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Magazine Chapter 11: Problem 20
An equation of a parabola is given. (a) Find the vertex, focus, and directrix of the parabola. (b) Sketch a graph showing the parabola and its directrix. $$x^{2}+2 x-20 y+41=0$$
Short Answer
Expert verified
The vertex is (-1, 2), focus is (-1, 7), and directrix is y = -3.
Step by step solution
01
Rewrite the Equation in Standard Form
Start with the given equation \(x^2 + 2x - 20y + 41 = 0\). Rearrange it to isolate the terms involving \(x\): \(x^2 + 2x = 20y - 41\). Next, complete the square for the \(x\) terms: \(x^2 + 2x\) can be rewritten as \((x + 1)^2 - 1\). Substitute this back into the equation: \((x + 1)^2 - 1 = 20y - 41\). Rearranging gives \((x + 1)^2 = 20y - 40\).
02
Express y in Terms of x
Now rewrite the equation to express \(y\) in terms of \(x\), resulting in the vertex form of the parabola: \((x + 1)^2 = 20(y - 2)\). This equation is now in the standard form \((x-h)^2 = 4p(y-k)\) for a parabola that opens vertically.
03
Identify the Vertex
From the vertex form \((x + 1)^2 = 20(y - 2)\), identify the vertex \((h, k)\) as \((-1, 2)\).
04
Find the Focus
In the equation \((x - h)^2 = 4p(y - k)\), the value of \(4p\) is 20, so \(p = 5\). This means the focus is located at \((h, k + p)\). Calculate \((h, k + p) = (-1, 2 + 5)\), giving the focus as \((-1, 7)\).
05
Determine the Directrix
The directrix is a horizontal line given by \(y = k - p\). Thus, calculate \(y = 2 - 5 = -3\). So, the directrix is \(y = -3\).
06
Sketch the Parabola
To sketch the parabola, plot the vertex \((-1, 2)\) and the focus \((-1, 7)\). Draw the parabola opening upwards, equidistant from the focus and directrix \(y = -3\). Indicate the points of intersection of the parabolic curve with the vertical axis and sketch the symmetric curve.
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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 context of our parabola equation, the vertex can be found from the standard form \((x - h)^2 = 4p(y - k)\). By completing the square and rewriting the given equation \(x^2 + 2x - 20y + 41 = 0\) to its vertex form, we discovered that the vertex \((h, k)\) is \((-1, 2)\). Here, the vertex represents the point <-1, 2> where the parabola reaches its minimum point along the vertical opening.
- The vertex is often visualized as the peak or bottom point of a parabola.
- It is helpful in identifying the line of symmetry for the parabola, which in this case is \(x = -1\).
Focus
The focus of a parabola is a significant point that helps define the parabola's shape. Each point on a parabola is equidistant from the focus and a line known as the directrix. For the equation given, the focus can be calculated through the value of \(p\), found from the equation \((x-h)^2 = 4p(y-k)\). In our instance, with \(4p = 20\), this implies \(p = 5\). With a vertex at <-1, 2>, the focus is then located at <-1, 7> by calculating \((h, k + p)\).
- The focus lies on the axis of symmetry of the parabola.
- This point plays a critical role in determining the path of the parabola's curve.
Directrix
The directrix of a parabola is a straight line used alongside the focus in defining and graphing a parabola. It is the counterpart to the focus, helping form the parabola by maintaining equidistance from points on the curve. For our parabola, the directrix can be calculated using the formula \(y = k - p\). With a calculated focus and a known vertex, the directrix is determined as \(y = -3\).
- The directrix is always perpendicular to the axis of symmetry of the parabola.
- This line is equally significant as it affects how "opening" or "narrow" the parabola appears.
Standard Form of Parabola
The standard form of a parabola equation is a crucial foundation for understanding and manipulating parabolic curves. The general form \((x - h)^2 = 4p(y - k)\) is specifically used for parabolas that open vertically, such as in the exercise's equation. By rearranging and completing the square in the given equation \(x^2 + 2x - 20y + 41 = 0\), we arrive at its standard form \((x + 1)^2 = 20(y - 2)\).
- "h" and "k" are the coordinates for the vertex of the parabola.
- "p" defines the distance from the vertex to the focus and the vertex to the directrix.
Graph Sketching of Parabola
Graph sketching of a parabola involves visualizing a set of points that satisfies the parabolic equation. Here, starting with known entities like the vertex, focus, and directrix allows a straightforward sketch of the parabola. The vertex of <-1, 2> serves as the starting point. The placement of the focus at <-1, 7> and the directrix line at \(y = -3\) forms the structural basis.
- Begin by marking the vertex and focus on the graph.
- Draw the directrix as a horizontal line beneath the vertex.
- Sketch the parabola opening upwards, ensuring that every point is equidistant to the focus and directrix.
