Problem 32 - Precalculus Mathematics for Calculus

Question

Use a graphing device to graph the ellipse. $$x^{2}+2 y^{2}=8$$

Step-by-Step Solution

Answer

The ellipse is horizontal with a major axis length of \( 4\sqrt{2} \) and a minor axis length of 4.

Step 1: Identify the Standard Form of an Ellipse

The general equation of an ellipse centered at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). We need to convert the given equation, \( x^2 + 2y^2 = 8 \), to this form.

Step 2: Divide by the Constant

To match the standard form, divide the entire equation \( x^2 + 2y^2 = 8 \) by 8: \[ \frac{x^2}{8} + \frac{2y^2}{8} = 1 \] Simplifying gives us: \[ \frac{x^2}{8} + \frac{y^2}{4} = 1 \]

Step 3: Identify the Values of a and b

From the equation \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \), we can see that \( a^2 = 8 \) and \( b^2 = 4 \). Thus, \( a = \sqrt{8} = 2\sqrt{2} \) and \( b = \sqrt{4} = 2 \).

Step 4: Determine the Orientation and Dimensions of the Ellipse

Since \( a > b \), the ellipse is stretched more along the x-axis, making it a horizontal ellipse. The major axis length is \( 2a = 2 \times 2\sqrt{2} \) and the minor axis length is \( 2b = 4 \).

Step 5: Graph the Ellipse

Use a graphing calculator or software. Plot the ellipse centered at the origin with the major axis along the x-axis having a length of \( 2a \) and the minor axis v along the y-axis having a length of \( 2b \). This shows the horizontal ellipse extended more along the x-axis.

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse equation provides a template for writing and analyzing ellipses in a clear manner. Generally, the standard form is written as: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]This equation describes an ellipse centered at the origin. Here, \(a^2\) and \(b^2\) are the denominators representing the squares of the semi-axis lengths. To convert the given equation \(x^2 + 2y^2 = 8\) into the standard form, we divide the entire equation by 8 to isolate 1 on the right side:\[ \frac{x^2}{8} + \frac{2y^2}{8} = 1 \]Simplifying, the equation becomes:\[ \frac{x^2}{8} + \frac{y^2}{4} = 1 \]With this standard form established, it sets the stage to analyze the size and orientation of the ellipse.

Ellipse Orientation

The orientation of an ellipse describes the direction the major axis is aligned with respect to the coordinate system. For an ellipse in standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the orientation is determined by comparing \(a^2\) and \(b^2\):- If \(a > b\), the ellipse is horizontal, stretching more along the x-axis.- If \(b > a\), the ellipse is vertical, stretching more along the y-axis.In the equation \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \), we notice that \(a^2 = 8\) and \(b^2 = 4\), leading to \(a = \sqrt{8}\) and \(b = 2\). Here, since \(a > b\), the ellipse is oriented horizontally.The horizontal orientation implies that the major axis runs parallel to the x-axis, which influences how the graph of the ellipse appears when plotted.

Ellipse Dimensions

Understanding the dimensions of an ellipse is crucial for graphing and interpreting its physical representation. In the standard form\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]the values \(a\) and \(b\) are the semi-major and semi-minor axes, respectively:- The length of the major axis is \(2a\).- The length of the minor axis is \(2b\).For the ellipse \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \), the calculations are:- \(a = 2\sqrt{2}\) so the major axis length is \(2a = 4\sqrt{2}\).- \(b = 2\) so the minor axis length is \(2b = 4\).These measurements tell us the extent of the ellipse in each direction on a graph. By knowing the exact lengths, we can accurately plot and understand the shape's perimeter and the space it occupies.