Problem 35 - College Algebra with Modeling and Visualization

Question

Sketch a graph of the ellipse. $$ \frac{(x+1)^{2}}{16}+\frac{(y+2)^{2}}{25}=1 $$

Step-by-Step Solution

Answer

Center at (-1, -2); semi-major axis length 10, vertical; semi-minor axis length 8.

Step 1: Identify the Equation Type

The given equation is \( \frac{(x+1)^2}{16} + \frac{(y+2)^2}{25} = 1 \). This is the standard form of an ellipse equation, given as \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h, k)\) is the center and \(a\) and \(b\) are the semi-major and semi-minor axes respectively.

Step 2: Determine the Center of the Ellipse

Compare the given ellipse equation with its standard form. From \( \frac{(x+1)^2}{16} + \frac{(y+2)^2}{25} = 1 \), we identify \(h = -1\) and \(k = -2\). Thus, the center of the ellipse is \((-1, -2)\).

Step 3: Identify the Semi-major and Semi-minor Axes

From the equation, \( \frac{(x+1)^2}{16} + \frac{(y+2)^2}{25} = 1 \), \(a^2 = 16\) gives \(a = 4\) and \(b^2 = 25\) gives \(b = 5\). Since \(b\) is greater than \(a\), the semi-major axis is vertical.

Step 4: Plot the Center on a Graph

Start by plotting the center of the ellipse \((-1, -2)\) on a coordinate plane. This will be the reference point for drawing the ellipse.

Step 5: Draw the Axes of the Ellipse

From the center \((-1, -2)\), plot a vertical line segment of length \(2b = 10\) centered at \((-1, -2)\). Similarly, plot a horizontal line of length \(2a = 8\). These represent the ellipse's semi-major and semi-minor axes respectively.

Step 6: Sketch the Ellipse

Using the plotted axes, sketch the ellipse by smoothly connecting the endpoints of the axes to form an oval shape. The longer axis (vertical) will dictate the overall orientation of the ellipse.

Key Concepts

Conic Sections

Conic sections are curves that are formed by intersecting a plane with a cone. These sections are fundamental in the study of geometry and include shapes such as circles, ellipses, parabolas, and hyperbolas. Each of these shapes has unique properties and can be described by a specific equation.

Ellipses, in particular, are an important type of conic section. They are formed when the intersecting plane cuts through both nappes of the cone, but not too steeply. This creates an oval shape.

The general equation of an ellipse in its standard form is:

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

where \(h, k\) represents the center of the ellipse.

The characteristics of an ellipse depend on the values of \(a\) and \(b\). Understanding these equations helps in precisely graphing and identifying the shape and orientation of ellipses.

Graphing Ellipses

Graphing ellipses involves understanding their mathematical properties and visually presenting them on the coordinate plane. One key aspect to consider is the center of the ellipse, which is given by the values \((h, k)\) in its standard equation form.

To graph an ellipse like \( \frac{(x+1)^2}{16} + \frac{(y+2)^2}{25} = 1 \), you first need to identify its center. Here, since \(h = -1\) and \(k = -2\), the center is at the point \((-1, -2)\).

Next, determine the semi-major and semi-minor axes. The length of these axes influences the size and direction of the ellipse:

• For the revised equation, \(a^2 = 16\) and \(b^2 = 25\).
• This tells us the semi-minor axis \(a = 4\) and the semi-major axis \(b = 5\).

Since \(b > a\), the semi-major axis is vertical.

Plot the center on the graph, and then draw the axes using the lengths \(2a\) and \(2b\). Lastly, sketch the shape of the ellipse around these axes by smoothly connecting the endpoints.

Semi-major and Semi-minor Axes

The semi-major and semi-minor axes of an ellipse are two crucial components that define its shape and size. An ellipse has two axes of symmetry: the longer is the semi-major axis, and the shorter is the semi-minor axis.

To calculate the semi-major and semi-minor axes from the ellipse's equation, observe the denominators under the squared terms:

• If \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), and \(b > a\), then \(b\) represents the semi-major axis, while \(a\) is the semi-minor axis.
• The lengths of the axes are \(2a\) for the semi-minor and \(2b\) for the semi-major.

In our ellipse \(\frac{(x+1)^2}{16} + \frac{(y+2)^2}{25} = 1\), we have \(a = 4\) and \(b = 5\).

This means the semi-major axis is vertical since \(b > a\). Knowing which axis is longer is essential for plotting the ellipse accurately.