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Chapter 11: Problem 51

Find the vertex, focus, and directrix of each parabola. Graph the equation. \(x^{2}+8 x=4 y-8\)

Short Answer

Expert verified
Vertex: \((-4, -2)\), Focus: \((-4, -1)\), Directrix: \(y = -3\)

Step by step solution

01

- Rewrite the Equation in Vertex Form

First, rewrite the given equation \(x^{2}+8x=4y-8\) in standard form: \(x^2 + 8x = 4y - 8\). Rewrite it as \(x^2 + 8x + 16 = 4y - 8 + 16\), making a perfect square trinomial on the left side: \((x + 4)^2 = 4y + 8\). Then simplify to vertex form: \((x + 4)^2 = 4(y + 2)\).
02

- Identify the Vertex

From the vertex form equation \((x + 4)^2 = 4(y + 2)\), identify the vertex \((h, k)\). Here, \(h = -4\) and \(k = -2\). Therefore, the vertex is \((-4, -2)\).
03

- Determine the Focus

For a parabola with vertex form \((x - h)^2 = 4p(y - k)\), the focus is \((h, k + p)\). Here, \(4p = 4\) so \(p = 1\). Therefore, the focus is at \((-4, -2 + 1) = (-4, -1)\).
04

- Find the Directrix

The equation of the directrix for a parabola in the form \((x - h)^2 = 4p(y - k)\) is \(y = k - p\). With \(k = -2\) and \(p = 1\), the directrix is \(y = -2 - 1 = -3\). Therefore, the directrix is \(y = -3\).
05

- Graph the Parabola

Plot the vertex \((-4, -2)\), the focus \((-4, -1)\), and the directrix \(y = -3\) on the graph. Sketch the parabola opening upwards, using the vertex and symmetric points around it.

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Key Concepts

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

Vertex
The vertex of a parabola is a crucial point where the curve changes direction. In the given equation, we first aim to convert it into vertex form from its original standard form. The vertex form of a parabola can be written as \((x - h)^2 = 4p(y - k)\). Here, \(h, k\) represents the vertex coordinates.
To convert the equation \(x^2 + 8x = 4y - 8\) into vertex form, complete the square on the \(x\)-terms: \(x^2 + 8x + 16 = 4y + 8\). Simplifying, we get \((x + 4)^2 = 4(y + 2)\). Hence, the vertex \((h, k)\) is \(-4, -2\).
In summary:
  • The vertex indicates the peak or lowest point of a parabola.
  • Its \( (x, y)\) coordinates are taken from the modification of the equation into vertex form.
  • For our exercise, the vertex is found to be \(-4, -2\).
Focus
The focus of a parabola is a special point that lies along the axis of symmetry of the parabola. For the vertex form equation \((x - h)^2 = 4p(y - k)\), the focus has coordinates \((h, k + p)\).
We find \({4p = 4, hence \ p = 1\})\) in the given equation. Using \(h = -4\) and \(k = -2\), we calculate the focus by adding \(p\) to the \ y \-coordinate of the vertex.
The focus for the given parabola is \((-4, -2 + 1) = (-4, -1)\).
Key points:
  • The focus lies inside the parabola and influences its shape.
  • It's one unit away from the vertex along the axis of symmetry.
  • For our exercise, the focus is located at \(-4, -1\)\.
Directrix
The directrix of a parabola is a line that lies outside the curve. It helps describe the parabola's shape and orientation. For a parabola in the form \( (x - h)^2 = 4p(y - k)\), the directrix is given by \( y = k - p\).
In our exercise, \( k = -2 \) and \( p = 1 \), thus the directrix is \( y = -2 - 1 = -3\).
Important aspects of the directrix:
  • The directrix is always perpendicular to the axis of symmetry of the parabola.
  • It is one unit away from the vertex but in the opposite direction of the focus.
  • For the exercise at hand, the directrix is the horizontal line \ (y = -3) .
Vertex Form
The vertex form of a parabola is a very useful way to express its equation. This format makes it easy to identify the key attributes of the parabola, such as the vertex, focus, and directrix. The vertex form is \( (x - h)^2 = 4p(y - k) \) for vertical parabolas.

In this format:
  • \(h, k\) are the coordinates of the vertex.
  • \(p\) is the distance from the vertex to the focus and from the vertex to the directrix.
For the given equation \(x^{2}+8 x=4 y-8\), converting it to vertex form involves completing the square:

  • Rewrite \(x^2 + 8x = 4y - 8\) to \(x^2 + 8x + 16 = 4y + 8\).
  • This transforms to \((x + 4)^2 = 4(y + 2)\).
Therefore, the vertex form displays the vertex as \(-4, -2 \).
Quadratic Equations
Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0 \). These are essential in defining parabolas and form the basic understanding of their geometry.
Our original exercise involves transforming \( x^2 + 8x = 4y - 8\) into a standard quadratic form.
Steps to rewrite it include:

  • Rearranging terms and completing the square.
  • Identifying the vertex form parameters from the quadratic equation.
  • Solving for the vertex, focus, and directrix using quadratic transformations.
To sum up:
  • Quadratic equations help in defining a parabola's structure and orientation.
  • They guide the transformation into vertex form for easier interpretation and solution.
  • In our exercise, recognizing the quadratic patterns facilitated accurate vertex identification and graphing.

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Most popular questions from this chapter

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