Problem 37 - College Algebra with Modeling and Visualization

Question

Sketch a graph of the ellipse. $$ \frac{(x+2)^{2}}{4}+y^{2}=1 $$

Step-by-Step Solution

Answer

The ellipse has a center at (-2, 0), with axes lengths 2 (horizontal) and 1 (vertical).

Step 1: Identify the equation components

The given equation \( \frac{(x+2)^{2}}{4}+y^{2}=1 \) is already in the standard form of an ellipse equation, \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). From the equation, you can identify that \( h = -2 \), \( k = 0 \), \( a^2 = 4 \), and \( b^2 = 1 \). So, the center of the ellipse is at \( (-2, 0) \), \( a = 2 \), and \( b = 1 \).

Step 2: Determine the axes lengths

The values of \( a \) and \( b \) represent the semi-major and semi-minor axes lengths, respectively. In this equation, since \( a = 2 \) and \( b = 1 \), the semi-major axis is 2 units long and the semi-minor axis is 1 unit long.

Step 3: Sketch the ellipse

Start by plotting the center of the ellipse at \( (-2, 0) \). Since the semi-major axis length \( a = 2 \) is horizontal (as it is with the \( x \)-term), draw the major axis along the \( x \)-axis, extending 2 units to the left and right of the center. Plot the points at \( (-2 - 2, 0) = (-4, 0) \) and \( (-2 + 2, 0) = (0, 0) \). The semi-minor axis \( b = 1 \) is vertical, so plot points 1 unit up and down from the center at \( (-2, 1) \) and \( (-2, -1) \). Finally, draw a smooth, oval-shaped curve connecting these points.

Key Concepts

Ellipse Equation

To sketch an ellipse, it's important to understand its equation. The standard form of an ellipse equation is: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] Here,

• \( (h, k) \) represents the center of the ellipse, which determines its position on the graph.
• \( a^2 \) and \( b^2 \) are the denominators under the squared terms, which correspond to the lengths of the semi-major and semi-minor axes when squared.

In our given equation, \( \frac{(x+2)^2}{4} + y^2 = 1 \), we can gather the following:

• \( h = -2 \) and \( k = 0 \), indicating the ellipse is shifted left by 2 units, with the center at \((-2, 0)\).
• \( a^2 = 4 \) and \( b^2 = 1 \), giving us the lengths of the axes.

Recognizing these parts of the ellipse equation helps identify the dimensions and location of the ellipse on a graph.

Semi-Major Axis

The semi-major axis is an essential feature of an ellipse that represents its longest radius. In the general equation \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), the semi-major axis is identified by the larger of \( a \) or \( b \).
For our specific equation, \( a^2 = 4 \) which makes \( a = 2 \). This implies that the semi-major axis is 2 units long.
Since it's positioned under the \((x-h)^2\) term, our major axis is horizontal, extending from the center in both directions on the x-axis. The points defining this axis are found by moving 2 units to the left and right of the center, \(-2\). Thus, our semi-major axis stretches from \((-4, 0)\) to \((0, 0)\).
Understanding the semi-major axis is crucial as it dictates the stretch and orientation of the ellipse.

Semi-Minor Axis

The semi-minor axis of an ellipse is its shortest radius, perpendicular to the semi-major axis. In an ellipse's equation, \( b \) represents this axis, derived from \( b^2 \).
In the given equation \( \frac{(x+2)^2}{4} + y^2 = 1 \), we find \( b^2 = 1 \), leading to \( b = 1 \). Thus, the semi-minor axis has a length of 1 unit.
This axis is aligned with the y-axis, perpendicular to our horizontal major axis. From the center \((-2, 0)\), we mark the semi-minor axis by moving 1 unit up and down, creating points at \((-2, 1)\) and \((-2, -1)\).
The combination of the semi-major and semi-minor axes forms the complete structure of the ellipse, giving it its distinct oval shape.

Graphing Techniques

Sketching an ellipse requires correctly positioning and drawing it according to its equation. Here's a simple guide for graphing:

• First, locate the center of the ellipse on the graph. For our equation, the center is \((-2, 0)\).
• Identify the length and orientation of the semi-major axis. Place points extennding from the center for the full length. In this case, to \((-4, 0)\) and \((0, 0)\). This gives us the ellipse's widest spread.
• Next, determine the semi-minor axis and its direction. For our ellipse, mark points at \((-2, 1)\) and \((-2, -1)\).
• Draw a smooth oval curve connecting these points, making sure it symmetrically rounds through the axes' endpoints.

With these steps, you create an accurate visual representation of the ellipse, showing how the shapes are formed based on their equations.