Problem 64 - Algebra 2

Question

Graph each equation. Describe each graph and its lines of symmetry. Give the domain and range for each graph. $$ x^{2}+3 y^{2}=36 $$

Step-by-Step Solution

Answer

The graph is an ellipse with a horizontal length of 12 units and a vertical length of \(4\sqrt{3}\) units. It has lines of symmetry along the x-axis and y-axis. The domain is \(-6 \leq x \leq 6\) and the range is \(-2\sqrt{3} \leq y \leq 2\sqrt{3}\).

Step 1: Identify the Type of Conic Section

Recognize the equation \(x^{2}+3 y^{2}=36\) as an ellipse since it is in the general form of the equation of an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\). To put the equation in this standard form, each side of the equation is divided by 36 to achieve \(\frac{x^{2}}{36}+\frac{y^{2}}{12}=1\), where \(a^{2} = 36\) and \(b^{2} = 12\).

Step 2: Graph the Ellipse

Using the standard form of the equation, graph the ellipse with a horizontal axis of length \(2a = 2\sqrt{36} = 12\) and a vertical axis of length \(2b = 2\sqrt{12} = 4\sqrt{3}\) centered at the origin.

Step 3: Identify the Lines of Symmetry

The lines of symmetry for an ellipse centered at the origin are the x-axis and the y-axis.

Step 4: Define Domain and Range

The domain (possible x-values) for this graph is \(-6 \leq x \leq 6\), and the range (possible y-values) is \(-2\sqrt{3} \leq y \leq 2\sqrt{3}\).

Key Concepts

Lines of Symmetry

In geometry, the concept of lines of symmetry plays a crucial role in understanding shapes and graphs. A line of symmetry is a line through a shape that divides it into two identical halves, such that if you fold the shape along this line, the parts will match perfectly. When dealing with ellipses, lines of symmetry are always a key feature.

For an ellipse centered at the origin, such as the one defined by the equation \(x^2+3y^2=36\), the lines of symmetry are typically the coordinate axes: the x-axis and the y-axis. These lines pass through the center of the ellipse and split it into symmetrical sections:

• The x-axis divides the ellipse into identical top and bottom halves.
• The y-axis divides it into identical left and right halves.

These symmetrical properties help easily predict the shape and configuration of the ellipse, simplifying both its graphing and analysis.

Domain and Range

In mathematics, the domain and range of a function or graph describe the set of possible input values (x-values) and output values (y-values) respectively. They are an essential consideration when graphing any kind of function, including ellipses.

For an ellipse, the domain and range are determined by examining its extents along the x and y axes. The domain takes into account the extreme left and right points on the graph, while the range considers the extreme top and bottom points.

• In the example \(x^2+3y^2=36\), after converting it to standard form, we find the horizontal extent is from \(-6\) to \(6\), due to its semi-major axis along the x-axis.
• Simultaneously, the range extends from \(-2\sqrt{3}\) to \(2\sqrt{3}\), given the shorter semi-minor axis along the y-axis.

Understanding the domain and range aids in setting up a Cartesian plot and validates that the ellipse will fully contain all coordinates inside these limits.

Graphing Ellipses

Graphing ellipses involves plotting the ellipse on a coordinate plane and requires a grasp of its general form and dimensions. To graph the ellipse given by the equation \(x^2+3y^2=36\), we start by converting it to the standard form of an ellipse:
\[ \frac{x^2}{36} + \frac{y^2}{12} = 1 \]

Each ellipse has a major and minor axis that can be horizontally or vertically aligned. Here, the semi-major axis has a length \(2a = 12\) along the x-axis, and the semi-minor axis is \(2b = 4\sqrt{3}\) along the y-axis.

• First, locate the center of the ellipse, which is at the origin in this case \((0, 0)\).
• Then, mark the vertices at a distance \(6\) units along the x-axis (both positive and negative).
• Similarly, mark points \(2\sqrt{3}\) units along the y-axis (both positive and negative).

Connect these points smoothly to form an oval shape.
Graphing helps visualize the geometric symmetry and appreciate the elegant structure of the ellipse.