Problem 31 - Algebra A Combined Function

Question

Graph each system. $$ \left\\{\begin{array}{r} x^{2}-y^{2} \geq 1 \\ y \geq 0 \end{array}\right. $$

Step-by-Step Solution

Answer

Shade the region outside the hyperbola \(x^2 - y^2 = 1\) and above the x-axis.

Step 1: Identify the Regions

First, let's identify the regions represented by each inequality. For the first inequality \(x^2 - y^2 \geq 1\), this represents the region outside or on the branches of the hyperbola defined by \(x^2 - y^2 = 1\). For the second inequality \(y \geq 0\), this indicates that we are looking at the region above the x-axis.

Step 2: Sketch the Hyperbola

Draw the hyperbola \(x^2 - y^2 = 1\). It opens in the horizontal direction, with vertices at (\(1,0\)) and (\(-1,0\)). The asymptotes of this hyperbola are the lines \(y = x\) and \(y = -x\).

Step 3: Determine the Relevant Region of the Hyperbola

The inequality \(x^2 - y^2 \geq 1\) includes all points on or outside the hyperbola, which means shading the region outside the hyperbola branches on the graph.

Step 4: Apply the Constraint for Non-negative y

Since we have \(y \geq 0\), we will only consider the above-the-x-axis portion of the shaded region. Any part of the shaded area below the x-axis should be excluded from the final graph.

Step 5: Graph the Final Solution

Combine the two inequalities by shading only the portions of the outside region of the hyperbola that are also above the x-axis. This shaded area represents the solution to the given system of inequalities.

Key Concepts

Understanding Hyperbolas

A hyperbola is a type of conic section that is formed by intersecting a double cone with a plane. Unlike circles or ellipses, which are closed curves, a hyperbola consists of two separate branches. In general, a hyperbola is defined by an equation of the form \(x^2/a^2 - y^2/b^2 = 1\). This tells us that the hyperbola opens about the x-axis.
For the exercise at hand, the hyperbola given by \(x^2 - y^2 = 1\) opens horizontally. This means there are two distinct branches extending indefinitely in the positive and negative x-directions. The vertices, key points directly on the x-axis that are closest to the center, are located at (1,0) and (-1,0).
Hyperbolas are important because they help model various real-world situations where two entities are related in a way that follows this unique shape. They are especially prevalent in physics and engineering.

Concept of Inequalities in Graphing

Inequalities are mathematical expressions that show the relationship between two values, indicating if one is greater than, less than, or equal to the other. When graphing inequalities, you're often asked to shade a region of the coordinate plane that represents the solution to the inequality system.
In the given exercise, we're dealing with the inequality \(x^2 - y^2 \geq 1\), which means we shade the region outside or on the hyperbola branches. Additionally, for the inequality \(y \geq 0\), we focus on the upper half plane—above the x-axis.
By graphing these inequalities, we identify the common region that satisfies both conditions. This common region is what represents the valid solutions to the system. Inequalities like these are crucial in optimization and decision-making problems where constraints define permissible solutions.

Navigating the Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane divided by a horizontal axis (x-axis) and a vertical axis (y-axis). It is the fundamental tool for graphing equations and inequalities. Each point on the plane is identified by an ordered pair (x, y), which tells us its location relative to the axes.
In the exercise, we first graph the given hyperbola on this coordinate plane. After that, we apply the inequality conditions to determine which region to shade. The inequality \(y \geq 0\) means we only consider the top half of the plane, including the line on the x-axis.
Understanding how to navigate and interpret graphs on the coordinate plane is essential in algebra and geometry. It allows us to visually parse complex relationships and analyze solutions to systems of equations.

Role of Asymptotes in Hyperbolas

Asymptotes are lines that a graph approaches but never actually touches or crosses. They provide a boundary that the graph of a hyperbola will get infinitely close to but never intersect. These lines are crucial for accurately sketching hyperbolas.
For the hyperbola \(x^2 - y^2 = 1\) in the given problem, the asymptotes are the lines \(y = x\) and \(y = -x\). These lines are found by setting the hyperbola equation to zero, or by analyzing the slopes derived from the equation's denominators when simplified into a standard form.
Visually, asymptotes serve as guides to understanding the direction in which the branches of the hyperbola extend. Asymptotes maintain the hyperbola's structure and provide insight into the graph's behavior far from the center.