Problem 13 - Algebra 2

Question

Graph each equation. Identify the conic section and describe the graph and its lines of symmetry. Then find the domain and range. $$ x^{2}+9 y^{2}=1 $$

Step-by-Step Solution

Answer

The given equation represents a vertically oriented ellipse with a domain of \(-1 \leq x \leq 1\) and a range of \(-\frac{1}{3} \leq y \leq \frac{1}{3}\). The lines of symmetry are the lines x=0 and y=0.

Step 1: Identifying the Conic Section

From the form of the equation \(x^{2}+9 y^{2}=1\), we recognize it as the standard equation of an ellipse with the form \(\frac{x^{2}}{a^2} + \frac{y^{2}}{b^2} = 1\). Whereas a and b are the semi major and semi minor axes respectively and here \(a^2 = 1\) and \(b^2 = \frac{1}{9}\). Therefore, the conic section in this problem is an ellipse.

Step 2: Describing the Graph and Its Lines of Symmetry

Since 'a' is along x-axis and 'b' is along y-axis, hence, the ellipse stretches more along the y-axis than x-axis. This indicates that it is a vertically oriented ellipse. The lines of symmetry for an ellipse are its major and minor axes. In this case, lines of symmetry are x=0 and y=0.

Step 3: Finding the Domain and Range

The domain (set of all x-values) and range (set of all y-values) can be read directly from the equation of the ellipse. The maximum x-value is 'a' and the minimum x-value is '-a', so the domain is \(-1 \leq x \leq 1\). The maximum y-value is 'b' and the minimum y-value is '-b', so the range is \(-\frac{1}{3} \leq y \leq \frac{1}{3}\).

Key Concepts

Ellipses

Ellipses are a type of conic section that appear as an elongated circle. Imagine stretching a circle until it looks more like an oval. In mathematical terms, ellipses are defined by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a\) and \(b\) denote the lengths of the semi-axes. The larger value between \(a\) and \(b\) decides whether the ellipse stretches more along the x-axis or the y-axis. This can make the ellipse either wider or taller.
For the given equation \(x^{2}+9 y^{2}=1\), we identified it as an ellipse. By rewriting the equation in standard form, we see that \(a^2=1\) and \(b^2=\frac{1}{9}\). Thus, this ellipse is taller rather than wider, indicating that it is stretched along the y-axis.
Understanding ellipses is crucial because they model many real-world phenomena, such as planetary orbits.

Lines of Symmetry

Lines of symmetry are imaginary lines that pass through an object or shape, creating a mirror-like reflection on either side. For ellipses, these lines simplify the complexities of the shape.
Ellipses have two lines of symmetry: the major and the minor axes. These axes are the longest and shortest diameters, respectively. For the equation \(x^{2}+9 y^{2}=1\), the lines of symmetry can be found at \(x=0\) and \(y=0\). These axes cut through the ellipse's center, reflecting the equal curves on either side.

• Horizontal symmetry (along the x-axis) is where the top reflects to the bottom.
• Vertical symmetry (along the y-axis) reflects the left half to the right.

By understanding this concept, you can anticipate the ellipse's shape even before plotting it.

Domain and Range

In mathematics, the domain refers to all the possible x-values, while the range refers to all the possible y-values that a function can take. For conic sections like ellipses, these values define the shape's boundaries on a graph.
For the ellipse \(x^{2}+9 y^{2}=1\), observing the equation helps to identify the stretching limits on both axes:

• The domain is restricted to \(-1 \leq x \leq 1\), as this reflects the farthest left and right the ellipse reaches.
• The range is \(-\frac{1}{3} \leq y \leq \frac{1}{3}\), found by calculating the limits of vertical stretching.

Understanding domain and range helps visualize where the ellipse lies in the coordinate plane, making graph plotting informative and easier.

Graphing Equations

Graphing equations, especially for conic sections like ellipses, provides a visual representation of equations, helping in understanding the shape and orientation.
When graphing the equation \(x^{2}+9 y^{2}=1\), it is essential to ensure the re-formatted equation guides the plot:

• Reform the equation to \(\frac{x^2}{1} + \frac{y^2}{(1/3)^2} = 1\).
• Recognize that the x-axis runs from -1 to 1, and the y-axis from -\(\frac{1}{3}\) to \(\frac{1}{3}\).
• Mark these onto the coordinate plane, then sketch the ellipse through these points, respecting the symmetry lines at \(x=0\) and \(y=0\).

Properly graphing the equation not only showcases understanding of ellipses, symmetry, domain, and range but also enhances comprehension of these mathematical concepts in visual formats.