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Chapter 13: Problem 1

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ y^{2}=8 x $$

Short Answer

Expert verified
The vertex is \((0, 0)\), the focus is \((2, 0)\), and the directrix is \(x = -2\).

Step by step solution

01

Identify the Parabola Form

The given equation is \(y^2 = 8x\). This can be compared with the standard form of a parabola \(y^2 = 4px\) where the parabola opens to the right or left. From here, we can identify that \(4p = 8\), so \(p = 2\).
02

Determine the Vertex

The vertex form of the parabola \(y^2 = 8x\) is \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex. Since there are no values subtracted from \(x\) or \(y\), the vertex is at the origin: \((0, 0)\).
03

Find the Focus

The focus of a parabola \((y-k)^2 = 4p(x-h)\) is \((h+p, k)\). Using our parabola, this gives: \((0+2, 0) = (2, 0)\). Therefore, the focus is at \((2, 0)\).
04

Identify the Directrix

The directrix of a parabola \((y - k)^2 = 4p(x - h)\) is \(x = h - p\). For our parabola, this becomes \(x = 0 - 2 = -2\). So, the directrix is the line \(x = -2\).
05

Sketch the Graph

To sketch the graph, plot the vertex at \((0, 0)\), the focus at \((2, 0)\), and draw the directrix as a vertical line at \(x = -2\). The parabola will open to the right, with the vertex as the starting point.

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Key Concepts

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

Vertex
The vertex is an essential part of understanding the geometry of a parabola. It is the point where the parabola changes direction, and in many cases, it represents the maximum or minimum value of the parabola. In the given equation, \( y^2 = 8x \), the vertex is located at the origin, \((0, 0)\). This is because the equation is already in a simplified form where there is nothing subtracted from \( x \) or \( y \). This tells us that the parabola is symmetric about this point.
  • The vertex is a fixed point from which the parabola curves away.
  • It serves as a reference for plotting both the focus and the directrix.
Understanding the vertex helps in sketching and analyzing the parabola's behavior.
Focus
The focus is a special point on the interior of a parabola that plays a vital role in its construction. All points on the parabola are equidistant to the focus and a corresponding point on the directrix. For the equation \( y^2 = 8x \), we determined that \( p = 2 \), meaning the distance from the vertex to the focus is 2 units in the direction in which the parabola opens.
  • The focus can be thought of as a guiding point, influencing the shape of the parabola.
  • For a parabola opening to the right, the focus is located to the right of the vertex.
In this exercise, the focus is at \((2, 0)\). Understanding the location of the focus is critical when trying to derive the path the curve takes.
Directrix
The directrix is a fixed line used in tandem with the focus to define and describe parabolas geometrically. It is perpendicular to the axis of symmetry of the parabola and provides a constant distance relationship with all points on the parabola. In our example, the directrix is given by the equation \( x = -2 \).
  • The directrix serves as a reference line for measuring the distance from points on the parabola.
  • It is always located on the opposite side of the vertex from the focus.
This alignment means that while the focus is at \((2, 0)\), the directrix at \( x = -2 \) balances the parabola's internal geometry.
Graph of a Parabola
Graphing a parabola involves plotting not just the curve itself, but also indicating the key geometric components, such as the vertex, focus, and directrix. With the information derived from the equation \( y^2 = 8x \), we can sketch a comprehensive graph:- Begin by marking the vertex at \((0, 0)\).- Next, plot the focus at \((2, 0)\).- Draw the directrix as a vertical line at \( x = -2 \).The graph of this parabola opens to the right, meaning it curves away from the vertex in the direction of the focus.
  • The vertex serves as the anchor point for the parabola.
  • The placement of both the focus and directrix facilitates a symmetrical graph around the axis perpendicular to the direction of opening.
Understanding the intricacies of plotting these components supports visual comprehension of the parabolic structure.

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Most popular questions from this chapter

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ x^{2}=8 y $$

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