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33. Find the directrix. (y- 9)2 - 8 O y = -9 O y = -7 O x = -7 | x = -9

33. Find the directrix. (y- 9)2 - 8 O y = -9 O y = -7 O x = -7 | x = -9

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Problem 33: Find the Directrix** Given the equation of a parabola: \[ x = \frac{1}{4}(y - 9)^2 - 8 \] Determine the directrix from the following options: - \( \circ \; y = -9 \) - \( \circ \; y = -7 \) - \( \circ \; x = -7 \) - \( \circ \; x = -9 \) **Explanation:** The equation provided is in vertex form for a parabola that opens horizontally. The general form for such a parabola is: \[ x = a(y - k)^2 + h \] Where \((h, k)\) is the vertex of the parabola. In this case, the vertex is \((-8, 9)\). This specific parabola has \(a = \frac{1}{4}\), which informs us about the direction and width of the parabola's opening. For a horizontally opening parabola, the directrix is a vertical line given by the equation: \[ x = h - \frac{1}{4a} \] Substituting the values: \[ x = -8 - \frac{1}{4 \times \frac{1}{4}} \] \[ x = -8 - 1 \] \[ x = -9 \] Therefore, the directrix is \(x = -9\).
Transcribed Image Text:**Problem 33: Find the Directrix** Given the equation of a parabola: \[ x = \frac{1}{4}(y - 9)^2 - 8 \] Determine the directrix from the following options: - \( \circ \; y = -9 \) - \( \circ \; y = -7 \) - \( \circ \; x = -7 \) - \( \circ \; x = -9 \) **Explanation:** The equation provided is in vertex form for a parabola that opens horizontally. The general form for such a parabola is: \[ x = a(y - k)^2 + h \] Where \((h, k)\) is the vertex of the parabola. In this case, the vertex is \((-8, 9)\). This specific parabola has \(a = \frac{1}{4}\), which informs us about the direction and width of the parabola's opening. For a horizontally opening parabola, the directrix is a vertical line given by the equation: \[ x = h - \frac{1}{4a} \] Substituting the values: \[ x = -8 - \frac{1}{4 \times \frac{1}{4}} \] \[ x = -8 - 1 \] \[ x = -9 \] Therefore, the directrix is \(x = -9\).
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