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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Question
Write the standard equation for the hyperbola graphed above.
![### Understanding Hyperbolas
In this graph, we see a hyperbola graphed on a coordinate plane. The hyperbola is represented by the two blue curves opening left and right, situated on either side of the vertical and horizontal asymptotes (represented by red dashed lines).
### Key Features of the Graph:
- **Axes:**
- The horizontal axis is labeled "x."
- The vertical axis is labeled "y."
- **Grid Lines:**
- The grid lines are spaced in intervals of 3 units along both axes.
- **Asymptotes:**
- The hyperbola's asymptotes are shown as red dashed lines. These lines are diagonal and pass through the origin, forming an "X" shape. The equations for these asymptotes typically take the form \( y = kx \) and \( y = -kx \) for some constant \( k \).
- **Vertices:**
- The vertices of the hyperbola appear to be at points where the hyperbola intersects the horizontal line \( y=y_0 \), which in this case seem to be \( x = \pm 5 \).
### Equation of a Hyperbola:
For a hyperbola centered at the origin with horizontal orientation, the standard equation is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Here:
- \( a \) and \( b \) are distances related to the shape and size of the hyperbola.
- \( a \) represents the distance from the center to the vertices on the x-axis.
Given the positioning of the vertices at \( x = \pm 5 \):
\[ a = 5 \]
Since the graph does not specify the value of \( b \) directly, we must assume it to match the hyperbola's curvature as typically labeled in an analytical problem.
### Prompt:
**Write the standard equation for the hyperbola graphed above:**
\[ \frac{x^2}{5^2} - \frac{y^2}{b^2} = 1 \]
Note:
To fully determine the value of \( b \), additional information such as coordinates of the foci or specific properties of the hyperbola would be needed. However, based on standard hyperbola properties:
\[ \frac{x^2}{25} - \frac{y^2}{b^2} = 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a7c8b7c-6421-40e7-b33a-5c89a0da3579%2F6591f88f-a0fc-4e3b-a8c3-5719c585d94a%2F2rzyud.png&w=3840&q=75)
Transcribed Image Text:### Understanding Hyperbolas
In this graph, we see a hyperbola graphed on a coordinate plane. The hyperbola is represented by the two blue curves opening left and right, situated on either side of the vertical and horizontal asymptotes (represented by red dashed lines).
### Key Features of the Graph:
- **Axes:**
- The horizontal axis is labeled "x."
- The vertical axis is labeled "y."
- **Grid Lines:**
- The grid lines are spaced in intervals of 3 units along both axes.
- **Asymptotes:**
- The hyperbola's asymptotes are shown as red dashed lines. These lines are diagonal and pass through the origin, forming an "X" shape. The equations for these asymptotes typically take the form \( y = kx \) and \( y = -kx \) for some constant \( k \).
- **Vertices:**
- The vertices of the hyperbola appear to be at points where the hyperbola intersects the horizontal line \( y=y_0 \), which in this case seem to be \( x = \pm 5 \).
### Equation of a Hyperbola:
For a hyperbola centered at the origin with horizontal orientation, the standard equation is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Here:
- \( a \) and \( b \) are distances related to the shape and size of the hyperbola.
- \( a \) represents the distance from the center to the vertices on the x-axis.
Given the positioning of the vertices at \( x = \pm 5 \):
\[ a = 5 \]
Since the graph does not specify the value of \( b \) directly, we must assume it to match the hyperbola's curvature as typically labeled in an analytical problem.
### Prompt:
**Write the standard equation for the hyperbola graphed above:**
\[ \frac{x^2}{5^2} - \frac{y^2}{b^2} = 1 \]
Note:
To fully determine the value of \( b \), additional information such as coordinates of the foci or specific properties of the hyperbola would be needed. However, based on standard hyperbola properties:
\[ \frac{x^2}{25} - \frac{y^2}{b^2} = 1
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