Problem 32 - Algebra 2

Question

Graph each equation. Describe the graph and its lines of symmetry. Then find the domain and range. $$ 25 x^{2}+16 y^{2}-320=0 $$

Step-by-Step Solution

Answer

The graph is an ellipse with center at the origin (0,0) and with a horizontal symmetry about the x-axis and vertical symmetry about the y-axis. The length of the major axis is 40 units and minor axis is 25.6 units. The domain of this function is [-12.8, 12.8] and the range is [-20,20].

Step 1: Rewrite the Equation in Standard Form

Firstly, rearrange the equation to match with the standard form of the ellipse equation. Thus, the rewritten equation becomes: \(25x^2 + 16y^2 = 320\). We can further simplify this by dividing the entire equation by 320 to get: \(\frac{x^{2}}{(320/25)} + \frac{y^{2}}{(320/16)} = 1\), which simplifies to \(\frac{x^{2}}{12.8^{2}} + \frac{y^{2}}{20^{2}} = 1\). This is the standard form of the equation of ellipse.

Step 2: Determine the length of Major and Minor Axes

From the standard form, we can tell that the semi-major axis (a) is 20 and the semi-minor axis (b) is 12.8. The major axis length is therefore 2a = 40 and the minor axis length is 2b = 25.6.

Step 3: Graph the Equation and Identify Symmetry

The graph of the equation is an ellipse centered at the origin (0,0) because there is no h or k in the equation. It has vertical symmetry about the y-axis and horizontal symmetry about the x-axis because the equation includes squared quantities of both x and y.

Step 4: Determine the Domain and Range

The domain of this function, represented in interval notation, is given by the extreme x-values. Since the x-values range from -12.8 to 12.8, the domain is [-12.8, 12.8]. Similarly, the y-values range from -20 to 20. So, the range is [-20,20].

Key Concepts

Exploring Conic Sections

Conic sections are shapes created when a plane intersects a double cone. Depending on the angle of the cut, different types of conic sections are formed: circles, ellipses, parabolas, and hyperbolas. Each conic section has unique characteristics.

Ellipses are one such conic section and are oval-shaped. Their standard form equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively. This form reveals important properties of the ellipse, such as its orientation and dimensions.

In the given exercise, the equation is transformed into a standard ellipse form: \( \frac{x^2}{12.8^2} + \frac{y^2}{20^2} = 1 \), indicating an ellipse centered at the origin. The ellipse has its major axis along the y-axis, as \( b > a \). Understanding conic sections involves recognizing how they graphically represent different types of curves based on their equation.

The Role of Symmetry in Ellipses

Symmetry is an essential aspect of many mathematical shapes and particularly conic sections like ellipses. Symmetry in an ellipse indicates the balance with respect to the central points or lines.

For the ellipse given in the exercise, it is centered at (0,0) and demonstrates two types of symmetry:

• **Vertical Symmetry:** Along the y-axis, because of the squared \( y \) term.
• **Horizontal Symmetry:** Across the x-axis, due to the squared \( x \) term.

The symmetry of an ellipse makes it easier to understand and graph. Knowing the axes of symmetry helps identify the ellipse's orientation. These lines of symmetry essentially cut the ellipse into mirror-image halves, making calculations more intuitive. Recognizing symmetry is a helpful tool when solving and graphing ellipses.

Finding Domain and Range in an Ellipse

The domain and range of a function are crucial concepts that describe the set of possible input values (domain) and possible output values (range). For conic sections like ellipses, the domain and range are derived from the maximum extents of the ellipse along the x and y axes.

In the context of the provided ellipse equation \( \frac{x^2}{12.8^2} + \frac{y^2}{20^2} = 1 \), the domain and range can be found as follows:

• **Domain:** Since the x-values span from \(-12.8\) to \(12.8\), the domain is expressed as \([-12.8, 12.8]\).
• **Range:** The y-values range from \(-20\) to \(20\), giving a range of \([-20, 20]\).

This approach to determining domain and range is straightforward due to the inherent symmetry of ellipses. The size of the axes from the center outwards directly provides these intervals. Knowing the domain and range helps to understand the full span of the ellipse on any graph.