Problem 33 - Algebra 2

Question

Solve each equation for \(y .\) Graph each relation on your graphing calculator. Use the TRACE feature to locate the vertices. $$ x^{2}-2 y^{2}=4 $$

Step-by-Step Solution

Answer

The solution to the equation is \( y = ± \sqrt{(x^{2}-4)/2} \). The vertices are located at (-2, 0) and (2, 0).

Step 1: Solve the Equation for \(y\)

First, we need to rearrange the equation to get \( y \) by itself on one side. The original equation is \( x^{2}-2 y^{2}=4 \). \n To isolate \( y \), we add \( 2y^{2} \) to both sides, then divide by 2 to get \( y^{2}= (x^{2}-4) / 2 \). And to solve y, we take the square root of both sides, yielding ±\( y = \sqrt{(x^{2}-4)/2} \).

Step 2: Graphing the Equation Using a Graphing Calculator

Next, we can graph this equation, recognizing that there will be two lines because of the '±' in the equation. This means there is one line for \( +\sqrt{(x^{2}-4)/2} \) and one line for \( -\sqrt{(x^{2}-4)/2} \). Plot both lines to complete the graph.

Step 3: Locating the Vertices

Finally, use the TRACE feature of your graphing calculator to locate the vertices, you should find them at (-2, 0) and (2, 0) for this equation.

Key Concepts

Graphing Calculator

A graphing calculator is a handy tool for visualizing mathematical equations, especially conic sections like the hyperbola. When graphing the equation \(x^2 - 2y^2 = 4\), a key step involves understanding the two resulting graphs from \(y = \pm \sqrt{(x^2 - 4)/2}\).

This tool allows you to enter equations directly and produces a graph, helping to see both lines' shapes. By plotting, you can visually confirm key features like symmetry and intercepts. Another useful aspect is the ability to adjust the view window to better understand how the graph behaves for different values of \(x\).

Utilizing the TRACE feature will track points along the graph which can be very helpful to identify and confirm specific points and vertices' coordinates. This feature is crucial for learning about conic section properties visually.

Equation Solving

Solving equations for \(y\) involves rearranging and manipulating to make \(y\) the subject. In the equation \(x^2 - 2y^2 = 4\), this requires isolating terms involving \(y\).

Start by moving terms around to simplify: first, add \(2y^2\) across to separate it from \(x^2\). Dividing by 2 gets: \[y^2 = \frac{x^2 - 4}{2}\]
Taking the square root gives the final solution: \[y = \pm \sqrt{\frac{x^2 - 4}{2}}\].

This indicates there are two potential values for \(y\) for each \(x\), corresponding to the positive and negative roots, which contribute to the formation of the hyperbola on the graph.

Vertices

Vertices are significant points on conics, such as hyperbolas, marking their turning points. For the given equation \(x^2 - 2y^2 = 4\), finding the vertices involves analyzing critical points where the graph changes direction.

Using your graphing calculator's TRACE feature, locate the points where the graph reaches maximum or minimum height. In this case, the vertices are located at \((-2, 0)\) and \((2, 0)\).

These are the x-intercepts for this particular equation and reveal how the hyperbola stretches along the x-axis. Identifying vertices helps in comprehending the overall shape and size of the graph, which is essential for deeper understanding of conic sections and their properties.