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

검증된 단계별 안내

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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Identification of Conic Sections from Equations

Conic sections are curves obtained by intersecting a plane with a cone, resulting in parabolas, ellipses, or hyperbolas. Recognizing the type involves rewriting the equation in a standard form and analyzing the degree and signs of variables. For example, an equation with one squared term and one linear term often represents a parabola.

Properties and Features of Parabolas

A parabola is defined as the set of points equidistant from a fixed point (focus) and a line (directrix). Its graph is symmetric about an axis. Key features include the vertex, focus, directrix, and axis of symmetry, which can be found by rewriting the equation in vertex form or standard form.

Graphing and Analyzing Conic Sections

Graphing conics requires identifying key points such as vertices, foci, and asymptotes (for hyperbolas). For parabolas, locating the focus and directrix helps in sketching. Understanding how to extract these features from the equation enables accurate graphing and interpretation of the curve's shape and orientation.