Problem 32 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The ellipse is centered at (0,0), extending about 2.83 units horizontally and 2 units vertically.

Step 1: Write the Standard Form Equation

The general form of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). To transform the given equation \( x^2 + 2y^2 = 8 \) to the standard form, divide every term by 8: \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \). This shows that \( a^2 = 8 \) and \( b^2 = 4 \).

Step 2: Identify Parameters of the Ellipse

From \( a^2 = 8 \), calculate \( a = \sqrt{8} = 2\sqrt{2} \). From \( b^2 = 4 \), calculate \( b = \sqrt{4} = 2 \). This defines the semi-major axis along the x-axis as \( a = 2\sqrt{2} \) and the semi-minor axis along the y-axis as \( b = 2 \).

Step 3: Graph the Ellipse

Plot the ellipse using the parameters identified. Center the ellipse at the origin \((0,0)\). The ellipse stretches horizontally to \( \pm 2\sqrt{2} \) (approximately \( \pm 2.83 \)) and vertically to \( \pm 2 \). Use a graphing device to assist with drawing the precise shape.

Key Concepts

Standard Form Equation for Ellipses

The standard form of an ellipse is crucial for understanding its geometry and graphing it accurately. An ellipse's standard form equation is given by:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]This equation allows us to identify the key parameters of the ellipse, specifically the semi-major and semi-minor axes. Notice that this form assumes the center of the ellipse is at the origin ((0,0)). In the exercise's equation, \( x^2 + 2y^2 = 8 \), there is a slight transformation required.
By dividing every term in the original equation by 8, we align it with the standard form:\[ \frac{x^2}{8} + \frac{y^2}{4} = 1 \]Now, the parameters \( a^2 \) and \( b^2 \) are visible: they are 8 and 4, respectively. Recognizing these values is essential as they determine the lengths of the axes and, thus, the exact shape of the ellipse.

Understanding the Semi-Major Axis

The semi-major axis is the longest radius of an ellipse, running from its center to the furthest point on the perimeter. In our equation, it is associated with the larger of the two denominators found in the standard form.
After transforming the equation, we find that \( a^2 = 8 \). This leads us to calculate the semi-major axis, \( a \), as:\[ a = \sqrt{8} = 2\sqrt{2} \]This axis indicates that the ellipse stretches horizontally in its longest dimension. Knowing the semi-major axis is vital when graphing the ellipse, as it helps to determine the horizontal extent from the center, here located at \( ext{origin } (0,0) \).

• The ellipse will extend from the origin to both \( +2\sqrt{2} \) and \(-2\sqrt{2} \) along the x-axis.
• This translates approximately to the ellipse reaching \( ext{almost } \pm 2.83 \).

Explaining the Semi-Minor Axis

While the semi-major axis is the longest stretch of an ellipse, the semi-minor axis is the shortest. It grows from center to the edge perpendicularly to the major axis. In the transformed standard form, this shorter radius corresponds to the smaller denominator. In this situation, we find:
\( b^2 = 4 \)This gives us a semi-minor axis, \( b \), of:\[ b = \sqrt{4} = 2 \]

• The ellipse will extend vertically from the origin \( ext{upward at } +2 \).
• And also \( ext{downward to } -2 \) along the y-axis.

Having both axes, you can now sketch a more accurate shape of the ellipse. The semi-minor axis is important as it defines how slim or wide the ellipse appears vertically.

Graphing Ellipses

Graphing an ellipse might seem complex, but understanding its parameters simplifies the task. Once the axes are identified, the sketch becomes much clearer. For the ellipse given by\( \frac{x^2}{8} + \frac{y^2}{4} = 1 \):

• Center the ellipse at the origin ((0,0)).
• Mark the horizontal reach at \( +2\sqrt{2} \) and \(-2\sqrt{2} \).
• Mark the vertical reach at \( +2 \) and \(-2 \).

Next, simply connect these bounds smoothly to form the ellipse's path. The semi-major axis and the semi-minor axis give the ellipse its distinct shape. A precise graphing device is recommended for more accurate results, as ellipses require precise curves for an accurate depiction.