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Which graph corresponds to the equation y^2/49 - x^2/45 = 1

Which graph corresponds to the equation y^2/49 - x^2/45 = 1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 41E
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Which graph corresponds to the equation y^2/49 - x^2/45 = 1

### Identifying Hyperbolas #### Question: Which graph corresponds to the equation \(\frac{y^2}{49} - \frac{x^2}{25} = 1\)? #### Explanation: This equation represents the equation of a hyperbola. Notice the vertices, asymptotes, and orientation. **Analysis of the Graphs:** 1. **First Graph**: - Orientation: Horizontal hyperbola. - Equation Format: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). - Characteristics: Opens left and right. 2. **Second Graph**: - Orientation: Vertical hyperbola. - Equation Format: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). - Characteristics: Opens up and down. ### Matching the Equation: The given equation \(\frac{y^2}{49} - \frac{x^2}{25} = 1\) matches the form of a vertical hyperbola. **Conclusion**: The correct graph corresponding to the given equation is the **second graph**. ### Detailed Graph Description: **Second Graph**: - The hyperbola opens upwards and downwards. - The vertices are positioned at \((0, \pm7)\). - The asymptotes intersect at the origin (0, 0). The slopes of the asymptotes are determined by the coefficients under \(x^2\) and \(y^2\) (i.e., \(\pm\frac{7}{5}\)). ### Learning Objective: Understand the identification and characteristics of hyperbolas and how their graphs correspond to their equations.
Transcribed Image Text:### Identifying Hyperbolas #### Question: Which graph corresponds to the equation \(\frac{y^2}{49} - \frac{x^2}{25} = 1\)? #### Explanation: This equation represents the equation of a hyperbola. Notice the vertices, asymptotes, and orientation. **Analysis of the Graphs:** 1. **First Graph**: - Orientation: Horizontal hyperbola. - Equation Format: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). - Characteristics: Opens left and right. 2. **Second Graph**: - Orientation: Vertical hyperbola. - Equation Format: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). - Characteristics: Opens up and down. ### Matching the Equation: The given equation \(\frac{y^2}{49} - \frac{x^2}{25} = 1\) matches the form of a vertical hyperbola. **Conclusion**: The correct graph corresponding to the given equation is the **second graph**. ### Detailed Graph Description: **Second Graph**: - The hyperbola opens upwards and downwards. - The vertices are positioned at \((0, \pm7)\). - The asymptotes intersect at the origin (0, 0). The slopes of the asymptotes are determined by the coefficients under \(x^2\) and \(y^2\) (i.e., \(\pm\frac{7}{5}\)). ### Learning Objective: Understand the identification and characteristics of hyperbolas and how their graphs correspond to their equations.
### Understanding Graphs of Specific Functions #### Introduction In this section, we will study and analyze the shapes and characteristics of several mathematical functions as represented on a Cartesian plane. This will aid in understanding their behaviors and applications in various fields. #### Graph 1: Parabolic Curve (Upward Opening) The first graph displays a parabolic curve opening upwards. This is typical of a quadratic function of the form: \[ y = ax^2 \] where \( a \) is a positive constant. The vertex of the parabola is at the origin (0,0), indicating this is a standard parabolic curve without horizontal or vertical shifts. **Characteristics:** - Symmetric about the y-axis. - Minimum point at the origin. - As \( x \) increases or decreases, \( y \) increases. #### Graph 2: Inverted Parabolic Curve The second graph depicts an inverted parabolic curve, which graphs a quadratic function of the form: \[ y = -ax^2 \] where \( a \) is a positive constant. The vertex, like the first parabola, is at the origin (0,0). **Characteristics:** - Symmetric about the y-axis. - Maximum point at the origin. - As \( x \) increases or decreases, \( y \) decreases. #### Graph 3: Hyperbolic Curve The third graph showcases a hyperbolic curve representing a hyperbola of the form: \[ xy = c \] or a standard hyperbola such as: \[ y = \frac{a}{x} \] where \( a \) and \( c \) are constants. The graph is characterized by two symmetrical curved branches. **Characteristics:** - Asymptotes along the coordinate axes (x=0 and y=0). - The branches head towards infinity as \( x \) increases or decreases. - Not symmetric about the y-axis or x-axis but has reflective symmetry about the origin. ### Conclusion By interpreting and graphing these functions, one can gain a hands-on understanding of their geometric properties and mathematical implications. It’s crucial to be able to visualize these graphs to predict function behavior in practical and theoretical applications.
Transcribed Image Text:### Understanding Graphs of Specific Functions #### Introduction In this section, we will study and analyze the shapes and characteristics of several mathematical functions as represented on a Cartesian plane. This will aid in understanding their behaviors and applications in various fields. #### Graph 1: Parabolic Curve (Upward Opening) The first graph displays a parabolic curve opening upwards. This is typical of a quadratic function of the form: \[ y = ax^2 \] where \( a \) is a positive constant. The vertex of the parabola is at the origin (0,0), indicating this is a standard parabolic curve without horizontal or vertical shifts. **Characteristics:** - Symmetric about the y-axis. - Minimum point at the origin. - As \( x \) increases or decreases, \( y \) increases. #### Graph 2: Inverted Parabolic Curve The second graph depicts an inverted parabolic curve, which graphs a quadratic function of the form: \[ y = -ax^2 \] where \( a \) is a positive constant. The vertex, like the first parabola, is at the origin (0,0). **Characteristics:** - Symmetric about the y-axis. - Maximum point at the origin. - As \( x \) increases or decreases, \( y \) decreases. #### Graph 3: Hyperbolic Curve The third graph showcases a hyperbolic curve representing a hyperbola of the form: \[ xy = c \] or a standard hyperbola such as: \[ y = \frac{a}{x} \] where \( a \) and \( c \) are constants. The graph is characterized by two symmetrical curved branches. **Characteristics:** - Asymptotes along the coordinate axes (x=0 and y=0). - The branches head towards infinity as \( x \) increases or decreases. - Not symmetric about the y-axis or x-axis but has reflective symmetry about the origin. ### Conclusion By interpreting and graphing these functions, one can gain a hands-on understanding of their geometric properties and mathematical implications. It’s crucial to be able to visualize these graphs to predict function behavior in practical and theoretical applications.
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