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Chapter 10: Problem 14

Sketching a Parabola In Exercises \(11-16,\) find the vertex, focus, and directrix of the parabola, and sketch its graph. $$y^{2}+6 y+8 x+25=0$$

Short Answer

Expert verified
The vertex of the parabola is at (2, -3), the focus is at (0, -3) and the directrix is the line x = 4. The parabola opens to the left.

Step by step solution

01

Transform Given Equation into Standard Form

Firstly, re-arrange the given equation \(y^{2}+6 y+8 x+25=0\) to the form \((y - k)^{2} = 4a(x - h)\). Completing the square for the y term gives us \((y + 3)^{2} - 9 + 8x + 25 = 0\). Further simplifying, we get \((y + 3)^{2} = - 8x + 16\), or \((y + 3)^{2} = - 8(x - 2)\). So, h = 2, k = -3, and 4a = -8.
02

Identify the Vertex, Focus and Directrix

From the values obtained above, the vertex is (2, -3) since h = 2, and k = -3. Since 4a = -8, then a = -2. We can find the focus by adding 'a' to the x-coordinate of the vertex giving us (2 - 2, -3), or (0, -3). The directrix is found by subtracting 'a' from the x-coordinate of the vertex, providing the line x = 2 - (-2) = 4.
03

Sketch the Graph

Plot the vertex, (2, -3), on the graph. Because the 'a' value is negative, we know that the parabola opens to the left. Also plot the focus, (0, -3), inside the parabola, and draw the directrix, x = 4, outside the parabola. Sketch the parabola such that it wraps around the focus and is equidistant from the focus and the directrix at the vertex.

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Key Concepts

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

Vertex
In a parabola, the vertex is a key point that defines its shape and position. It's the point where the parabola changes direction.
The equation we started with was \[ y^{2} + 6y + 8x + 25 = 0 \].
To find the vertex, we completed the square for the equation, transforming it into \[ (y + 3)^{2} = -8(x - 2) \].
This can be compared to the standard form \[ (y - k)^{2} = 4a(x - h) \], where the vertex is \( (h, k) \).
Here, \( h = 2 \) and \( k = -3 \), so the vertex is at \( (2, -3) \).

The vertex is crucial because it guides the shape of the parabola.
  • It tells us where the parabola is located on the graph.
  • It indicates the parabola's highest or lowest point.
Focus
The focus of a parabola is a special point that defines its shape and path. It's located inside the parabola and determines its width and steepness.
Using the equation from the vertex form \[ (y + 3)^{2} = -8(x - 2) \], the value of \( a \) is derived from \( 4a = -8 \), meaning \( a = -2 \).
The focus is calculated by adjusting the vertex.
Since our parabola opens left, subtracting \( a \) from the x-coordinate gives \( (2 - 2, -3) = (0, -3) \).

Key aspects of the focus:
  • The focus is the point around which the parabola "wraps."
  • It's always inside the parabola.
  • The parabola is equidistant from the focus and directrix at any given point.
Directrix
A parabola's directrix is a fixed line used in its geometric definition and helps in its construction.
It's located outside the parabola and reflects its shape along with the focus.
From the vertex form of the parabola \[ (y + 3)^{2} = -8(x - 2) \], we find \( a = -2 \).
Using this value, the directrix is calculated by adding \( a \) to the x-coordinate of the vertex:\[ x = 2 - (-2) = 4 \].

Important points about the directrix:
  • It's perpendicular to the axis of symmetry of the parabola.
  • The directrix, along with the focus, defines the parabola as the set of points equidistant from both.
  • The vertex is exactly in the middle between the focus and the directrix.
Completing the Square
Completing the square is an algebraic technique to transform quadratic equations into a more usable form, such as the standard form of a parabola.
This process makes it easy to identify attributes like the vertex.
Starting with the equation \[ y^{2} + 6y + 8x + 25 = 0 \], we focus on rewriting \[ y^{2} + 6y \].
To complete the square, we take half of 6, which is 3, square it to get 9, and use this to transform the equation: \[ (y + 3)^{2} - 9 \].
Substituting back gives us \[ (y + 3)^{2} = -8(x - 2) \].

Benefits of completing the square:
  • Simplifies the quadratic and reveals key features like the vertex.
  • Makes graphing parabolas straightforward.
  • Helps solve quadratic equations easily by working with perfect squares.

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Most popular questions from this chapter

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