Problem 57 - College Algebra

Question

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 functions are \(y = \pm \sqrt{9 + 9x^2}\).

Step 1: Understand the Hyperbola Equation

The given equation is \(\frac{y^2}{9} - \frac{x^2}{1} = 1\), which represents a vertical hyperbola centered at the origin, with the equation \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). Here, \(a^2 = 9\) and \(b^2 = 1\).

Step 2: Solve for \(y^2\)

Multiply both sides by 9 to get rid of the fraction: \(y^2 - 9x^2 = 9\).

Step 3: Isolate \(y^2\)

Add \(9x^2\) to both sides: \(y^2 = 9 + 9x^2\).

Step 4: Solve for \(y\) - Positive Solution

Take the square root of both sides for the positive solution: \(y = \sqrt{9 + 9x^2}\).

Step 5: Solve for \(y\) - Negative Solution

Take the square root of both sides for the negative solution: \(y = -\sqrt{9 + 9x^2}\).

Step 6: Write the Functions

Combine both solutions to express \(y\) as two functions of \(x\): \(y = \pm \sqrt{9 + 9x^2}\).

Step 7: Graph the Functions

Using a graphing calculator, input the two functions \(y = \sqrt{9 + 9x^2}\) and \(y = -\sqrt{9 + 9x^2}\) to sketch the hyperbola on the same axes. The graph should display two curves opening upwards and downwards centered at the origin.

Key Concepts

Vertical Hyperbola

A vertical hyperbola is a type of hyperbola where the major axis, or the line connecting the vertices, is oriented vertically. This differs from a horizontal hyperbola, where the major axis is horizontal.
In the standard equation for a vertical hyperbola, \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] the terms involve \(y^2\) and \(x^2\), with the \(y\) part having a positive coefficient. The square root of \(a^2\) gives the distance from the center to the vertices along the y-axis. The center of this hyperbola is usually at the origin (0,0), unless otherwise specified.
Key characteristics of vertical hyperbolas include:

• The transverse axis is vertical.
• The asymptotes are oblique lines intersecting at the center.
• Vertices and co-vertices define the shape and span of the hyperbola.

The graph showcasing these traits is structured such that the curves open upwards and downwards.

Equation of Hyperbola

Understanding the equation of a hyperbola helps in identifying its type and structure. For vertical hyperbolas, the equation is \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] where:

• \(a^2\) is the denominator of \(y^2\) and defines the distance related to the vertical (transverse) axis.
• \(b^2\) is the denominator of \(x^2\) and defines the distance related to the horizontal (conjugate) axis.

This specific equation results in a hyperbola centered at the origin with one pair of opposite branches opening up and down. Adjusting \(a\) and \(b\) modifies the scale but not the fundamental shape.
This representation helps when converting the original equation to express \(y\) as a function of \(x\), paving the way for easy graphing.

Graphing Hyperbolas

Graphing hyperbolas involves plotting the curves defined by the hyperbola's equation. It is beneficial to first express the equation in terms of functions of \(x\), as done by solving for \(y\) in the given hyperbola equation.

• Identify the center, vertices, and asymptotes based on \(a\) and \(b\).
• Use the two possible equations for \(y\) like \( y = \pm \sqrt{9 + 9x^2} \).

These functions display the characteristic hyperbolic shape on a graph, opening in this case vertically on a graph. Both the positive and negative solutions are needed to fully represent the hyperbola because it naturally occurs in two parts.
Graphical tools, such as graphing calculators, simplify the plotting process and help visualize the curves accurately.

Functions of x

Expressing \(y\) as a function of \(x\) is crucial for visualizing the hyperbola. From the given hyperbola equation:\[ \frac{y^2}{9} - \frac{x^2}{1} = 1 \] Manipulating this equation yields two functions:

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

These derived functions of \(x\) simplify graphing, as they allow plotting points on a Cartesian coordinate system where each \(x\) corresponds to two possible \(y\) values. This characteristic, having both a positive and negative variant, ensures both parts of the hyperbola are captured.
These functions assist not only in graph construction but also in understanding how variations in \(x\) influence \(y\), reflecting the nature of a hyperbolic curve in terms of the relationships between its variables.