Problem 23 - Intermediate Algebra

Question

Graph each equation. \(16 x^{2}+4 y^{2}=64\)

Step-by-Step Solution

Answer

Graph the ellipse centered at the origin with vertices at (0, ±4) and co-vertices at (±2, 0).

Step 1: Identify the Type of Conic Section

The given equation is \(16x^2 + 4y^2 = 64\). This is a form of the equation of an ellipse, as it contains different coefficients for \(x^2\) and \(y^2\).

Step 2: Rewrite in Standard Form

To rewrite the equation in standard form for an ellipse, which is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), we first divide every term by 64:\[\frac{16x^2}{64} + \frac{4y^2}{64} = \frac{64}{64}\]This simplifies to:\[\frac{x^2}{4} + \frac{y^2}{16} = 1\]

Step 3: Identify the Parameters of the Ellipse

From the standard form \(\frac{x^2}{4} + \frac{y^2}{16} = 1\), we identify the parameters:- \(a^2 = 4\) so \(a = 2\)- \(b^2 = 16\) so \(b = 4\)This ellipse is centered at the origin (0,0) with semi-major axis length of 4 (vertical direction) and semi-minor axis length of 2 (horizontal direction).

Step 4: Plot the Ellipse on the Coordinate Plane

1. Draw the axes of symmetry: the major axis (vertical) goes through the origin along the y-axis and the minor axis (horizontal) goes through the origin along the x-axis.2. Mark the vertices at \((0, \pm4)\) and the co-vertices at \((\pm2, 0)\).3. Sketch a smooth curve to connect these points, forming an ellipse.

Key Concepts

Conic Sections

Conic sections are figures that can be formed by the intersection of a plane and a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. Conic sections have unique properties and equations that characterize their shape and position in the coordinate plane. Understanding these sections is key to exploring more complex geometric figures.
Ellipses, in particular, are one common type of conic where the plane cuts through the cone at an angle, not parallel to the base. This creates an oval-like figure that has interesting mathematical properties and applications. By analyzing the equation, one can discern which form of conic section is being described.

Standard Form of Ellipse

The standard form of an ellipse's equation is given by:

• displaystyle \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)

This represents an ellipse centered at \((h, k)\), where \(a\) and \(b\) are the distances from the center to the ellipse along the x-axis and y-axis, respectively. These distances are also known as the semi-major and semi-minor axes. The semi-major axis is longer.

• If \(a > b\), the ellipse is stretched along the x-axis, creating a horizontal orientation.
• If \(b > a\), the ellipse is stretched vertically.

Writing the equation in this form helps understand the positioning and dimensions of the ellipse on a graph.
For example, rewriting the equation \(16x^2 + 4y^2 = 64\) into its standard form shows the ellipse is centered at the origin since the equation includes only \(x^2\) and \(y^2\) (i.e., \(h = 0, k = 0\)).

Ellipse Parameters

Identifying the parameters of an ellipse is crucial for graphing and understanding its geometry. The parameters of an ellipse from its standard form equation \(\frac{x^2}{4} + \frac{y^2}{16} = 1\) include:

• Semi-Major Axis (\(b\)): The longest radius (4 units in this case), indicating a vertical stretch.
• Semi-Minor Axis (\(a\)): The shortest radius (2 units), indicating a smaller spread along the horizontal with vertices at \((0, \pm4)\) and co-vertices at \((\pm2, 0)\).
• Center: Located at the origin \((0, 0)\), giving the base point from which the axes extend.

Markers along these axes help to accurately sketch the ellipse, highlighting its symmetry and ensuring a precise representation. Recognizing these parameters guides the construction of the graph and provides information about the ellipse's physical properties, like its eccentricity which describes the "ovalness" based on the ratio of \(a\) and \(b\).