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find the equations of the asymptotes and graph

find the equations of the asymptotes and graph

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 equations of the asymptotes and graph
The image depicts a Cartesian coordinate grid. The graph is a standard 10 by 10 unit grid with horizontal and vertical axes. Here are some details about the graph: - **Axes:** The horizontal axis (x-axis) and vertical axis (y-axis) intersect at the origin (0,0). - **Gridlines:** There are darker gridlines every 5 units, both horizontally and vertically, which help in easily locating points. - **Scale:** The scale on both axes ranges from -10 to 10. - **Numbering:** The axes are labeled with numbers indicating the scale at intervals of 1 unit, while every gridline marking 5 units is emphasized for clarity. This grid is typically used to plot points and lines in mathematics and can represent various mathematical functions or data sets.
Transcribed Image Text:The image depicts a Cartesian coordinate grid. The graph is a standard 10 by 10 unit grid with horizontal and vertical axes. Here are some details about the graph: - **Axes:** The horizontal axis (x-axis) and vertical axis (y-axis) intersect at the origin (0,0). - **Gridlines:** There are darker gridlines every 5 units, both horizontally and vertically, which help in easily locating points. - **Scale:** The scale on both axes ranges from -10 to 10. - **Numbering:** The axes are labeled with numbers indicating the scale at intervals of 1 unit, while every gridline marking 5 units is emphasized for clarity. This grid is typically used to plot points and lines in mathematics and can represent various mathematical functions or data sets.
### Hyperbola Equation and Graph Analysis **Equation:** 5. \(\frac{{(x+4)^2}}{9} - \frac{{(y+3)^2}}{16} = 1\) **Components to Identify:** - **Center:** - The center of the hyperbola can be identified from the equation \(\frac{{(x-h)^2}}{a^2} - \frac{{(y-k)^2}}{b^2} = 1\). - Here, \((h, k) = (-4, -3)\). - **Vertices:** - Since the transverse axis is horizontal, the vertices are located along the x-axis. - Vertices are at \((h \pm a, k)\) where \(a^2 = 9\) implies \(a = 3\). - Thus, vertices are \((-4 \pm 3, -3)\) = \((-1, -3)\) and \((-7, -3)\). - **Foci:** - The foci are found using the formula \(c^2 = a^2 + b^2\). - Here, \(b^2 = 16\), so \(c^2 = 9 + 16 = 25\) and \(c = 5\). - Foci are at \((h \pm c, k)\) = \((-4 \pm 5, -3)\) = \((1, -3)\) and \((-9, -3)\). - **Asymptotes:** - For a horizontal transverse axis hyperbola, asymptotes are of the form \(y - k = \pm \frac{b}{a} (x - h)\). - Here, \(\frac{b}{a} = \frac{4}{3}\). - Asymptotes equations are: \(y + 3 = \pm \frac{4}{3}(x + 4)\). **Graph Description:** - The graph is a centered grid with x-axis and y-axis marked. - The axes show increments from \(-10\) to \(10\). - The hyperbola is centered at \((-4, -3)\) and opens horizontally. - The graph highlights key aspects like vertices and asymptotes visually reflected on the grid
Transcribed Image Text:### Hyperbola Equation and Graph Analysis **Equation:** 5. \(\frac{{(x+4)^2}}{9} - \frac{{(y+3)^2}}{16} = 1\) **Components to Identify:** - **Center:** - The center of the hyperbola can be identified from the equation \(\frac{{(x-h)^2}}{a^2} - \frac{{(y-k)^2}}{b^2} = 1\). - Here, \((h, k) = (-4, -3)\). - **Vertices:** - Since the transverse axis is horizontal, the vertices are located along the x-axis. - Vertices are at \((h \pm a, k)\) where \(a^2 = 9\) implies \(a = 3\). - Thus, vertices are \((-4 \pm 3, -3)\) = \((-1, -3)\) and \((-7, -3)\). - **Foci:** - The foci are found using the formula \(c^2 = a^2 + b^2\). - Here, \(b^2 = 16\), so \(c^2 = 9 + 16 = 25\) and \(c = 5\). - Foci are at \((h \pm c, k)\) = \((-4 \pm 5, -3)\) = \((1, -3)\) and \((-9, -3)\). - **Asymptotes:** - For a horizontal transverse axis hyperbola, asymptotes are of the form \(y - k = \pm \frac{b}{a} (x - h)\). - Here, \(\frac{b}{a} = \frac{4}{3}\). - Asymptotes equations are: \(y + 3 = \pm \frac{4}{3}(x + 4)\). **Graph Description:** - The graph is a centered grid with x-axis and y-axis marked. - The axes show increments from \(-10\) to \(10\). - The hyperbola is centered at \((-4, -3)\) and opens horizontally. - The graph highlights key aspects like vertices and asymptotes visually reflected on the grid
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