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Chapter 11: Problem 15

Graph the parabolas. In each case, specify the focus, the directrix, and the focal width. Also specify the vertex. $$x^{2}-8 x-y+18=0$$

Short Answer

Expert verified
The vertex is (4, 2), the focus is (4, \(\frac{9}{4}\)), the directrix is \(y = \frac{7}{4}\), and the focal width is 1.

Step by step solution

01

Rearrange the Equation

Start with the given equation: \[ x^2 - 8x - y + 18 = 0 \] Rearrange it to make completing the square easier: \[ y = x^2 - 8x + 18 \] We'll complete the square to get the equation into vertex form.
02

Complete the Square

Focus on the quadratic part \( x^2 - 8x \). To complete the square, take half of the coefficient of \( x \), which is \( -8 \), divide by 2 to get \( -4 \), and then square it to get \( 16 \). Add and subtract 16 in the equation: \[ y = (x^2 - 8x + 16) + 18 - 16 \] This simplifies to: \[ y = (x - 4)^2 + 2 \] Now the equation is in the form \( (x-h)^2 = 4p(y-k) \).
03

Identify the Vertex

From the vertex form \( y = (x - 4)^2 + 2 \), we identify that the vertex \((h, k)\) is \((4, 2)\).
04

Write in Standard Parabola Form

The equation \( y = (x - 4)^2 + 2 \) is equivalent to \((x - 4)^2 = y - 2\). This matches the standard upward-opening parabola form \((x - h)^2 = 4p(y - k)\), which tells us \( 4p = 1 \). So, \( p = \frac{1}{4} \).
05

Find the Focus

The focus is \((h, k + p)\). Substitute \((h, k) = (4, 2)\) and \(p = \frac{1}{4}\): \[ Focus = (4, 2 + \frac{1}{4}) = (4, \frac{9}{4}) \]
06

Find the Directrix

The directrix is \(y = k - p\). Using \(k = 2\) and \(p = \frac{1}{4}\), we find: \[ Directrix: y = 2 - \frac{1}{4} = \frac{7}{4} \]
07

Calculate the Focal Width

The focal width is given by \(|4p|\). Since \(4p = 1\), the focal width is \(1\).

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Key Concepts

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

Focus
In the context of parabolas, the focus is a crucial point that helps define the curve's shape. Imagine a satellite dish; the focus is similar to where all signals come together. For the parabola \[ (x-4)^2 = y-2 \], the focus point is located directly above the vertex on the parabola's axis of symmetry. The steps to find the focus involve identifying the distance \( p \) from the vertex to the focus. For this parabola:
  • The vertex is \( (4, 2) \).
  • The value of \( p \) is determined from \( 4p = 1 \), so \( p = \frac{1}{4} \).
  • Therefore, the focus is at \( (4, 2 + \frac{1}{4}) \), which simplifies to \( (4, \frac{9}{4}) \).
This point \((4, \frac{9}{4})\) is where all the lines reflected off the parabola intersect.
Directrix
The directrix is an important line in understanding a parabola's geometry. It serves as a reference point alongside the focus, ensuring that each point on the parabola is equidistant from both the focus and this line. For the equation \((x-4)^2 = y-2\), you derive the directrix from the vertex and the value \( p \). This directrix is horizontal for upward- or downward-facing parabolas.
  • Start with the vertex point \((4, 2)\).
  • Using the equation \( y = k - p \), and knowing \( k = 2 \) and \( p = \frac{1}{4} \), compute the directrix at \( y = 2 - \frac{1}{4} = \frac{7}{4} \).
This line, \( y = \frac{7}{4} \), runs parallel to the x-axis, ensuring that the parabola remains symmetric and follows the correct geometric properties.
Vertex
The vertex of a parabola acts as either the utmost point or the lowest point, depending on its orientation. It is the central point that all the other elements of a parabola revolve around. To find the vertex of the parabola described by the original equation, convert it into its vertex form through the process of completing the square.
  • Given the equation \( y = (x-4)^2 + 2 \), the vertex can directly be read as \((h, k)\).
  • In this problem, the vertex \((h, k)\) is \((4, 2)\).
This means that the parabola is symmetric around the vertical line \( x = 4 \), and the vertex, being \( (4, 2) \), indicates it opens upwards from this point. The transformation into vertex form not only highlights the location of the vertex but also simplifies plotting or analyzing the parabola.

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