Problem 34 - College Algebra with Modeling and Visualization

Question

Sketch a graph of the ellipse. $$ \frac{(x+1)^{2}}{16}+\frac{(y+3)^{2}}{9}=1 $$

Step-by-Step Solution

Answer

Draw an ellipse centered at \((-1, -3)\) with horizontal axis length 8 and vertical axis length 6.

Step 1: Identify the Standard Form

The given equation \(\frac{(x+1)^{2}}{16} + \frac{(y+3)^{2}}{9} = 1\) is in the standard form of an ellipse, \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\), where \((h, k)\) is the center of the ellipse, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.

Step 2: Determine the Center

The center \((h, k)\) can be identified from the standard form of the ellipse equation. Here, \(h = -1\) and \(k = -3\) from the terms \((x+1)^{2}\) and \((y+3)^{2}\). Hence, the center of the ellipse is \((-1, -3)\).

Step 3: Identify the Axes Lengths

From the equation, \(a^{2} = 16\) and \(b^{2} = 9\). Therefore, \(a = \sqrt{16} = 4\) and \(b = \sqrt{9} = 3\). Since \(a > b\), this ellipse has a horizontal orientation with a semi-major axis of length 4 and a semi-minor axis of length 3.

Step 4: Sketch the Axes

From the center \((-1, -3)\), draw the horizontal axis with a length of 8 (since the full length is \(2\times 4 = 8\)) and the vertical axis with a length of 6 (\(2\times 3 = 6\)). Extend the major axis 4 units to the right and left of the center, and the minor axis 3 units up and down from the center.

Step 5: Draw the Ellipse

Utilizing the drawn axes as guides, sketch the ellipse ensuring it smoothly connects through the endpoints of the axes, maintaining the elliptical shape.

Key Concepts

Standard Form of an Ellipse

When dealing with ellipses, you'll frequently encounter the standard form of the equation. For an ellipse centered at \((h, k)\) with axes lengths defined by the terms in the equation, it generally looks like:

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

The values \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively. The point \((h, k)\) is the center of the ellipse.
In the given problem, the equation \(\frac{(x+1)^2}{16} + \frac{(y+3)^2}{9} = 1\) fits this form.
This allows us to identify the important information about the ellipse's center and axes lengths.

Ellipse Graphing

Graphing an ellipse involves understanding its equation and identifying key characteristics like the center, axes lengths, and orientation.
In the given equation, \(\frac{(x+1)^2}{16} + \frac{(y+3)^2}{9} = 1\), the center is found by inspecting the terms: \((x+1)^2\) indicates a shift 1 unit to the left (since \(x + 1\)), and \((y+3)^2\) a shift 3 units down.
Therefore, the center of this ellipse is located at \((-1, -3)\).
To graph the ellipse:

• Identify the major and minor axes based on the values of \(a\) and \(b\).
• Draw these axes extending from the center.
• Sketch the ellipse to smoothly pass through the extremities of these axes.

With this approach, you ensure an accurate representation of the ellipse.

Axes Lengths

The axes lengths are crucial for defining the size and shape of an ellipse.
From the equation \(\frac{(x+1)^2}{16} + \frac{(y+3)^2}{9} = 1\), you determine \(a^2 = 16\) and \(b^2 = 9\).
Calculating these gives us \(a = 4\) and \(b = 3\), meaning:

• The semi-major axis is 4 units long.
• The semi-minor axis is 3 units long.

Since \(a > b\), the ellipse is horizontally oriented.
The full axes will be twice these lengths:

• Major axis: 8 units (4 to the right and left of the center).
• Minor axis: 6 units (3 above and below the center).

Use these axes as guides to shape the ellipse accurately.

Conic Sections

Ellipses are one of the four conic sections — the others being circles, parabolas, and hyperbolas. These curves are formed by the intersection of a plane with a cone.
An ellipse specifically occurs when the intersecting plane is at an angle to the cone but doesn't cross the base.
This is why ellipses are often referred to in terms of their 'conic section' properties.
Each conic section has distinct characteristics:

• Circles: Special kind of ellipse where both axes are equal.
• Ellipses: Distinct major and minor axes, as we've discussed.
• Parabolas: Open shape with one focal point and directrix.
• Hyperbolas: Two symmetric open curves.

Understanding these distinctions helps in identifying and graphing complex curves in mathematics.