What is the Equation of the Vertical Asymptote of the Hyperbola ( y frac{2}{x-3} )?

Introduction to Asymptotes in Hyperbolas

Hyperbolas are a type of conic section characterized by their distinctive pair of curved branches. These branches extend towards and draw near to two lines that never touch them; these lines are known as the asymptotes. The asymptotes play a crucial role in defining the behavior of a hyperbola as it approaches infinity.

For the hyperbola given by the equation ( y frac{2}{x-3} ), we can determine its asymptotes and, more specifically, the equation of the vertical asymptote.

Understanding the Hyperbola Equation

The given hyperbola equation is ( y frac{2}{x-3} ) . This form of the hyperbola is centered at a point other than the origin, specifically at (3,0). To further understand the transformation, it can be compared to the standard form ( frac{x^2}{a^2} - frac{y^2}{b^2} 1 ) or ( frac{y^2}{b^2} - frac{x^2}{a^2} 1 ) .

Identifying the Asymptotes

The asymptotes of a hyperbola can be found by setting the numerator or the denominator of the fraction to zero, depending on the form of the equation. For the given equation ( y frac{2}{x-3} ) , the denominator ( x-3 ) dictates the location of the vertical and horizontal asymptotes.

Vertical Asymptote

The vertical asymptote occurs when the denominator is equal to zero, as it indicates a value that the variable ( x ) cannot take without causing the function to become undefined. Setting the denominator equal to zero:

( x - 3 0 )

Solving for ( x ) gives:

( x 3 )

Thus, the vertical asymptote of the given hyperbola is ( x 3 ).

Horizontal Asymptote

The horizontal asymptote can be determined by analyzing the behavior of the function as ( x ) approaches infinity (or negative infinity). For the given hyperbola, as ( x ) becomes very large (whether positively or negatively), the value of ( frac{2}{x-3} ) approaches zero. Therefore, the horizontal asymptote is:

( y 0 )

Graphing the Hyperbola

To visualize the hyperbola and its asymptotes, we can plot the equations ( y frac{2}{x-3} ) , ( y 0 ) , and ( x 3 ) . Here is a brief description of how to plot these:

Graph of ( y frac{2}{x-3} ) : This is a hyperbola shifted three units to the right from the y-axis. Graph of ( y 0 ) : This is the x-axis. Graph of ( x 3 ) : This is a vertical line passing through the point (3, 0).

These lines will help visualize how the hyperbola approaches but does not touch the vertical and horizontal asymptotes.

Conclusion

In conclusion, understanding the vertical and horizontal asymptotes of a hyperbola is essential for comprehending the behavior of the function and its graphical representation. For the hyperbola ( y frac{2}{x-3} ), the vertical asymptote is ( x 3 ). This line serves as a guide for the hyperbola's behavior, showing where it becomes undefined and how it approaches but does not cross this boundary.