Parabolas

Exercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.

x = −3y²

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Rewrite the given equation $x = -3y^2$ in the standard form of a parabola that opens horizontally. The general form for a parabola opening left or right is $ (y - k)^2 = 4p(x - h) $, where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus (and also to the directrix).

Identify the vertex by comparing the given equation to the standard form. Since the equation is $x = -3y^2$, it can be rewritten as $y^2 = \frac{1}{-3} x$, or $ (y - 0)^2 = 4p(x - 0) $. This shows the vertex is at the origin $(0,0)$.

Determine the value of \$4p$ by matching coefficients. From $y^2 = 4p x$, we see that $4p = -\frac{1}{3}$, so $p = -\frac{1}{12}$. The negative sign indicates the parabola opens to the left.

Find the focus using the vertex and $p$. Since the parabola opens horizontally, the focus is at $(h + p, k)$, which is $(0 + p, 0) = \left(-\frac{1}{12}, 0\right)$.

Find the directrix, which is a vertical line located at $x = h - p$. Using $h=0$ and $p = -\frac{1}{12}$, the directrix is at $x = 0 - \left(-\frac{1}{12}\right) = \frac{1}{12}$. Then, sketch the parabola opening left with vertex at the origin, focus at $\left(-\frac{1}{12}, 0\right)$, and directrix at $x = \frac{1}{12}$.

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주요 개념

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Standard Form of a Parabola

A parabola can be expressed in standard form as either (x - h)² = 4p(y - k) or (y - k)² = 4p(x - h), where (h, k) is the vertex. This form helps identify the parabola's orientation and key features like the focus and directrix.

Focus and Directrix of a Parabola

The focus is a fixed point inside the parabola, and the directrix is a line outside it. The parabola consists of points equidistant from the focus and directrix. The parameter p determines the distance from the vertex to both the focus and directrix.

Graphing Parabolas

To sketch a parabola, plot the vertex, focus, and directrix, then draw a smooth curve equidistant from the focus and directrix. Understanding the parabola's orientation (horizontal or vertical) guides the shape and direction of the graph.

ParabolasExercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.x = −3y²