Textbook Question
Find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). (y - 1)2 = 4(x - 1)
a. b. c. d.
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8. Conic Sections
Parabolas
3:03 minutesProblem 37
Textbook Question
Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. (x + 1)2 = - 8(y + 1)
Verified step by step guidance1
Identify the form of the given equation. The equation is \( (x + 1)^2 = -8(y + 1) \), which matches the standard form of a vertical parabola: \( (x - h)^2 = 4p(y - k) \), where \( (h, k) \) is the vertex.
From the equation, determine the vertex \( (h, k) \). Here, \( h = -1 \) and \( k = -1 \), so the vertex is at \( (-1, -1) \).
Find the value of \( p \) by comparing \( 4p \) to the coefficient on the right side. Since \( 4p = -8 \), solve for \( p \) to get \( p = -2 \). The negative value indicates the parabola opens downward.
Use the vertex and \( p \) to find the focus. The focus lies \( p \) units away from the vertex along the axis of symmetry (vertical line \( x = h \)). So, the focus is at \( (h, k + p) = (-1, -1 - 2) \).
Find the directrix, which is a horizontal line \( p \) units in the opposite direction from the vertex. The directrix is \( y = k - p = -1 - (-2) = -1 + 2 \). This line is above the vertex.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of a Parabola
The standard form of a parabola's equation helps identify its orientation and key features. For vertical parabolas, the form is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p determines the distance to the focus and directrix. Recognizing this form allows you to extract important information directly from the equation.
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Parabolas as Conic Sections
Vertex, Focus, and Directrix
The vertex is the parabola's turning point, located at (h, k). The focus is a point inside the parabola at a distance p from the vertex, and the directrix is a line perpendicular to the axis of symmetry, also p units from the vertex but in the opposite direction. These elements define the parabola's shape and position.
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Graphing Parabolas
Graphing a parabola involves plotting the vertex, focus, and directrix, then sketching the curve that is equidistant from the focus and directrix. Understanding the orientation (upward, downward, left, or right) based on the equation's form is essential for accurate graphing.
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