Problem 2 - 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. $$ 2 x^{2}+y^{2}=36 $$

Step-by-Step Solution

Answer

The given equation represents an ellipse with the center at the origin. The lines of symmetry are the x-axis and the y-axis. The domain of the equation is \(-\sqrt{18} \leq x \leq \sqrt{18}\) and the range is \(-6 \leq y \leq 6\).

Step 1: Identifying the Conic

Looking at the given equation \(2x^{2} + y^{2} = 36\), it can be identified as the equation of an ellipse as it satisfies the general equation \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) where \(a\neq b\).

Step 2: Rewriting in Standard Form

Rewrite the equation in standard form to identify the values of \(a\) and \(b\). When it's arranged, the expression becomes \(\frac{x^{2}}{18} + \(\frac{y^{2}}{36}=1\). Hence, \(a = \sqrt{18}\) and \(b = 6\).

Step 3: Drawing the Ellipse

Next, draw the graph of an ellipse with the corresponding values of \(a\) and \(b\). On a graph, the center of the ellipse is at the origin (0,0). The major axis is along the y-axis while the minor axis is along the x-axis.

Step 4: Identifying Lines of Symmetry

Alignment of axes defines the lines of symmetry. For this case, lines of symmetry are the x-axis and the y-axis.

Step 5: Determining the Domain and Range

The domain or ‘x’ values for the ellipse will be all real numbers between \(-a\) and \(a\), thereby making domain \(-\sqrt{18} \leq x \leq \sqrt{18}\). Similarly, ‘y’ values or range will be all real numbers between \(-b\) and \(b\), thereby making range \(-6 \leq y \leq 6\).

Key Concepts

Ellipse

An ellipse is a fascinating shape, a kind of stretched circle. It consists of all the points such that the sum of the distances from two fixed points, called foci, is constant. Ellipses occur naturally in many contexts, including in the orbits of planets and moons. They are part of a group called conic sections, which also includes circles, parabolas, and hyperbolas. In an equation, an ellipse generally appears in the form of \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). When you see an equation with both quadratic terms adding up as above, you're likely looking at an ellipse. In the example given, the equation of the ellipse is \( 2x^2 + y^2 = 36 \). By recognizing this as an ellipse, we know that we're dealing with an oval shape. Understanding this basic concept helps us know why and how to graph it.

Lines of Symmetry

Lines of symmetry in geometric shapes help in understanding their structure and in sketching. For an ellipse, these lines are the axes that run along the longest and shortest paths through its center. For the specific ellipse from the example \( 2x^2 + y^2 = 36 \), once rewritten as \( \frac{x^2}{18} + \frac{y^2}{36} = 1 \), we see it graphically extends along the y-axis more than the x-axis. Thus, its major axis, or longest line of symmetry, is the y-axis, while the minor axis, or shortest, is the x-axis. Both of these lines of symmetry pass through the center of the ellipse, and they help illustrate that the ellipse is evenly balanced around both the x and y axes.

Domain and Range

Understanding the domain and range of an ellipse is essential for graphing it accurately. The **domain** refers to all the possible x-values the ellipse can have. In an ellipse, this is all the x-values from the leftmost point to the rightmost point. For the ellipse described by \( \frac{x^2}{18} + \frac{y^2}{36} = 1 \), the span across the x-axis is defined as: \(-\sqrt{18} \leq x \leq \sqrt{18}\). Conversely, the **range** of an ellipse indicates all the possible y-values from the lowest to the highest point. For our ellipse, where the major axis is vertical, the range is \(-6 \leq y \leq 6\). This tells us that within these boundaries on a graph, every x-value has a matched y-value that maintains the shape of the ellipse.

Standard Form of an Ellipse

The standard form of an ellipse is a simplified version of its equation that makes it easier to understand and graph. This form is usually given as: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). The values of \(a\) and \(b\) determine the ellipse's shape and orientation. For instance, when you convert the equation \(2x^2 + y^2 = 36\), you rearrange it as \(\frac{x^2}{18} + \frac{y^2}{36} = 1\). Here, \(a = \sqrt{18}\) and \(b = 6\), showing that our ellipse vertically stretches longer than it does horizontally. Knowing the standard form and values helps you identify and plot the ellipse correctly on a graph, defining both its position and dimensions efficiently.