Problem 13 - Intermediate Algebra

Question

Graph each inequality. See Examples 1 and \(2 .\) $$ \frac{x^{2}}{4}+\frac{y^{2}}{9} \leq 1 $$

Step-by-Step Solution

Answer

Graph an ellipse with center at origin and axes 2 and 3; shade inside including the boundary.

Step 1: Identify the Inequality's Form

The inequality \( \frac{x^{2}}{4}+\frac{y^{2}}{9} \leq 1 \) is in the standard form of an ellipse. Specifically, it represents an ellipse centered at the origin \((0,0)\) with semi-major and semi-minor axes along the coordinate axes.

Step 2: Determine the Axes Lengths

From the equation \( \frac{x^{2}}{4}+\frac{y^{2}}{9} = 1 \), we identify that the lengths of the semi-major and semi-minor axes are \( \sqrt{9} = 3 \) (along the y-axis) and \( \sqrt{4} = 2 \) (along the x-axis), respectively.

Step 3: Sketch the Ellipse

Draw an ellipse centered at the origin \((0,0)\) going through the points \((2,0), (-2,0), (0,3), (0,-3)\), which are determined by the lengths of the semi-axes.

Step 4: Graph the Inequality

Since the inequality is \( \leq \), shade the region inside the ellipse, including the boundary, because points inside satisfy the inequality. Use a solid line for the ellipse boundary to indicate the inclusion of points on the ellipse.

Key Concepts

Graphing Inequalities

Graphing inequalities like \( \frac{x^{2}}{4}+\frac{y^{2}}{9} \leq 1 \) involves more than just plotting a curve or shape. It requires understanding how to not only draw the boundary of a shape but also determine which area it encompasses based on the inequality sign. In our equation,

• The symbol \( \leq \) indicates "less than or equal to."
• This means that we're looking for all the points that lie inside or exactly on the ellipse.

Start by drawing the ellipse, which acts as the boundary. Then, because of the "equal to" part of "less than or equal to," shade the inside area. This shaded area shows every possible solution — every \( (x, y) \) pair that makes the inequality true. Use a solid line to draw the boundary to indicate that points on the ellipse are also included in the solution set.

Ellipse Equations

An ellipse equation, such as \( \frac{x^{2}}{4}+\frac{y^{2}}{9} = 1 \), is crucial in visualizing and understanding the shape's geometry. Ellipses are unique, oval-shaped figures defined clearly by their mathematical equation. Key features of an ellipse equation include:

• The denominators, which determine the ellipse's width and height.
• The numerator always being \( x^{2} \) and \( y^{2} \) at the origin.

The general form \( \frac{x^{2}}{a^2} + \frac{y^{2}}{b^2} = 1 \) outlines an ellipse centered at the origin. Here, the fraction forms represent the dilation effect on both axes, shaping the ellipse's form. For our inequality, the ellipse equation helps identify its size and orientation, playing a crucial role in correct plotting.

Standard Form of an Ellipse

The standard form of an ellipse provides a systematic way to write its equation. It looks like this: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \). The essence of this format is in its parts:

• \( a^2 \) and \( b^2 \) are the squares of the lengths of the semi-axes.
• The numerator \( x^2 \) and \( y^2 \) indicate the center is at the origin (\(0,0\)).

For our specific ellipse, \( a^2 = 4 \) and \( b^2 = 9 \), translating to semi-axis lengths of 2 and 3. The form immediately tells us the ellipse's dimensions and its center’s location. This standardized way of writing it enables easy understanding and allows us to quickly sketch or interpret the ellipse's features by plugging into this formula.

Axes Lengths

Axes lengths are fundamental in defining an ellipse’s size and shape. They are portrayed through the terms \( a \) and \( b \), representing half the lengths of the major and minor axes of the ellipse. To find them, you take the square roots of the denominators from the standard form equation:

• For the x-axis, you'd determine \( a \) from \( a^2 \); here, \( a^2 = 4 \) so \( a = 2 \).
• For the y-axis, \( b \) is from \( b^2 \); here, \( b^2 = 9 \) so \( b = 3 \).

These axes dictate how far the ellipse extends in each direction from its center. In this scenario, the semi-major axis is of length 3 along the y-axis, and the semi-minor axis is length 2 along the x-axis. This knowledge is essential when sketching as it ensures accuracy in portraying the ellipse's dimensions.