Problem 65 - Algebra 2

Question

Graph each equation. Describe each graph and its lines of symmetry. Give the domain and range for each graph. $$ x^{2}-y^{2}=25 $$

Step-by-Step Solution

Answer

The given equation is a hyperbola centered at the origin with vertices at (5,0) and (-5,0). Lines of symmetry are y=x and y=-x. The domain and range are \((-\infty, -5)\) and \((5, \infty)\) respectively.

Step 1: Identify the Type of Equation

The equation \(x^{2}-y^{2}=25\) is a type of conic section called a hyperbola. Conic sections also include circles, ellipses and parabolas. Observing the equation, it can be written as \((x)^{2} - (y)^2 = 5^{2}\). Which is a hyperbola equation of the format \(x^2 - y^2 = a^2\). Here a= 5.

Step 2: Graph the Equation

Generate points that satisfy the equation then plot these points. The hyperbola is centered at the origin (0,0) because there are no adjustments to x and y. By substituting x values, you get two corresponding y values due to the square root (one positive and one negative). For example, when x = 0, y =-5, 5. Continue for more values and plot these points. The points will create two branches of the hyperbola, one in the top right and bottom left quadrants and the other in the top left and bottom right quadrants.

Step 3: Identify the Lines of Symmetry

Consider where the graph mirrors itself. For this hyperbola, the lines of symmetry are the lines y=x and y=-x. These lines cut through the origin and form a X shape.

Step 4: Determine the Domain and Range

The domain and range are all the possible x-values and y-values for the graph respectively. Here, because the hyperbola extends infinitely, the domain is \((-\infty, -5)\) and \((5, \infty)\): all real numbers except those between -5 and 5. Similarly, the range is also \((-\infty, -5)\) and \((5, \infty)\): all real numbers except those between -5 and 5.

Key Concepts

Equation of a Hyperbola

The equation given in the exercise is \(x^{2} - y^{2} = 25\). This is an example of a hyperbola, a type of conic section. Conic sections are the curves which result from the intersection of a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas.
For hyperbolas, the general form of the equation is \(x^2 - y^2 = a^2\), where \(a\) is a constant that determines the distance between the center of the hyperbola and the vertices along the x-axis. In our equation, \(a = 5\), so the distance from the center (which is at the origin, (0,0)) to each vertex on the x-axis is 5 units.
To identify hyperbolas, watch for the equation structure: a difference of squares, where one variable is subtracted from the other. Understanding this form helps in visualizing and plotting the hyperbola.

Graphing Conic Sections

Graphing a hyperbola involves determining the points that satisfy its equation and then plotting them. Begin by selecting several values for \(x\), then calculate the corresponding \(y\) values. Remember, due to the square term, each \(x\) will have two \(y\) values (positive and negative).
For example, set \(x = 0\), then solve for \(y\):

• \(0^2 - y^2 = 25\)
• \(y^2 = -25\)
• Thus, \(y = -5\) and \(y = 5\)

This means when \(x = 0\), the points \((0, 5)\) and \((0, -5)\) are on the graph.
Proceed with other values of \(x\) to find more points, and plot them to sketch the hyperbola's branches, which appear in opposite quadrants: top right and bottom left, as well as top left and bottom right. These branches stretch outward toward infinity, indicating that hyperbolas have no bounds.

Domain and Range

The domain and range of the hyperbola are vital to understanding its behavior in a graph. The domain includes all possible \(x\) values. For the hyperbola \(x^2 - y^2 = 25\), the domain is split into two intervals: \((-\infty, -5)\) and \((5, \infty)\). This indicates that \(x\) can take any value except those between -5 and 5, as the hyperbola does not exist in this gap.
Similarly, the range, representing all possible \(y\) values, is also \((-\infty, -5)\) and \((5, \infty)\). This pattern matches the domain due to the symmetric nature of the hyperbola.Understanding the domain and range helps clarify where the hyperbola is positioned on the plane and which parts of it can be traced.

Lines of Symmetry

In mathematics, symmetry helps identify where and how figures mirror over particular lines. For hyperbolas, symmetry reveals the balanced and mirror-like nature of their graph. The hyperbola given in this exercise is symmetric with respect to the lines \(y = x\) and \(y = -x\).

• The line \(y = x\) is a diagonal line passing through the origin with a slope of 1.
• The line \(y = -x\) is also a diagonal through the origin but has a slope of -1.

For the hyperbola \(x^2 - y^2 = 25\), these lines of symmetry bisect the hyperbola, creating an X-shaped reflection. Understanding these lines assists in predicting the shape and orientation of the hyperbola on the graph, as each branch mirrors across these lines.