Problem 34 - Algebra 2

Question

Solve each equation for \(y .\) Graph each relation on your graphing calculator. Use the TRACE feature to locate the vertices. $$ x^{2}-y^{2}=1 $$

Step-by-Step Solution

Answer

The solutions are \(y = sqrt{x^{2} - 1}\) and \(y = -sqrt{x^{2} - 1}\). The vertices can be found by graphing the solutions and using the TRACE feature on the graphing calculator.

Step 1: Isolate y

To isolate \(y\) from the equation \(x^{2} - y^{2} = 1\), first subtract \(x^{2}\) from both sides. This gives \(-y^{2} = 1 - x^{2}\). Multiplying by -1 to reverse the sign will result in \(y^{2} = x^{2} - 1\). To solve for \(y\), take the square root of both sides, but remember that it can be either positive or negative. This will give two solutions for \(y\): \(y = sqrt{x^{2} - 1}\) and \(y = -sqrt{x^{2} - 1}\).

Step 2: Graphing the equation using a graphing calculator

We now have two expressions for \(y\), \(y = sqrt{x^{2} - 1}\) and \(y = -sqrt{x^{2} - 1}\). Input these independently as two different functions in the graphing calculator. Ensure to use the TRACE feature of the calculator when the graph is displayed.

Step 3: Locating the vertices

After graphing, use the TRACE function to identify where the curve reaches its highest and lowest points on the y-axis, these are the vertices. Since the original equation is a hyperbola, the graph will have two distinct curves (called branches), one for each solution for \(y\). The vertices are the points at the tips of each branch.

Key Concepts

Conic Sections

Conic sections are fundamental in understanding geometry and algebra. They are the curves obtained by intersecting a plane with a cone in different ways. Conic sections include shapes like circles, ellipses, parabolas, and hyperbolas. Each shape has unique properties and applications.

In the context of the equation given, we are dealing with a hyperbola. Hyperbolas are created when the plane cuts through both nappes of the cone. The standard form of a hyperbola is \(x^2 - y^2 = 1\), as seen in our exercise. Remember:

• Hyperbolas have two separate branches.
• They can look like two mirrored curves facing opposite directions.

The focus of this exercise is to understand the structure and graphing of hyperbolas through equations like the one provided.

Graphing Calculator

A graphing calculator is a versatile tool for visualizing mathematical equations, including conic sections like hyperbolas. In this exercise, the graphing calculator helps us plot the two branches of the hyperbola equation \(x^2 - y^2 = 1\).

Here’s how to graph a hyperbola using a graphing calculator:

• Enter the equations for both branches of the hyperbola, \(y = \sqrt{x^2 - 1}\) and \(y = -\sqrt{x^2 - 1}\), into the graphing calculator.
• View the graph to see the two mirrored curves.
• Use the TRACE feature on your calculator to explore different points on the curves, especially to find the vertices.

A graphing calculator simplifies observing the geometric properties of conic sections, making it an invaluable tool for students learning about hyperbolas.

Vertices

Vertices are key features of conic sections like hyperbolas. In the equation provided, the vertices are the highest and lowest points on each branch of the hyperbola formed by \(x^2 - y^2 = 1\). Identifying these points helps in understanding the scale and orientation of the graph.

To locate these using a graphing calculator, follow these steps:

• Graph both \(y = \sqrt{x^2 - 1}\) and \(y = -\sqrt{x^2 - 1}\).
• Use the TRACE feature to move along the curve, watching for where the curve peaks or dips at the extremities.

Vertices provide insight into how the hyperbola opens and how wide the branches spread, key elements for fully describing the graph's shape and position.

Solution of Equations

Solving the equation \(x^2 - y^2 = 1\) for \(y\) involves isolating \(y\) using algebraic operations. The process highlights the importance of manipulative algebra in understanding mathematical solutions.

Start by rewriting the equation to isolate \(y^2\):

• Subtract \(x^2\) from both sides: \(-y^2 = 1 - x^2\)
• Multiply by -1 to get \(y^2 = x^2 - 1\)

At this point, taking the square root of both sides gives two potential solutions for \(y\): \(y = \sqrt{x^2 - 1}\) and \(y = -\sqrt{x^2 - 1}\). These solutions result from the nature of square roots (having both positive and negative outcomes) and represent the two branches of the hyperbola. Recognizing this dual solution is critical for comprehending hyperbolic equations.