Problem 18 - Beginning and Intermediate Algebra

Question

Identify the center of each ellipse and graph the equation. $$(x+3)^{2}+\frac{(y+4)^{2}}{9}=1$$

Step-by-Step Solution

Answer

The center of the given ellipse, \((x+3)^{2}+\frac{(y+4)^{2}}{9}=1\), is \((-3, -4)\). To graph the ellipse, plot the center point at \((-3, -4)\), draw the major and minor axes by moving 1 unit in the x-direction and 3 units in the y-direction, and then sketch the elongated oval shape using these points as guides.

Step 1: Identifying the Center of the Ellipse

Comparing the given equation, \[(x+3)^{2}+\frac{(y+4)^{2}}{9}=1\] with the standard form of an ellipse (\(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\)), we can see that the center of the ellipse is the point (h, k) where: \(h = -3\) \(k = -4\) So, the center of the ellipse is \((-3, -4)\).

Step 2: Graphing the Ellipse and Its Center

Now that we have identified the center of the ellipse, we will proceed to graph the ellipse along with its center. To do this, we first recognize that: \(a^2 = 1\), so \(a = 1\) \(b^2 = 9\), so \(b = 3\) Using these values and the center of the ellipse, we can graph the ellipse as follows: 1. Plot the center point \((-3, -4)\) on the coordinate plane. 2. From the center, move 1 unit in both the positive and negative x-directions to draw the major axis (since it's horizontal in this case). 3. From the center, move 3 units in both the positive and negative y-directions to draw the minor axis. 4. Sketch the ellipse by drawing an elongated oval shape, using the points from steps 2 and 3 as guides. Now, the ellipse has been graphed along with its center, \((-3, -4)\).

Key Concepts

ellipse equation

An ellipse is a fascinating geometrical shape that extends along two axes. Its mathematical representation is derived from its standard form equation. The ellipse equation looks like this: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] In this equation:

• (h, k) is the center of the ellipse.
• a is the semi-major axis length.
• b is the semi-minor axis length.

The values of \(a\) and \(b\) decide the stretch of the ellipse along the x and y axes, respectively. The larger value between \(a\) and \(b\) determines whether the ellipse is horizontal or vertical.

graphing ellipses

Graphing an ellipse involves plotting it on the coordinate plane based on its equation. First, identify the center, which acts as a reference point for the entire shape. Next, determine the lengths of the major and minor axes using the values under \((x - h)^2\) and \((y - k)^2\) in the standard ellipse equation. Here,

• The major axis aligns with the direction of the larger value among \(a\) and \(b\).
• The minor axis aligns perpendicular to the major axis.

Mark each endpoint of the axes based on the calculated distances from the center. Lastly, connect these points smoothly in an elongated oval to reflect the elliptical shape.

center of ellipse

The center of an ellipse is crucial as it serves as the anchor around which the entire ellipse is graphed. To find the center from the equation: \((x+3)^{2} + \frac{(y+4)^{2}}{9} = 1\), compare it with the standard form \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] From this comparison:

• h comes from shifting \(x\), identified by setting \(h = -3\).
• k comes from shifting \(y\), identified by setting \(k = -4\).

Thus, the center of the ellipse is \((-3, -4)\). Knowing the center lets you position the axes accurately and graph the ellipse correctly around this point.

algebraic graphing

Algebraic graphing is a powerful tool in visualizing mathematical ideas such as ellipses. It involves plotting equations onto a coordinate grid to provide a visual context for abstract concepts. To graph algebraically:

• Start by analyzing the given equation and extracting necessary components like the center, axes length, etc.
• Plot these components accurately on the graph.
• Use these plotted points to form the shape described by the equation.

For the ellipse in our problem, we’ve identified the center, calculated axis lengths (1 unit and 3 units), and then used these to depict the elliptical shape. Visual graphing through algebra helps in solidifying understanding of geometric properties and their algebraic relationships.