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Chapter 10: Problem 9

Find the vertex, the focus, and the directrix. Then draw the graph. $$y^{2}=-6 x$$

Short Answer

Expert verified
Vertex at \((0,0)\), focus at \((-\frac{3}{2},0)\), directrix is \(x=\frac{3}{2}\).

Step by step solution

01

Rewrite the equation in standard form

The equation given is \(y^2 = -6x\). This can be compared to the standard form of a parabola with the equation \(y^2 = 4px\).
02

Identify the value of \(p\)

From the equation \(y^2 = 4px\), we can see that \(4p = -6\). Solving for \(p\) gives us \(p = -\frac{6}{4} = -\frac{3}{2}\).
03

Find the vertex

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

Find the focus

The focus is located \(p\) units away from the vertex. Since \(p = -\frac{3}{2}\), the focus is at \((-\frac{3}{2}, 0)\).
05

Find the directrix

The directrix of a parabola given by \(y^2 = 4px\) is a vertical line located \(p\) units away from the vertex. The directrix is given by the equation \(x = -p\). Since \(p = -\frac{3}{2}\), the directrix is \(x = \frac{3}{2}\).
06

Draw the graph

To draw the graph of the parabola, plot the vertex at \((0, 0)\), the focus at \((-\frac{3}{2}, 0)\), and the directrix as the vertical line \(x = \frac{3}{2}\). The parabola opens to the left since \(p < 0\).

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Key Concepts

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

vertex
In mathematics, the **vertex** of a parabola is a crucial concept. It is the point where the parabola changes direction. For the equation given in the exercise, which is \(\text{y}^2 = -6x\), the parabola is in the form \(\text{y}^2 = 4px\). Here, the vertex is at the origin, \(0,0\).

Key Points:
  • The vertex represents the minimum or maximum point of the parabola, depending on its orientation.
  • In our case, the vertex is at \(0,0\) because the equation is centered at the origin.
  • Understanding the position of the vertex helps in graphing and determining other properties of the parabola.
focus
The **focus** of a parabola is another fundamental property. It is a point inside the parabola where all the reflected rays originating from the parabola converge. For the equation \(y^2 = -6x\), this can be rewritten as \(y^2 = 4px\) to determine the value of \(p\).

Calculation:
  • Compare \(y^2 = 4px\) to \(y^2 = -6x\) to find \(4p = -6\).
  • Solving for \(p\) gives \(p = -3/2\).
  • The focus is \(p\) units away from the vertex, which in this case is at \((-3/2, 0)\).
Key Points:
  • The focus helps in determining the shape and orientation of the parabola.
  • By understanding where the focus is located, it becomes easier to graph the parabola accurately.
directrix
The **directrix** of a parabola is a straight line used in the geometric definition of the parabola. It is located parallel to the axis of symmetry and serves as a reference to ensure the parabola鈥檚 symmetrical properties.

Calculation:
  • The equation for a parabola in standard form \(y^2 = 4px\) helps us find the directrix.
  • From our exercise with the equation \(y^2 = -6x\), we have \(p = -3/2\).
  • The directrix is a vertical line given by the equation \(x = -p\), which results in \(x = 3/2\).
Key Points:
  • The directrix is essential in understanding the geometric properties of the parabola.
  • It helps to maintain the correct orientation when graphing the parabola, ensuring accuracy.
graphing parabolas
When **graphing parabolas**, a step-by-step method is highly effective. For the equation \(y^2 = -6x\), the following steps can be observed:

  • Identify the vertex: The vertex is at \(0, 0\).
  • Determine the focus: The focus is at \((-3/2), 0)\).
  • Find the directrix: The directrix is given by the line \(x = 3/2\).
  • Graph the vertex, directrix, and focus: Plot these on your graph.
  • Draw the parabola: Make sure to curve it in the correct direction; for \(p < 0\), it opens to the left.
Key Points:
  • Graphing enables a visual understanding of the properties and orientation of the parabola.
  • Following these steps ensures accuracy in plotting and helps solidify the understanding of key concepts.
standard form of a parabola
The **standard form of a parabola** is essential in understanding and solving parabolic equations. The standard forms are different for parabolas opening horizontally and vertically:

  • Vertical Parabolas: \(y^2 = 4px\), where \(p\) is the distance from the vertex to the focus.
  • Horizontal Parabolas: \(x^2 = 4py\), with \(p\) again representing the distance from the vertex to the focus.
For the given equation \(y^2 = -6x\), it can be compared to the horizontal form \(y^2 = 4px\). By simplifying, we can identify the values of important properties like the vertex, directrix, and focus.

Key Points:
  • Understanding the standard forms of parabolas aids in quick identification and solving of related problems.
  • These forms are fundamental in plotting parabolas accurately and understanding their geometric properties.

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