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Chapter 10: Problem 6

Exer. \(1-12\) : Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$(x-3)^{2}=\frac{1}{2}(y+1)$$

Short Answer

Expert verified
Vertex: (3, -1), Focus: (3, -7/8), Directrix: y = -9/8.

Step by step solution

01

Write the Equation in Standard Form

The given equation is \((x - 3)^2 = \frac{1}{2}(y + 1)\). This is in the form of \((x-h)^2 = 4p(y-k)\), where the parabola opens either upwards or downwards. In this form, \(h = 3\) and \(k = -1\), giving us the vertex of the parabola as \((3, -1)\).
02

Identify Parameter p

The equation \((x-3)^2 = \frac{1}{2}(y+1)\) must be compared to \((x-h)^2 = 4p(y-k)\). Here, \(4p = \frac{1}{2}\), so to find \(p\), we solve \(4p = \frac{1}{2}\), giving \(p = \frac{1}{8}\).
03

Find the Focus

For a parabola of the form \((x-h)^2 = 4p(y-k)\), the focus is located at \((h, k + p)\). Substituting the values, we get \((3, -1 + \frac{1}{8}) = (3, -\frac{7}{8})\).
04

Determine the Directrix

The directrix of the parabola is given by the horizontal line \(y = k - p\). Substituting the known values, we have \(y = -1 - \frac{1}{8} = -\frac{9}{8}\).
05

Sketch the Graph

To sketch the parabola, plot the vertex \((3, -1)\), the focus \((3, -\frac{7}{8})\) and draw the directrix line at \(y = -\frac{9}{8}\). The parabola opens upwards since \(p > 0\). Ensure the parabola is symmetric about the line \(x = 3\), spreading from the vertex while passing closer to the focus.

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Key Concepts

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

Vertex of a Parabola
The vertex is a special point on the parabola, often considered its turning point. For equations in the form \((x-h)^2 = 4p(y-k)\), the vertex is situated at \((h, k)\).
In our given equation \((x-3)^2 = \frac{1}{2}(y+1)\), we see that \(h = 3\) and \(k = -1\). Thus, the vertex is \((3, -1)\).
This vertex forms the midpoint in the direction which the parabola opens, acting as the reference point for symmetry.
  • The vertex changes based on \(h\) and \(k\), which are derived directly from the equation.
  • It often represents the minimum or maximum point, depending on the direction the parabola opens.
Understanding the vertex aids in visualizing how the parabola is oriented on the graph.
Focus of a Parabola
The focus of a parabola is a point located inside it, determining its shape. For the form \((x-h)^2 = 4p(y-k)\), the focus is found at \((h, k + p)\).
In our example, we have \(h = 3\), \(k = -1\), and \(p = \frac{1}{8}\), so the focus becomes \((3, -\frac{7}{8})\).
The focus is crucial:
  • It helps in understanding how the parabola converges, as all points on the parabola are equidistant from the focus and the directrix.
  • It affects how the parabola's width and steepness are perceived.
Being aware of the focus gives insight into the parabola's depth and reach on the graph.
Directrix of a Parabola
The directrix is a line, which alongside the focus, helps define a parabola.
In the equation \((x-h)^2 = 4p(y-k)\), the directrix is given by \(y = k - p\).
With \(k = -1\) and \(p = \frac{1}{8}\), we attain the directrix \(y = -1 - \frac{1}{8} = -\frac{9}{8}\).
  • The directrix is perpendicular to the axis of symmetry of the parabola.
  • It acts as a mirror line for shaping the parabola, guiding the curve's symmetry.
Both the focus and directrix are essential for understanding the parabola's balanced structure.
Graphing Parabolas
Graphing a parabola involves plotting key points such as the vertex, focus, and the directrix. Let's break it down:
1. Plot the vertex \((3, -1)\) on your graph as a starting point.
2. Locate the focus at \((3, -\frac{7}{8})\), slightly above the vertex since \(p > 0\).
3. Draw the directrix line parallel to the x-axis at \(y = -\frac{9}{8}\).
4. Use these elements to sketch the parabola, opening upwards as indicated by the positive \(p\) value.
5. Ensure the curve is symmetric about the line \(x = 3\).
  • Having the vertex and focus helps narrow down where the parabola starts and how it bends.
  • The directrix offers contrast, showing the distance from the focus.
  • Points along the parabola maintain equal lengths from the focus to the line of the directrix.
Graphing allows visualization of mathematical relationships, making them easier to comprehend.

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