Problem 19 - Intermediate Algebra

Question

Graph each equation. \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\)

Step-by-Step Solution

Answer

Graph is an ellipse centered at the origin with vertices at (0,3) and (0,-3).

Step 1: Identify the Equation Type

The given equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) is the standard form of an ellipse. It can be identified by comparing it to the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). In this case, \( a^2 = 4 \) and \( b^2 = 9 \) which corresponds to \( a = 2 \) and \( b = 3 \).

Step 2: Determine the Orientation of the Ellipse

For the given ellipse equation, \( b^2 = 9 \) is greater than \( a^2 = 4 \), indicating that \( b = 3 \) is associated with the \( y \)-axis. Therefore, the major axis is vertical. The center of the ellipse is at the origin \((0, 0)\).

Step 3: Identify the Vertices and Co-Vertices

Since the major axis is vertical (along the \( y \)-axis), the vertices, which are at a distance \( b \) from the center, are located at \( (0, 3) \) and \( (0, -3) \). The co-vertices, along the \( x \)-axis, are at a distance \( a \) from the center, located at \( (2, 0) \) and \( (-2, 0) \).

Step 4: Draw the Ellipse

Using the calculated vertices and co-vertices, plot these points on the Cartesian plane. Draw a smooth, elongated circle connecting these points, ensuring that the ellipse extends further along the \( y \)-axis compared to the \( x \)-axis, reflecting the longer vertical major axis.

Key Concepts

Ellipses

Ellipses are fascinating geometric figures that look like flattened circles. In simple terms, an ellipse can be thought of as an oval shape. Ellipses are part of the conic sections, which also include circles, parabolas, and hyperbolas. What makes ellipses unique is their symmetrical appearance and the relationship between their axes. They have two axes: the major and the minor.

• The major axis is the longest diameter of the ellipse.
• The minor axis is perpendicular to the major at the center point and is the shortest diameter.

The center of an ellipse is the midpoint of both axes. While drawing or graphing an ellipse, attention is particularly paid to these axes, as they define the overall shape and orientation of the ellipse. Each ellipse has two focal points, or foci, located along the major axis, and one interesting property is that the sum of the distances from any point on the ellipse to these foci is constant.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, plays a key role in understanding shapes like ellipses on a plane. It combines algebra and geometry using graphs and equations. For an ellipse, the coordinate plane helps locate points, lines, and curves by using ordered pairs.

• Coordinates describe the position of points in the space using a pair \(x,y\).
• In the given problem, the ellipse is centered at the origin \(0, 0\).

The x-axis and y-axis are used to determine the horizontal and vertical position of points, respectively. By using these coordinates, we can plot key features of the ellipse, like vertices and co-vertices, and interpret its overall shape effortlessly. Understanding coordinate geometry is fundamental for graphing and solving geometric problems involving ellipses.

Standard Form of an Ellipse

The standard form of an ellipse equation is essential for easily identifying and graphing ellipses. It is typically written as:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Both \(a^2\) and \(b^2\) are positive numbers that determine the lengths of the axes:

• \(a^2\) is associated with the x-axis direction.
• \(b^2\) is linked to the y-axis direction.

In the given ellipse equation, \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), we identify \(a^2 = 4\) and \(b^2 = 9\). From these values, the lengths of semi-axes \(a\) and \(b\) can be calculated as \(a = 2\) and \(b = 3\). Since \(b\) is greater than \(a\), the ellipse's major axis is vertical. This insight is crucial when sketching the ellipse accurately, as it informs us about its orientation and dimensions on a Cartesian coordinate system.