Problem 22 - 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 positive and negative roots: \(y = \pm \sqrt{9x^2 + 1}\), which forms a hyperbola.

Step 1: Identify the Type of Equation

The equation given is \(y^2 - 9x^2 = 1\). This equation is a standard form of a hyperbola because it involves terms with both \(x\) and \(y\) squared with opposite signs.

Step 2: Rewrite the Equation

We need to solve for \(y\). Start by rearranging the equation: \(y^2 = 9x^2 + 1\).

Step 3: Solve for y

Take the square root of both sides to solve for \(y\):\[y = \pm \sqrt{9x^2 + 1}\].This gives us two separate functions to graph: \(y = \sqrt{9x^2 + 1}\) and \(y = -\sqrt{9x^2 + 1}\).

Step 4: Graph the Positive Root

Graph the function \(y = \sqrt{9x^2 + 1}\). Note that this represents the top half of the hyperbola. The graph will open upwards and continue indefinitely.

Step 5: Graph the Negative Root

Graph the function \(y = -\sqrt{9x^2 + 1}\). This is the reflection of the positive root across the \(x\)-axis, representing the bottom half of the hyperbola.

Key Concepts

Graphing Techniques

When creating a graph of a hyperbola, it's important to understand the relationship between its algebraic form and its visual representation. In the case of the equation \(y^2 - 9x^2 = 1\), we start by recognizing it as a hyperbola due to the difference in squared terms.

To successfully plot this hyperbola, we first solve the equation for \(y\), arriving at two equations: \(y = \sqrt{9x^2 + 1}\) and \(y = -\sqrt{9x^2 + 1}\). These two equations correspond to the two halves of the hyperbola.

• The positive root, \(y = \sqrt{9x^2 + 1}\), graphs the upper half of the hyperbola and opens upward.
• The negative root, \(y = -\sqrt{9x^2 + 1}\), graphs the lower half, a mirror image of the positive half across the \(x\)-axis.

Consider using graph paper or a graphing tool to accurately draw these curves. Plot points from both equations to guide your curves and trace the characteristic shape of the hyperbola.

Standard Form of Hyperbolas

The standard form of a hyperbola is crucial for identifying its properties and graphing it correctly. A typical representation of a hyperbola in standard form is \(Ax^2 - By^2 = C\) or \(Bx^2 - Ay^2 = C\), where \(A\), \(B\), and \(C\) are constants.

In our equation \(y^2 - 9x^2 = 1\), notice that the term involving \(y^2\) is positive while \(x^2\) is negative, fitting the standard hyperbola form. This indicates the orientation and opening of the hyperbola.

Key features that can be determined from the standard form include:

• Center: For this particular form, the center is at the origin \((0,0)\).
• Axes Orientation: Since \(y^2\) is positive, the transverse axis is vertical.
• Vertices: These are crucial points where the hyperbola intersects its principal axis.

Understanding these components allows you to sketch the hyperbola more accurately and helps in predicting the graph's behavior.

Solving Algebraic Equations

Solving algebraic equations is a fundamental step in representing hyperbolas graphically. The initial task is to isolate \(y\) to express it in terms of \(x\).

Starting with the equation \(y^2 - 9x^2 = 1\), we rearrange it to \(y^2 = 9x^2 + 1\). Taking the square root of both sides gives two solutions: \(y = \sqrt{9x^2 + 1}\) and \(y = -\sqrt{9x^2 + 1}\).

These solutions represent two functions that form the two branches of a hyperbola. When solving:

• Ensure each step is accurate, particularly when dealing with square roots.
• Remember to consider both the positive and negative solutions to cover both halves of the hyperbola.

Approaching these algebraic equations with care is necessary to maintain mathematical accuracy and achieve a precise graphical representation.

Coordinate Plane Representation

Representing hyperbolas on the coordinate plane involves plotting accurate points and understanding the hyperbola's symmetry. Given the equations \(y = \sqrt{9x^2 + 1}\) and \(y = -\sqrt{9x^2 + 1}\), plotting involves choosing a range of \(x\) values and calculating corresponding \(y\) points from both functions.

Steps for effective representation include:

• Identify the center at point \((0,0)\).
• Determine key points, such as vertices, to establish the overall shape.
• Use symmetry about the axes to guide plotting. The positive and negative solutions are symmetrical across the \(x\)-axis.

A well-labeled graph will show how the hyperbola extends infinitely and has two branches opening upward and downward. By integrating these steps, you can graphically represent hyperbolas accurately on the coordinate plane, enhancing comprehension and visualization.