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. ### Understanding Graphs of Specific Functions#### IntroductionIn 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 CurveThe 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 CurveThe 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.### ConclusionBy 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.

Answered: Image /qna-images/answer/6993bd43-71fb-4cec-8431-dceccc873561.jpg

Answered: Which graph corresponds to the equation…

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

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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