91Ó°ÊÓ

Asymptotes play a crucial role in graphing hyperbolas. They are straight lines that the hyperbola approaches but never touches. For a hyperbola in the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equations of the asymptotes are given by \ y = \frac{b}{a} x\ and \ y = -\frac{b}{a} x \. These lines guide the hyperbola's shape and ensure that its branches extend toward infinity in the correct direction. To find the equations of the asymptotes:

• Identify values of \(a\) and \(b\).
• Compute the slope \( \frac{b}{a} \).

For our exercise with the hyperbola equation \[ \frac{x^2}{25} - \frac{y^2}{9} = 1 \], we determined:\( a = 5 \) and \( b = 3 \), leading to asymptotes \( y = \frac{3}{5} x \) and \( y = -\frac{3}{5} x \). This will help you draw the hyperbola accurately.

It's important to write the hyperbola equation in its standard form to understand its properties easily. The standard form for a hyperbola that opens left and right is \ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \. For our given equation 9x² - 25y² = 225:

• Isolate the terms with variables: 9x² - 25y² = 225.
• Divide by 225: \frac{x^2}{25} - \frac{y^2}{9} = 1.

After these steps, the equation is in standard form: \ \frac{x^2}{25} - \frac{y^2}{9} = 1 \.
This form tells us a lot about the hyperbola, such as its orientation, values of \(a\) and \(b\), and helps with finding its asymptotes.

Identifying the center and vertices of a hyperbola is the next crucial step in graphing it. For a standard form equation like \ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \:

• The center is at the origin (0,0).
• Vertices are \(a\) units from the center along the x-axis for horizontal hyperbolas.

In our exercise:

• The center is easily spotted at (0,0) due to the form of the equation.
• \( a^2 = 25 => a = 5 \).

Thus, the vertices are at (5,0) and (-5,0).
With these points and the asymptotes, plotting the hyperbola becomes straightforward. Always start from the center and use the vertices to draw the basic outline, guided by the asymptotes.