Problem 12 - Intermediate Algebra

Question

Sketch the graph of each equation. \(x^{2}+4 y^{2}=16\)

Step-by-Step Solution

Answer

The graph is an ellipse centered at the origin with a major axis of length 8 along the x-axis and a minor axis of length 4 along the y-axis.

Step 1: Recognize the Equation Type

The equation \(x^2 + 4y^2 = 16\) is in the form of \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), an equation of an ellipse. We can rewrite it in the standard form of an ellipse by dividing through by 16.

Step 2: Rewrite the Equation in Standard Form

Divide each term of the equation \(x^2 + 4y^2 = 16\) by 16, which results in \(\frac{x^2}{16} + \frac{y^2}{4} = 1\). This matches the standard form of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) where \(a^2 = 16\) and \(b^2 = 4\).

Step 3: Identify the Parameters of the Ellipse

From the standard form \(\frac{x^2}{16} + \frac{y^2}{4} = 1\), we find that \(a^2 = 16\), so \(a = 4\) and \(b^2 = 4\), so \(b = 2\). This means the ellipse has a semi-major axis of length 4 and a semi-minor axis of length 2.

Step 4: Determine the Orientation of the Ellipse

Since the larger denominator is under \(x^2\), the major axis is horizontal, meaning the ellipse is wider along the x-axis.

Step 5: Plot the Major and Minor Axes

Draw the coordinate axes. Plot the endpoints of the major axis at (4, 0) and (-4, 0), and the endpoints of the minor axis at (0, 2) and (0, -2). These points are derived from adding and subtracting \(a\) and \(b\) from the origin respectively.

Step 6: Sketch the Ellipse

Using the major and minor axes endpoints, sketch the ellipse by connecting them smoothly into an oval shape that is symmetric about both the x-axis and y-axis.

Key Concepts

Conic Sections

Conic sections are fascinating curves that are formed by the intersection of a plane with a double-napped cone. Imagine slicing a cone with a flat surface at different angles. The shapes that appear can be rings, parabolas, hyperbolas, or ellipses, depending on the angle. These shapes are collectively known as conic sections.

The most basic conic sections include:

• Circle
• Ellipse
• Parabola
• Hyperbola

Each conic section has its own unique set of properties and equations. Ellipses, for example, appear when the intersecting plane cuts through both nappes of the cone in a manner that creates a closed, oval shape. This is what distinguishes ellipses from parabolas or hyperbolas, where the plane's angle allows more open curves. Learning to identify and sketch these curves aids in understanding many real-world phenomena and applications, such as satellite orbits and the paths of objects under the force of gravity.

Graphing Ellipses

Graphing an ellipse begins with understanding its equation and parameters. An ellipse is often represented by the equation \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \].

The values of \(a\) and \(b\) are crucial because they determine the lengths of the ellipse's axes.

• Semi-major axis is the longest radius of the ellipse, represented by \(a\).
• Semi-minor axis is the shortest radius of the ellipse, represented by \(b\).

The major axis always corresponds to the larger denominator between \(a^2\) and \(b^2\) and dictates the ellipse's orientation:

• If \(a > b\), the ellipse is stretched along the x-axis.
• If \(b > a\), it is stretched along the y-axis.

To graph, plot the endpoints of both axes by moving \(a\) units along the major axis direction from the center and \(b\) units along the minor axis. Connect these endpoints smoothly to form an oval shape, ensuring symmetry about the center. This procedure helps visualize ellipses in contexts ranging from planetary motion to statistics.

Standard Form of an Ellipse

The standard form of an ellipse is the key to identifying and graphing them accurately. The standard form equation is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \].

This form clearly shows the center at the origin (0,0), unless otherwise translated, and helps determine the lengths of the axes:

• \(a^2\) signifies the distance from center to vertex along the x-axis.
• \(b^2\) indicates the distance along the y-axis.

To convert a given ellipse equation like \(x^2 + 4y^2 = 16\) into standard form, divide each term by the constant on the right side, often resulting in an expression where the right side equals 1. This process highlights the values of \(a^2\) and \(b^2\), providing necessary measures for the graph.

Understanding the standard form makes complex contexts more accessible, enabling easier identification of axes lengths, orientation, and potential translations or rotations.