Problem 57 - College Algebra

Question

For the following exercises, express the equation for the hyperbola as two functions, with \(y\) as a function of \(x .\) Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes. $$ \frac{y^{2}}{9}-\frac{x^{2}}{1}=1 $$

Step-by-Step Solution

Answer

The two functions are \( y = \sqrt{9 + x^2} \) and \( y = -\sqrt{9 + x^2} \).

Step 1: Identify the Hyperbola Equation

The given equation is \( \frac{y^2}{9} - \frac{x^2}{1} = 1 \). This is the standard form of a hyperbola centered at the origin with the transverse axis along the y-axis.

Step 2: Rewrite the Equation for Two Functions

To express \( y \) as a function of \( x \), first isolate \( y \) by solving the equation for \( y^2 \): \( y^2 = 9 + x^2 \). Then, take the square root of both sides to get two separate functions: \( y = \sqrt{9 + x^2} \) and \( y = -\sqrt{9 + x^2} \).

Step 3: Express the Functions in Their Simplest Form

The two functions derived from the hyperbola's equation are \( y_1(x) = \sqrt{9 + x^2} \) and \( y_2(x) = -\sqrt{9 + x^2} \). These represent the upper and lower halves of the hyperbola, respectively.

Step 4: Graph the Functions on a Calculator

Input \( y_1(x) = \sqrt{9 + x^2} \) and \( y_2(x) = -\sqrt{9 + x^2} \) into your graphing calculator. Plot both functions on the same set of axes to visualize the hyperbola as two separate curves.

Key Concepts

Graphing Calculator

Using a graphing calculator for plotting equations is a powerful tool to understand mathematical concepts visually. When dealing with hyperbolas, as in the equation \( \frac{y^2}{9} - \frac{x^2}{1} = 1 \), the calculator helps visualize the separate functions which form the hyperbola.

To begin:

• Input the functions into the calculator as: \( y_1(x) = \sqrt{9 + x^2} \) and \( y_2(x) = -\sqrt{9 + x^2} \).
• This will plot both the upper and lower sections of the hyperbola on the same graph.

Ensure you have set an appropriate range for both the x and y axes. This ensures you can see the entirety of the hyperbola. Adjusting the view window often leads to a clearer and more accurate graph.

Graphing calculators also offer the functionality to analyze the graph by inspecting points of interest, such as intercepts and symmetries. This supports deeper learning by providing a dynamic way to explore the properties of the hyperbola.

Functions with y as a Function of x

In order to represent a hyperbola as functions where \( y \) is expressed in terms of \( x \), one must understand the transformation of the given equation.

The original hyperbola equation is \( \frac{y^2}{9} - \frac{x^2}{1} = 1 \). To isolate \( y \), solve for \( y^2 \):

• This gives \( y^2 = 9 + x^2 \), an intermediary step illustrating the relationship between \( x \) and \( y \).

From here, obtain the square root to find the functions for \( y \):

• \( y = \sqrt{9 + x^2} \)
• \( y = -\sqrt{9 + x^2} \)

These two functions reflect the two halves of the hyperbola. \( y = \sqrt{9 + x^2} \) represents the upper curve, while \( y = -\sqrt{9 + x^2} \) represents the lower curve.

This distinction is how hyperbolas are typically graphed and analyzed.

Standard Form of a Hyperbola

Understanding the standard form of a hyperbola is crucial for recognizing how hyperbolas are structured in equations. In mathematical terms, a hyperbola can commonly be written as \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), where \( a \) and \( b \) represent distances from the center to vertices and co-vertices, respectively.

In the case of the exercise equation \( \frac{y^2}{9} - \frac{x^2}{1} = 1 \):

• The hyperbola is centered at the origin \( (0,0) \) because it is not shifted.
• The transverse axis is aligned along the y-axis as reflected by \( y^2 \) being positive and before \( x^2 \).
• The values \( a^2 = 9 \) and \( b^2 = 1 \) tell us that \( a = 3 \) and \( b = 1 \), which determine the shape and stretch of the hyperbola.

Comprehending the format allows you to quickly identify the arrangement of axes and size of the hyperbola simply by looking at the equation. This conceptual understanding is valuable for predicting the behavior and appearance of hyperbolic graphs.