $$y^2. $$Recognize the form of the equation: since $$x is $$expressed as a quadratic function of $$y$$, this is the equation of a parabola that opens either left or right. Identify the vertex of the parabola. Because the equation is in the form $$x = a y^2$ + h$$, the vertex is at the point $$(0,0). $$Determine the direction the parabola opens. Since the coefficient of $$y^2 is $$negative ($$-\frac{1}{4}$$), the parabola opens to the left along the $$x-$$axis. Find the focus and directrix using the standard parabola formula $$x = \frac{1}{4p} y^2$. Here, $$\frac{1}{4p} = -\frac{1}{4}$$, so solve for p$, then use p$ to locate the focus at (h+p$$, k)$$ and the directrix at x = h - p$.

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Key Concepts

Identification of Conic Sections from Equations

Properties and Features of Parabolas

Graphing and Analyzing Conic Sections