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Chapter 10: Problem 28

$$\text {Give the focus, directrix, and axis of symmetry for each parabola.}$$ $$x=-16 y^{2}$$

Short Answer

Expert verified
Focus: (-4, 0), Directrix: x = 4, Axis of Symmetry: y = 0.

Step by step solution

01

- Identify general form of the parabola

First, recognize the given equation of the parabola and write it in its standard form. The given equation is: \[x=-16y^2\].The general form for a parabola that opens horizontally (left or right) is \(x = ay^2\).
02

- Determine the direction and parameters

Since the coefficient of \(y^2\) is negative, the parabola opens to the left. The equation can now be written in the form \(x = -4ay^2\). Compare this with \(x = -16y^2\), we get \(-4a = -16\), thus \(a = 4\).
03

- Calculate the focus

The focus of a parabola given by \(x = a(y-k)^2 + h\) is located at \((h + \frac{1}{4a}, k)\). For the given equation \(x = -16y^2\), we have \(h = 0\) and \(k = 0\). Therefore, the focus is at \((-4, 0)\).
04

- Find the directrix

The directrix of a parabola \(x = a(y-k)^2 + h\) is given by \(x = h - \frac{1}{4a}\). Substituting \(h = 0\) and \(a = 4\), the directrix becomes \(x = 4\).
05

- Identify the axis of symmetry

The axis of symmetry for a horizontally opening parabola \(x = ay^2\) is the horizontal line given by \(y = k\). For the given parabola \(y = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

focus of a parabola
The focus of a parabola is a specific point that plays a crucial role in defining its shape and properties. Technically, it is the point from which distances are measured to determine the curve. In a parabola defined by the equation \(x = a(y - k)^2 + h\), the focus can be found using the formula: \((h + \frac{1}{4a}, k)\). For the given parabola, \(x = -16y^2\), we identify \(h = 0\) and \(k = 0\). We also found \(a = 4\), so substituting these values into the formula for the focus, we get: \((-4, 0)\). This means that for this particular parabola, the focus is located at the point (-4, 0).
directrix of a parabola
The directrix of a parabola is a line that helps to determine the curvature of the parabola. For a parabola described by the standard equation \(x = a(y - k)^2 + h\), the directrix is given by the equation \(x = h - \frac{1}{4a}\). In our example \(x = -16y^2\), we determined that \(a = 4\) and both \(h\) and \(k\) are zero. Substituting these values into the formula for the directrix provides: \(x = 0 - \frac{1}{16} = 4\). Therefore, the directrix for this parabola can be represented by the vertical line \(x = 4\). This line helps in shaping the parabola along with the focus.
axis of symmetry
The axis of symmetry of a parabola is the line that divides the parabola into two mirror images. It passes through both the vertex and the focus of the parabola. For a horizontally opening parabola like \(x = ay^2\), the axis of symmetry is a horizontal line given by the equation \(y = k\). For our given equation \(x = -16y^2\), it is clear that \(k = 0\). Hence, the axis of symmetry for this particular parabola is the horizontal line \(y = 0\). This line is crucial as it helps in understanding the mirror-like properties of the parabola.

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Most popular questions from this chapter

Prove that the parabola with focus \((p, 0)\) and directrix \(x=-p\) has the equation \(y^{2}=4 p x\).

Write an equation for each parabola. vertex \((4,3),\) focus \((4,5)\)

$$\text {Give the focus, directrix, and axis of symmetry for each parabola.}$$ $$x=-32 y^{2}$$

Graph each hyperbola. Give the domain, range, center, vertices, foci, and equations of the asymptotes for each figure. $$9 x^{2}-25 y^{2}=225$$

Determine the two equations necessary to graph each hyperbola with a graphing calculator, and graph it in the viewing window indicated. $$y^{2}-9 x^{2}=9 ; \quad[-10,10] \text { by }[-10,10]$$
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