Problem 27 - Precalculus: Mathematical for Calculus

Question

Use a graphing device to graph the ellipse. $$6 x^{2}+y^{2}=36$$

Step-by-Step Solution

Answer

Graph the equation \( \frac{x^2}{6} + \frac{y^2}{36} = 1 \); it's an ellipse centered at the origin with a vertical major axis.

Step 1: Write the Equation in Standard Form

We start with the given equation of the ellipse, which is in the form \( 6x^2 + y^2 = 36 \). To convert this into standard form, divide every term by 36 to get \( \frac{6x^2}{36} + \frac{y^2}{36} = 1 \). This simplifies to \( \frac{x^2}{6} + \frac{y^2}{36} = 1 \). This is the standard form of an ellipse centered at the origin.

Step 2: Identify the Semi-Axes Lengths

From the standard form \( \frac{x^2}{6} + \frac{y^2}{36} = 1 \), we identify the denominators of the fractions as the squares of the semi-axis lengths. The term \( \frac{x^2}{6} \) indicates that \( a^2 = 6 \) (implying \( a = \sqrt{6} \)), and the term \( \frac{y^2}{36} \) indicates that \( b^2 = 36 \) (implying \( b = 6 \)). Hence, the semi-major axis length is 6 (along the y-axis), and the semi-minor axis length is \( \sqrt{6} \) (along the x-axis).

Step 3: Determine the Orientation

Because \( b^2 > a^2 \), the major axis of the ellipse is vertical. This means the ellipse stretches more along the y-axis than the x-axis.

Step 4: Use a Graphing Device

Enter the standard form equation \( \frac{x^2}{6} + \frac{y^2}{36} = 1 \) into a graphing device, ensuring the device is set to plot in Cartesian coordinates. Visually confirm the orientation and lengths of the semi-axes, observing an elongated shape along the y-axis.

Key Concepts

Standard Form of an Ellipse

To graph an ellipse accurately, we need to convert its equation into standard form. The standard form of an ellipse centered at the origin is given by \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \(a\) and \(b\) are the semi-axes lengths. The given equation \(6x^2 + y^2 = 36\) can be rewritten by dividing each term by 36. This yields:

• \(\frac{6x^2}{36} + \frac{y^2}{36} = 1\)
• Simplifying gives \(\frac{x^2}{6} + \frac{y^2}{36} = 1\)

Now, the equation is in the desired standard form. This setup is crucial for identifying the ellipse's geometric characteristics, such as semi-axes lengths and orientation.

Semi-Axes Lengths

The semi-axes lengths of an ellipse are crucial for understanding its stretch. In the standard form \(\frac{x^2}{6} + \frac{y^2}{36} = 1\), the denominators represent the squares of the semi-axes lengths:

• The term \(\frac{x^2}{6}\) implies that \(a^2 = 6\), so \(a = \sqrt{6}\). This is the semi-minor axis along the x-axis.
• Similarly, \(\frac{y^2}{36}\) implies that \(b^2 = 36\), so \(b = 6\). This is the semi-major axis along the y-axis.

These lengths allow us to visualize how the ellipse stretches in different directions. The longer length (semi-major) dictates the primary orientation.

Orientation of Ellipse

The orientation of an ellipse describes how it is positioned in the plane. This is determined by comparing the lengths of the semi-major and semi-minor axes.

• If \(b^2 > a^2\), the ellipse is vertically oriented, meaning it stretches more along the y-axis. This is what happens for the given ellipse where \(b = 6\) and \(a = \sqrt{6}\).
• If \(a^2 > b^2\), the ellipse would stretch horizontally instead.

In this exercise, since \(b^2 = 36\) is greater than \(a^2 = 6\), the ellipse has a vertical orientation. This elongation, combined with semi-axes lengths, gives it its distinct oval shape when graphed.