Writing Equations of Hyperbolas Practice

1. Which of the following graphs can represent the hyperbola

   $\dfrac{x^2}{25}-\dfrac{y^2}{36}=1$?

   Choose 1 answer:
   

   1. 
   2. 
   3. 

   4. 

2. Which of the following graphs can represent the hyperbola

   $\dfrac{y^2}{25}-\dfrac{x^2}{1}=1$?

   Choose 1 answer:
   

   1. 
   2. 
   3. 

   4. 

3. Choose the equation that can represent the hyperbola graphed below.

   Choose 1 answer:
   

   1. $\dfrac{x^2}{25}-\dfrac{y^2}{9}=1$
   2. $\dfrac{x^2}{9}-\dfrac{y^2}{25}=1$
   3. $\dfrac{y^2}{25}-\dfrac{x^2}{9}=1$
   4. $\dfrac{y^2}{9}-\dfrac{x^2}{25}=1$

   
   

4. Which of the following graphs can represent the hyperbola

   $\dfrac{x^2}{9}-\dfrac{y^2}{25}=1$?

   Choose 1 answer:
   

   1. 
   2. 
   3. 
   4. 
      

Source: Khan Academy, https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:conics/x9e81a4f98389efdf:hyperb-foci/e/equation_of_a_hyperbola
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

1. The strategy

   In order to find the correct graph, we need to know the following.

   • Whether the hyperbola opens in the $x$-direction or the $y$-direction.
   • Where the vertices of the hyperbola are located.

   Determining the direction in which the hyperbola opens
   

   The positive term in the equation of a hyperbola determines whether the hyperbola opens in the $x$-direction or in the ‍$y$-direction. Our hyperbola has a positive ‍$x^2$ term, so it opens in the ‍$x$-direction (left and right).
   

   Since our hyperbola opens in the $x$-direction, we can narrow down our choices to graphs ‍B and C. Let's see if we can eliminate any more options based on the vertices of our hyperbola.

   Finding the vertices of the hyperbola

   If we denote the distance from each vertex to the center by $a$, then the coefficient of the ‍$x^2$ term is $\dfrac{1}{a^2}$. We are given that ‍ $a^2=25$, so ‍$a=5$, which means the vertices are located at ‍$(\pm 5,0)$. This corresponds to graph C.

   Summary

   Graph C can represent our hyperbola.

   ---

2. The strategy

   In order to find the correct graph, we need to know the following.

   • Whether the hyperbola opens in the $x$-direction or the $y$-direction.
   • Where the vertices of the hyperbola are located.

   Determining the direction in which the hyperbola opens
   

   The positive term in the equation of a hyperbola determines whether the hyperbola opens in the $x$-direction or in the ‍$y$-direction. Our hyperbola has a positive ‍$y^2$ term, so it opens in the ‍$y$-direction (up and down).
   

   Since our hyperbola opens in the $y$-direction, we can narrow down our choices to graphs ‍C and D. Let's see if we can eliminate any more options based on the vertices of our hyperbola.

   Finding the vertices of the hyperbola

   If we denote the distance from each vertex to the center by $a$, then the coefficient of the ‍$y^2$ term is $\dfrac{1}{a^2}$. We are given that ‍ $a^2=25$, so ‍$a=5$, which means the vertices are located at ‍$(0,\pm 5)$. This corresponds to graph C.

   Summary

   Graph D can represent our hyperbola.

   ---

3. The strategy

   In order to find the correct graph, we need to know the following.

   • Whether of the $x^2$ or $y^2$ terms is positive.
   • The distance from the center to a vertex.

   Determining the positive term
   

   Since the graphed hyperbola opens in the $x$-direction, it has $x$-intercepts. This means that the equation of the hyperbola has a positive ‍$x^2$ term.
   

   The general equation of such a hyperbola is $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$.

   Determining the distance from the center to a vertex

   According to the graph, the distance between the center and a vertex is ‍$5$ units. If we denote this distance by ‍$a$, then the coefficient of the $x^2$ term is $\dfrac{1}{a^2}$.

   So the equation of our hyperbola is of the form $\dfrac{x^2}{25}-\dfrac{y^2}{b^2}=1$.
   

   Summary

   The graphed hyperbola can be represented by the equation

   $\dfrac{x^2}{25}-\dfrac{y^2}{9}=1$.
   

   ---

4. The strategy

   In order to find the correct graph, we need to know the following.

   • Whether the hyperbola opens in the $x$-direction or the $y$-direction.
   • Where the vertices of the hyperbola are located.

   Determining the direction in which the hyperbola opens
   

   The positive term in the equation of a hyperbola determines whether the hyperbola opens in the $x$-direction or in the ‍$y$-direction. Our hyperbola has a positive ‍$x^2$ term, so it opens in the ‍$x$-direction (left and right).
   

   Since our hyperbola opens in the $x$-direction, we can narrow down our choices to graphs ‍B and D. Let's see if we can eliminate any more options based on the vertices of our hyperbola.

   Finding the vertices of the hyperbola

   If we denote the distance from each vertex to the center by $a$, then the coefficient of the ‍$x^2$ term is $\dfrac{1}{a^2}$. We are given that ‍ $a^2=9$, so ‍$a=3$, which means the vertices are located at ‍$(0,\pm 3)$. This corresponds to graph D.

   Summary

   Graph D can represent our hyperbola.