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Chapter 11: Problem 40

Find the vertex, focus, and directrix of each parabola. Graph the equation. \(y^{2}=8 x\)

Short Answer

Expert verified
Vertex: \((0,0)\), Focus: \((2,0)\), Directrix: \(x = -2\)

Step by step solution

01

Identify the parabola's equation form

The given equation is \(y^{2} = 8x\). This equation represents a parabola that opens to the right because the equation is in the form \(y^2 = 4ax\).
02

Identify the value of \(a\)

Compare \(y^2 = 8x\) with the standard form \(y^2 = 4ax\). Here, \(4a = 8\). Solve for \(a\) to find \(a = 2\).
03

Find the vertex

The vertex of the parabola \(y^2 = 4ax\) is at the origin, which is the point \((0,0)\).
04

Locate the focus

For a parabola \(y^2 = 4ax\), the focus is at \((a, 0)\). Since \(a = 2\), the focus is at \((2, 0)\).
05

Determine the directrix

The directrix of a parabola \(y^2 = 4ax\) is the line \(x = -a\). Since \(a = 2\), the directrix is the line \(x = -2\).
06

Graph the parabola

Plot the vertex at \(0,0\), the focus at \(2,0\), and draw the directrix line at \(x = -2\). Sketch the shape of the parabola opening to the right.

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Key Concepts

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

vertex of parabola
The vertex of a parabola is the point where the curve changes direction. For the given equation, \(y^2 = 8x\), the standard form is \(y^2 = 4ax\).
By comparing, we have \(4a = 8\) which gives us a value of \(a = 2\).
In this form, the vertex is always located at the point \((0, 0)\). This means for our given equation, the vertex lies at the origin, or \((0, 0)\).
Knowing the vertex is vital because it serves as the reference point for the graph and helps in identifying the shape and direction of the parabola.
focus of parabola
The focus of a parabola is a point from which distances are measured in a consistent fashion to form the shape. For the equation \(y^2 = 8x\), we found \(a = 2\).
In general, for a parabola in the form \(y^2 = 4ax\), the focus is located at \((a,0)\). This indicates that for our equation, the focus is at \((2, 0)\).
The focus lies on the axis of symmetry of the parabola, and every point on the parabola is equidistant from the focus and the directrix. Understanding the focus helps in accurately sketching the parabola and understanding its geometric properties.
directrix of parabola
The directrix of a parabola is a line that, together with the focus, helps in defining the curve. For the provided equation \(y^2 = 8x\), we earlier found \(a = 2\).
For parabolas of the form \(y^2 = 4ax\), the directrix is a vertical line given by the equation \(x = -a\).
Therefore, our directrix is the line \(x = -2\).
This line is crucial because, along with the focus, it defines how the parabola is oriented and helps in ensuring the distances from the focus to the parabola are equal to the distances from the parabola to the directrix.

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