Problem 30 - Algebra and Trigonometry

Question

Graph the equation \(y^{2}-9 x^{2}=1\) by solving for \(y\) and graphing the two equations corresponding to the positive and negative square roots. (This graph is called a hyperbola)

Step-by-Step Solution

Answer

Graph the two equations: \( y = \sqrt{9x^2 + 1} \) and \( y = -\sqrt{9x^2 + 1} \).

Step 1: Recognize the Type of Equation

The given equation is \( y^2 - 9x^2 = 1 \). This is a standard form of a hyperbola. Hyperbolas can be recognized by their general form \( Ax^2 + By^2 = C \) where \( A \) and \( B \) have opposite signs.

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

Start by isolating \( y^2 \) in the equation to simplify it for the next steps.\[ y^2 = 9x^2 + 1 \] This rearranges the original equation to express \( y^2 \) in terms of \( x^2 \).

Step 3: Solve for \( y \)

Take the square root of both sides to solve for \( y \). Recall that taking the square root introduces both positive and negative solutions.\[ y = \pm \sqrt{9x^2 + 1} \] Now you have two separate equations, \( y = \sqrt{9x^2 + 1} \) and \( y = -\sqrt{9x^2 + 1} \).

Step 4: Graph Each Equation

Graph the two equations found in the previous step separately on a coordinate system.1. Graph \( y = \sqrt{9x^2 + 1} \), which represents the upper branch of the hyperbola.2. Graph \( y = -\sqrt{9x^2 + 1} \), which gives the lower branch of the hyperbola.Both branches open vertically upwards and downwards, showing the distinctive shape of a hyperbola.

Key Concepts

Equations of Hyperbolas

When you encounter an equation like \( y^2 - 9x^2 = 1 \), you are dealing with a hyperbola. Hyperbolas are one of the conics, which also include circles, ellipses, and parabolas. These equations are important because they describe curves that have unique properties and visual shapes.

A hyperbola can generally be identified with the equation format \( Ax^2 + By^2 = C \), where the coefficients \( A \) and \( B \) have opposite signs. This indicates that one of the variables is subtracted from the other, distinguishing hyperbolas from other conics.

• Positive and Negative Parts: For a hyperbola, the equation will often split into two parts corresponding to positive and negative square roots, leading to two distinct curves or branches on a graph.
• Example Analysis: In our example \( y^2 - 9x^2 = 1 \), \( A = -9 \) and \( B = 1 \), indicating the opposite signs necessary for a hyperbola.

Graphing Hyperbolas

Graphing a hyperbola involves plotting each branch of the equation separately, which occurs due to the presence of square roots. In the example \( y = \pm \sqrt{9x^2 + 1} \), there are two parts to consider:

• Upper Branch: This is represented by the equation \( y = \sqrt{9x^2 + 1} \), where \( y \) will be positive or zero, creating the upper curve of the hyperbola.
• Lower Branch: This is represented by \( y = -\sqrt{9x^2 + 1} \), where \( y \) will be negative or zero, forming the lower curve.

To graph this hyperbola, you would typically:

- Plot points on a coordinate plane for each equation
- Connect these points smoothly to form the distinct curves

The hyperbola's shape is unique because it opens in two opposite directions, reflecting about the x-axis. Each branch curves away from the center point, which is referred to as the center of the hyperbola. This reflects the characteristic bow-tie shape of a hyperbola.

Standard Form of Hyperbolas

Understanding the standard form of a hyperbola is crucial as it simplifies the process of graphing and identifying key features. The standard form for hyperbolas centered at the origin is:

• Vertical Opening: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)
• Horizontal Opening: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)

In this form:

• a and b: These values determine the distance from the center to the vertices and co-vertices of the hyperbola. They are derived from the rearranged equation.
• Axes of Symmetry: For a hyperbola, these lines of reflection can provide guidance for where branches will be graphed.

Applying these principles to our example means understanding how \( 9x^2 \) and \( y^2 \) correspond to the values of \( a^2 \) and \( b^2 \). Identifying this can help in accurately graphing and predicting the hyperbola's shape and vertices. The standard form helps you to easily see properties like orientation and size.