Problem 34 - Intermediate Algebra

Question

Graph equation. \(x^{2}-9 y^{2}=9\)

Step-by-Step Solution

Answer

Horizontal hyperbola with vertices (3,0) and (-3,0), asymptotes are y=±(1/3)x.

Step 1: Recognizing the Equation Type

The given equation is \(x^2 - 9y^2 = 9\). Notice that it resembles the standard form \( ax^2 - by^2 = c \), which is an equation of a hyperbola.

Step 2: Rewriting in Standard Form

Start by dividing the entire equation by 9 to get it into standard form. \[ \frac{x^2}{9} - \frac{y^2}{1} = 1 \]. This shows the hyperbola centered at the origin \((0,0)\).

Step 3: Identifying the Orientation

The equation \(\frac{x^2}{9} - \frac{y^2}{1} = 1\) has a positive \(x^2\) term, indicating a horizontal hyperbola.

Step 4: Finding the Vertices

For the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the vertices are \((\pm a, 0)\). Here, \(a = 3\) and \(b = 1\), so the vertices are \((3, 0)\) and \((-3, 0)\).

Step 5: Plotting the Asymptotes

The formula for the asymptotes of a hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) is \(y = \pm \frac{b}{a}x\). With \(a=3\) and \(b=1\), the asymptotes are \(y = \pm \frac{1}{3}x\). Draw these lines through the center \((0, 0)\).

Key Concepts

Hyperbola Equation

Understanding the equation of a hyperbola is crucial when graphing, as it dictates the hyperbolic shape and its properties. A hyperbola appears when the difference in distances to two points (called foci) is constant.

In our scenario, we start with the equation \(x^2 - 9y^2 = 9\). This equation falls into the general form \(ax^2 - by^2 = c\), which signifies a hyperbola due to the difference of squares.

Recognizing this form is the first step in identifying and analyzing hyperbolas since it differentiates hyperbolas from other conic sections like parabolas or ellipses.

Standard Form of Hyperbola

To fully understand the properties of a hyperbola, placing the equation in the standard form is essential.

For hyperbolas, the standard form is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for a horizontal orientation, or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) for a vertical one.

Let's look at our equation: \(x^2 - 9y^2 = 9\). By dividing the whole equation by 9, we get:

• \(\frac{x^2}{9} - \frac{y^2}{1} = 1\).

This transformation makes it clear that the hyperbola is horizontally oriented, visible by the positive \(x^2\) term. The standard form helps us readily identify essential characteristics like the transverse and conjugate axes.

Asymptotes of Hyperbola

Asymptotes play a fundamental role in shaping a hyperbola. They provide boundary lines that the hyperbola approaches but never touches.

Once an equation is in standard form, finding the asymptotes becomes straightforward. For a hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), use the formula:

• \(y = \pm \frac{b}{a}x\)

In our example, where \(a = 3\) and \(b = 1\), the asymptotes are expressed as:

• \(y = \pm \frac{1}{3}x\)

These asymptotes intersect at the hyperbola's center, which is at the origin \((0,0)\) in this example.

Vertices of Hyperbola

Identifying the vertices of a hyperbola grants insight into its extent along the main axis.

For a hyperbola in standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the vertices are found at the points:

• \((\pm a, 0)\)

In our case, where \(a = 3\), the vertices are located at

• \((3, 0)\) and \((-3, 0)\)

These points lie along the transverse axis and are equidistant from the center. The vertices are crucial as they display the hyperbola's broadest width.