Problem 30 - College Algebra with Modeling and Visualization

Question

Translate the ellipse with the given equation so that it is centered at the given point. Find the new equation and sketch its graph. $$ \frac{x^{2}}{9}+\frac{y^{2}}{2}=1 ;(-3,7) $$

Step-by-Step Solution

Answer

New equation: \( \frac{(x+3)^2}{9} + \frac{(y-7)^2}{2} = 1 \).

Step 1: Understand the Current Equation

The given equation \( \frac{x^2}{9} + \frac{y^2}{2} = 1 \) represents an ellipse centered at the origin \((0,0)\). The semi-major axis is \(3\) units along the x-axis and the semi-minor axis is \(\sqrt{2}\) units along the y-axis.

Step 2: Identify the New Center

We are required to translate the ellipse so that it is centered at \((-3, 7)\). Translation of the center involves moving the center from \((0,0)\) to \((-3,7)\).

Step 3: Translate the Equation

To translate the ellipse, adjust the variables to reflect the new center \((-3, 7)\). This results in the transformed equation: \( \frac{(x+3)^2}{9} + \frac{(y-7)^2}{2} = 1 \).

Step 4: Sketch the Graph

Draw the new ellipse with its center at \((-3, 7)\). The ellipse retains its shape, with a semi-major axis of 3 units along the x-axis and a semi-minor axis of \(\sqrt{2}\) units along the y-axis.

Key Concepts

Equation of an Ellipse

An ellipse is a fascinating shape in mathematics. Its equation can describe its dimensions and location on a coordinate plane. A standard form of an ellipse centered at the origin is expressed as \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Here, \(a\) and \(b\) are the semi-major and semi-minor axes respectively. These axes measure the distance from the center of the ellipse to its perimeter along the x and y axes.
The given equation \( \frac{x^2}{9} + \frac{y^2}{2} = 1 \) reveals an ellipse centered at the origin \((0,0)\). In this version:

• The semi-major axis \(a\) is \(\sqrt{9} = 3\).
• The semi-minor axis \(b\) is \(\sqrt{2}\).

This equation form helps identify key characteristics such as the axes' lengths, which define the ellipse's size and shape.

Center of an Ellipse

The center of an ellipse is a pivotal point that determines the ellipse's position but not its shape. When an ellipse's center is moved from one point to another, it's called a translation. This change shifts the entire ellipse on the coordinate plane without altering its dimensions or orientation.
Originally, our ellipse is centered at the origin \((0,0)\). In the exercise, we aim to move this center to the new location \((-3,7)\). To achieve this translation, we adjust every occurrence of \(x\) and \(y\) in the ellipse equation by modifying their variables:

• Replace \(x\) with \(x+3\).
• Replace \(y\) with \(y-7\).

This leads us to the new ellipse equation \[\frac{(x+3)^2}{9} + \frac{(y-7)^2}{2} = 1\]Now, the center is at \((-3,7)\) as intended, and it maintains the same size and shape as before.

Graphing Ellipses

Graphing ellipses visually represents the solution to its equation, giving a clear view of its features. When graphing an ellipse, a few steps ensure accuracy and clarity:

• Identify the Center: Start with marking the ellipse's center based on the given or transformed equation. For our new ellipse, this point is \((-3, 7)\).
• Plot the Axes: Use the values of \(a\) and \(b\) to draw the semi-major and semi-minor axes originating from the center. This determines the size and orientation of the ellipse. The semi-major axis extends 3 units along the x-axis from \((-3, 7)\), while the semi-minor axis stretches approximately \(\sqrt{2}\) units along the y-axis.
• Sketch the Ellipse: Once the axes are plotted, draw a smooth, oval shape ensuring it intersects the endpoints of these axes, forming the ellipse.

Graphing reflects translating the ellipse's center from the origin to a new point while keeping its geometry intact. Through careful plotting, it retains its symmetric, oval form while shifting position on the coordinate plane.