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9. Find the standard form of the equation of the parabola with a focus at (-3, 4) and directrix y = 2.

9. Find the standard form of the equation of the parabola with a focus at (-3, 4) and directrix y = 2.

Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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find the standard form of the equation of the parabola with a focus at -3,4 and directrix at y=2
The image is of a Cartesian coordinate graph. The graph is a two-dimensional grid marked with both horizontal (x-axis) and vertical (y-axis) lines. Each axis is labeled with numbers ranging from -10 to 10. - **X-Axis (Horizontal):** - The center is labeled at 0. - Positive numbers increase to the right, going up to 10. - Negative numbers decrease to the left, going down to -10. - **Y-Axis (Vertical):** - The center is labeled at 0. - Positive numbers increase upwards, going up to 10. - Negative numbers decrease downwards, going down to -10. The graph consists of a grid made up of smaller squares, which allows for precise plotting of points. Each square represents one unit on both axes. The bold lines at every 5 units help in identifying larger values easily.
Transcribed Image Text:The image is of a Cartesian coordinate graph. The graph is a two-dimensional grid marked with both horizontal (x-axis) and vertical (y-axis) lines. Each axis is labeled with numbers ranging from -10 to 10. - **X-Axis (Horizontal):** - The center is labeled at 0. - Positive numbers increase to the right, going up to 10. - Negative numbers decrease to the left, going down to -10. - **Y-Axis (Vertical):** - The center is labeled at 0. - Positive numbers increase upwards, going up to 10. - Negative numbers decrease downwards, going down to -10. The graph consists of a grid made up of smaller squares, which allows for precise plotting of points. Each square represents one unit on both axes. The bold lines at every 5 units help in identifying larger values easily.
**Problem:** Find the standard form of the equation of the parabola with a focus at (-3, 4) and directrix \( y = 2 \). **Explanation:** The problem includes an empty coordinate grid with axes labeled. To solve the problem, consider the following: 1. **Key points:** - Focus: \((-3, 4)\) - Directrix: \(y = 2\) 2. **Determine the Vertex:** - The vertex of a parabola is equidistant from the focus and the directrix. The distance between the focus and the directrix is \(4 - 2 = 2\). - Hence, the vertex is 1 unit away from both the focus and the directrix. It lies on the line \(y = 3\) since that’s the midpoint between the focus and the directrix for the y-coordinate. Consequently, the vertex is at \((-3, 3)\). 3. **Equation of the Parabola:** - Since the parabola opens vertically (upwards due to the focus being above the directrix), use the standard form for vertical parabolas: \[ (x - h)^2 = 4p(y - k) \] where \((h, k)\) is the vertex. - Here, \(h = -3\) and \(k = 3\). - The distance \(p\) from the vertex to the focus is 1 (since 4 to 3 is 1 unit). - Substitute to get the equation: \[ (x + 3)^2 = 4(y - 3) \] This equation can be used to understand the parabola's position and shape with respect to the given focus and directrix.
Transcribed Image Text:**Problem:** Find the standard form of the equation of the parabola with a focus at (-3, 4) and directrix \( y = 2 \). **Explanation:** The problem includes an empty coordinate grid with axes labeled. To solve the problem, consider the following: 1. **Key points:** - Focus: \((-3, 4)\) - Directrix: \(y = 2\) 2. **Determine the Vertex:** - The vertex of a parabola is equidistant from the focus and the directrix. The distance between the focus and the directrix is \(4 - 2 = 2\). - Hence, the vertex is 1 unit away from both the focus and the directrix. It lies on the line \(y = 3\) since that’s the midpoint between the focus and the directrix for the y-coordinate. Consequently, the vertex is at \((-3, 3)\). 3. **Equation of the Parabola:** - Since the parabola opens vertically (upwards due to the focus being above the directrix), use the standard form for vertical parabolas: \[ (x - h)^2 = 4p(y - k) \] where \((h, k)\) is the vertex. - Here, \(h = -3\) and \(k = 3\). - The distance \(p\) from the vertex to the focus is 1 (since 4 to 3 is 1 unit). - Substitute to get the equation: \[ (x + 3)^2 = 4(y - 3) \] This equation can be used to understand the parabola's position and shape with respect to the given focus and directrix.
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