Problem 14 - Algebra A Combined Function

Question

Graph each inequality. $$ \frac{x^{2}}{25}+\frac{y^{2}}{4} \geq 1 $$

Step-by-Step Solution

Answer

Shade the region outside the ellipse centered at (0,0) defined by the ellipse equation.

Step 1: Identify the conic section

The inequality \( \frac{x^{2}}{25} + \frac{y^{2}}{4} = 1 \) is the equation of an ellipse with semi-major axis \( 5 \) along the x-axis and semi-minor axis \( 2 \) along the y-axis. It is centered at the origin (0,0). The inequality \( \frac{x^{2}}{25} + \frac{y^{2}}{4} \geq 1 \) represents the region outside the ellipse.

Step 2: Plot the boundary ellipse

First, graph the ellipse given by \( \frac{x^{2}}{25} + \frac{y^{2}}{4} = 1 \). It stretches from (-5,0) to (5,0) on the x-axis and from (0,-2) to (0,2) on the y-axis. Use a dashed line, as points on the boundary \( \frac{x^{2}}{25} + \frac{y^{2}}{4} = 1 \) are not included in \( \geq 1 \) (strict inequality).

Step 3: Shade the appropriate region

Since the inequality is \( \geq 1 \), shade the region outside the ellipse. Any point (x,y) that satisfies \( \frac{x^{2}}{25} + \frac{y^{2}}{4} > 1 \) lies in this region. Choose a test point such as (0,3): since \( \frac{0^{2}}{25} + \frac{3^{2}}{4} = \frac{9}{4} > 1 \), this point is outside, confirming the shading.

Key Concepts

Ellipses

An ellipse is a geometric shape that resembles an elongated circle. Think of it as a squashed or stretched circle. It's a type of conic section, which means that it can be formed by slicing a cone at a particular angle. In math, the standard form of the equation of an ellipse centered at the origin equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a\) is the length of the semi-major axis which is the maximum radius of the ellipse in one direction, and \(b\) is the semi-minor axis, the shortest radius in the perpendicular direction. When graphing, the semi-major axis stretches horizontally if \(a > b\), and vertically if \(b > a\). In our example, the ellipse stretches from -\(5\) to \(5\) along the x-axis and from -\(2\) to \(2\) along the y-axis. This means our semi-major axis is horizontal with length \(5\), and the semi-minor axis is vertical with length \(2\).

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. Depending on the angle and position of the intersection, different shapes such as ellipses, parabolas, and hyperbolas are formed.

• Ellipse: Formed when the plane cuts through a single nappe at an angle, creating a closed, oval shape.
• Parabola: Formed when the plane is parallel to a generator of the cone, resulting in an open curve.
• Hyperbola: Created when the plane intersects both nappes of the cone, forming two separate curves.

Conic sections are represented mathematically by quadratic equations, and each has a unique set of characteristics. For ellipses, one feature is the sum of the distances from any point on the ellipse to the two foci is constant. This distinct property allows ellipses to have applications in astronomy and engineering.

Inequalities

Inequalities are expressions that describe the relative size or order of two values. In mathematics, they extend the concept of an equation by allowing one side to be larger or smaller than the other, rather than equal. Common inequality symbols include:

• \(<\): Less than
• \(>\): Greater than
• \(\leq\): Less than or equal to
• \(\geq\): Greater than or equal to

In graphing, inequalities help describe regions rather than just lines or curves. For example, with \(\frac{x^2}{25} + \frac{y^2}{4} \geq 1\), the inequality describes the area outside and including an elliptical boundary. When solving such inequalities, shading is used to represent the solution area. You test regions using points to determine which areas satisfy the inequality.