Problem 31 - Algebra 2

Question

Graph each equation. Describe the graph and its lines of symmetry. Then find the domain and range. $$ -8 x^{2}+32 y^{2}-128=0 $$

Step-by-Step Solution

Answer

The graph of this equation is an ellipse with its center at the origin. Its major axis is along the y-axis and minor axis along the x-axis. It has lines of symmetry along the x and y axes. The domain is \[-4, 4\] and the range is \[-2, 2\].

Step 1: Identify the form of equation

First, we rewrite the given equation so it follows the standard form of an ellipse equation \((x/h)^2/a^2 + (y/k)^2/b^2 = 1\). This can be obtained by dividing the equation \(-8x^2+32y^2=128\) by 128, resulting in \(-x^2/16 + y^2/4 = 1\)

Step 2: Describe the graph and lines of symmetry

Given the equation we now have, we can tell that this is an equation of an ellipse that is elongated vertically (since \(b^2\) > \(a^2\)). The center of the ellipse is (0,0). The major axis is along the y-axis and the minor axis along the x-axis. The lines of symmetry for this ellipse are the x and y axes.

Step 3: Find the domain and range

For an ellipse, the domain and range are given by the distances from the center to the ellipse along the x and y axes respectively. Since \(a^2\) = 16, and \(a\) is the distance along the x axis, the domain is \(-a \leq x \leq a\), or \[-4, 4\]. Similarly, since \(b^2\) = 4, and \(b\) is the distance along the y axis, the range is \(-b \leq y \leq b\), or \[-2, 2\].

Key Concepts

Equation of an Ellipse

Ellipses are fascinating geometric shapes defined by a specific equation. The standard form is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h, k)\) represents the center of the ellipse. The parameters \(a\) and \(b\) determine the lengths of the major and minor axes, respectively. In the given exercise, we start with the equation \(-8x^2 + 32y^2 - 128 = 0\). Rewriting it involves manipulation to match the standard form. Once simplified to \(-\frac{x^2}{16} + \frac{y^2}{4} = 1\), we recognize how the values under the squared terms (\(16\) and \(4\)) inform about the axes. - \(h = 0\) and \(k = 0\) place the center at the origin.- \(a^2 = 16\) and \(b^2 = 4\) show that \(a = 4\) and \(b = 2\), making it a vertically stretched ellipse.

Graphing

Graphing an ellipse helps visualize its size, shape, and position. For the equation \(-\frac{x^2}{16} + \frac{y^2}{4} = 1\), drawing starts by marking the center \((0, 0)\). From the center, distances defined by \(a\) and \(b\) are measured: - Move \(4\) units in the \(x\) direction (left and right) for the x-axis boundaries.- Extend \(2\) units in the \(y\) direction (up and down) for the y-axis boundaries. From these four points, sketching the ellipse involves gently connecting them to form the smooth curved shape typical of ellipses. This sketch shows how it extends more along the y-axis than the x-axis. Using axis length and shape observation, the ellipse is identified as vertically elongated.

Domain and Range

In mathematics, domain and range are crucial for understanding how far and wide a graph extends. For an ellipse, these terms describe available \(x\) and \(y\) values the ellipse spans. In our case:- **Domain** reflects the spread along the x-axis. Since \(a = 4\), the domain of our ellipse is \([-4, 4]\).- **Range** highlights limits on the y-axis. Given \(b = 2\), the range stretches from \([-2, 2]\). These intervals are derived directly from the center point and the axis lengths, providing clear boundaries for ellipse plotting.

Symmetry in Ellipses

Symmetry is a unique feature of ellipses, making them elegant and predictable. For any ellipse centered at \((h, k)\), its lines of symmetry convey key geometric properties. In our exercise, the ellipse shows symmetry along:- The x-axis: Reflected horizontally, each point on one side has a mirror image.- The y-axis: This vertical symmetry shows similar reflection properties. Additionally, for ellipses centered at origin \((0, 0)\), the lines of symmetry also include the origins. Understanding this characteristic further strengthens graph comprehension, offering aesthetic uniformity recognizable at first glance.