Problem 22 - Algebra A Combined Function

Question

Graph each equation. $$ \frac{x^{2}}{36}=1-y^{2} $$

Step-by-Step Solution

Answer

The graph is an ellipse centered at the origin with major axis on x-axis ([-6, 6]) and minor axis on y-axis ([-1, 1]).

Step 1: Rearrange the Equation

First, we'll rearrange the given equation \( \frac{x^2}{36} = 1 - y^2 \) to express it in a standard form for graphing. Add \( y^2 \) to both sides to get \( \frac{x^2}{36} + y^2 = 1 \). This equation is now in the form of an ellipse centered at the origin.

Step 2: Identify the Type of Conic Section

The equation \( \frac{x^2}{36} + y^2 = 1 \) represents an ellipse. In this form, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a^2 = 36 \) and \( b^2 = 1 \). Here, \( a = 6 \) and \( b = 1 \). The larger denominator is under \( x^2 \), indicating a more stretched ellipse along the x-axis.

Step 3: Determine the Axes of the Ellipse

Since \( a = 6 \) and \( b = 1 \), the lengths of the axes of the ellipse are determined as follows: The major axis is along the x-axis with a span of \([-6, 6]\), and the minor axis is along the y-axis with a span of \([-1, 1]\).

Step 4: Plot the Ellipse

To graph the equation, plot the center at the origin (0,0). Mark endpoints of the major axis at (-6,0) and (6,0), and endpoints of the minor axis at (0,-1) and (0,1). Then, sketch the ellipse by smoothly connecting these points.

Key Concepts

Ellipse

An ellipse is a fascinating geometric shape that appears similar to a stretched circle. It forms one of the main types of conic sections, which are curves derived from slicing a cone at different angles.

• An ellipse can be described mathematically by its standard equation format, which is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]. In this equation, \(a\) and \(b\) represent the distances from the center to the ellipse's vertices along the x-axis and y-axis, respectively.
• The longest distance across the ellipse is the major axis, and the shortest is the minor axis. The lengths of these axes are determined by \(2a\) and \(2b\), providing a clear measure of the ellipse's "stretched" view.
• Ellipses are commonly seen in planetary orbits, or even in daily objects like racetrack shapes or the outline of a rugby ball.

Ellipses hold substantial importance in both mathematical theory and practical applications, representing an elegant and unique class of shapes.

Graphing Ellipses

Graphing an ellipse begins with understanding its equation and identifying key components like the center, lengths of the axes, and orientation. Let's explore how to break this down:

• The first step is to rearrange the equation to match the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), just like the given equation \(\frac{x^2}{36} + y^2 = 1\).
• Identify the center of the ellipse, which is often at the origin (0,0) unless shifted by additional terms in the equation.
• The lengths of the major and minor axes are determined by \(a\) and \(b\). Here, \(a\) is 6, reflecting the major axis's half-length along the x-axis, and \(b\) is 1, indicating the minor axis's half-length along the y-axis.
• Plot the center first, followed by the endpoints of the major axis at \((-a, 0)\) and \((a, 0)\), and the minor axis at \((0, -b)\) and \((0, b)\).
• Finally, sketch the ellipse by smoothly connecting these points, maintaining the oval shape.

This meticulous procedure ensures a clear and accurate graph, highlighting the beautiful symmetry of the ellipse.

Equation Rearrangement

The skill of equation rearrangement is crucial in mathematical problem-solving, enabling the conversion of equations into a form that is easier to analyze or graph. Let's see how it applies to ellipses:

• The initial given equation \(\frac{x^2}{36} = 1 - y^2\) was not immediately in the proper form to identify as an ellipse.
• The first step is to manipulate the equation by adding \(y^2\) to both sides, resulting in \(\frac{x^2}{36} + y^2 = 1\). This aligns the terms to showcase the properties of an ellipse, allowing further analysis or graphing.
• This technique highlights how equations can be transformed to better fit scientific models or graphical representations. It showcases the flexibility and importance of algebra in simplifying and organizing mathematical expressions.

Mastering equation rearrangement will bolster your ability to tackle a wide array of mathematical challenges, whether for academic purposes or real-world applications.