Graphing Conic Sections

Sketch the parabolas in Exercises 55–58. Include the focus and directrix in each sketch.

y² = −(8/3)x

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Identify the type of conic section given by the equation \(y^{2} = -\frac{8}{3}x\). Since the equation is in the form \(y^{2} = 4px\), it represents a parabola that opens either left or right.

Rewrite the equation to match the standard form \(y^{2} = 4px\). Here, \(4p = -\frac{8}{3}\), so solve for \(p\) by dividing both sides by 4: \(p = -\frac{8}{3} \times \frac{1}{4} = -\frac{2}{3}\).

Determine the vertex, focus, and directrix of the parabola. The vertex is at the origin \((0,0)\) because the equation is not shifted. The focus lies at \((p, 0)\), so substitute \(p = -\frac{2}{3}\) to get the focus at \(\left(-\frac{2}{3}, 0\right)\).

Find the equation of the directrix. For a parabola \(y^{2} = 4px\), the directrix is the vertical line \(x = -p\). Substitute \(p = -\frac{2}{3}\) to get the directrix \(x = \frac{2}{3}\).

Sketch the parabola opening to the left (since \(p\) is negative), plot the vertex at the origin, mark the focus at \(\left(-\frac{2}{3}, 0\right)\), and draw the directrix line \(x = \frac{2}{3}\). This completes the graph with all key features.

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주요 개념

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Parabola Definition

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Understanding this geometric definition helps in identifying the focus and directrix from the equation and in sketching the parabola accurately.

Standard Form of a Parabola

Parabolas can be expressed in standard forms such as y² = 4px or x² = 4py, where p represents the distance from the vertex to the focus (and directrix). Recognizing and rewriting the given equation into this form allows determination of the parabola's orientation and key features.

Focus and Directrix Calculation

From the standard form y² = 4px, the focus is located at (p, 0) and the directrix is the line x = -p (or similarly for x² = 4py). Calculating p from the equation's coefficients enables plotting these elements, which are essential for accurately sketching the parabola.

Graphing Conic SectionsSketch the parabolas in Exercises 55–58. Include the focus and directrix in each sketch.y² = −(8/3)x