Problem 29 - Algebra A Combined Function

Question

Graph each system. $$ \left\\{\begin{array}{l} \frac{x^{2}}{4}+\frac{y^{2}}{9} \geq 1 \\ x^{2}+y^{2} \geq 4 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph regions outside the ellipse and circle.

Step 1: Identify Each Inequality

The system consists of two inequalities. The first inequality is \( \frac{x^{2}}{4} + \frac{y^{2}}{9} \geq 1 \), representing an ellipse, and the second inequality is \( x^{2} + y^{2} \geq 4 \), representing a circle.

Step 2: Graph the Inequality of the Ellipse

To graph the ellipse \( \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 \), note that it's centered at the origin. It stretches 2 units along the x-axis (horizontal semi-axis) and 3 units along the y-axis (vertical semi-axis). The region where \( \frac{x^{2}}{4} + \frac{y^{2}}{9} \geq 1 \) includes all points outside and on the boundary of the ellipse.

Step 3: Graph the Inequality of the Circle

Graph the circle \( x^{2} + y^{2} = 4 \), which is also centered at the origin. The circle has a radius of 2. The region where \( x^{2} + y^{2} \geq 4 \) includes all points outside and on the boundary of the circle.

Step 4: Determine the Intersection of Inequalities

The solution to the system of inequalities is the intersection of the regions obtained from the inequalities. Shade the area that satisfies both \( \frac{x^{2}}{4} + \frac{y^{2}}{9} \geq 1 \) and \( x^{2} + y^{2} \geq 4 \). This is the region outside both the ellipse and the circle.

Key Concepts

Ellipse

An ellipse is a smooth, closed curve that resembles an elongated circle. It is centered at a point called the focal point, and its shape is determined by two axes: the major axis and the minor axis. The lengths of these axes define how "stretched" the ellipse appears.
To understand an ellipse, consider the standard equation for an ellipse centered at the origin: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

• "a" represents the semi-major axis length (half the longest diameter along the x-axis).
• "b" represents the semi-minor axis length (half the shortest diameter along the y-axis).

In the inequality \( \frac{x^2}{4} + \frac{y^2}{9} \geq 1 \), the ellipse stretches 2 units along the x-axis and 3 units along the y-axis. This equation includes all points outside the boundary of the ellipse.

Circle

A circle is a simple shape in mathematics. It's a special form of an ellipse where both axes are of equal length. This means the circle is perfectly round, unlike an ellipse which is stretched in one direction.
The equation of a circle centered at the origin is: \[ x^2 + y^2 = r^2 \]

• "r" represents the circle's radius, or half the diameter.

For the inequality \( x^2 + y^2 \geq 4 \), the circle has a radius of 2, indicating that all points lying on or outside this circle satisfy the inequality. Being centered at the origin simplifies its geometry, making it easy to plot in a standard coordinate system.

System of Inequalities

A system of inequalities consists of multiple inequalities that need to be satisfied simultaneously. When graphed, this involves finding the intersection of regions represented by each inequality.
For example, in the system given:

• \( \frac{x^2}{4} + \frac{y^2}{9} \geq 1 \)
• \( x^2 + y^2 \geq 4 \)

The solutions are points that meet both criteria. When graphing these two inequalities:- The ellipse defines a region outside and on its boundary.- The circle defines another region outside and on its boundary.The combined solution is the area outside both shapes, which satisfies both inequalities simultaneously. This concept helps in visualizing constraints in real-world problems by sketching reasonable solution spaces on a graph.