Problem 11 - Beginning and Intermediate Algebra

Question

Identify the center of each ellipse and graph the equation. $$\frac{(x-3)^{2}}{9}+\frac{(y+2)^{2}}{16}=1$$

Step-by-Step Solution

Answer

The center of the ellipse with the equation \(\frac{(x-3)^{2}}{9}+\frac{(y+2)^{2}}{16}=1\) is (3, -2), with a horizontal semi-major axis of 3 and a vertical semi-minor axis of 4. To graph the ellipse, plot the center, move 3 units right/left and 4 units up/down from the center to find the vertices, and then sketch an ellipse passing through these vertices.

Step 1: Identify the center of the ellipse

The equation in the standard form of an ellipse is: \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\) Given equation: \(\frac{(x-3)^{2}}{9}+\frac{(y+2)^{2}}{16}=1\) Comparing the given equation with the standard form, we can identify the center (h, k) as (3, -2).

Step 2: Identify the values of a and b

a and b are the semi-major axis and semi-minor axis of the ellipse, respectively. They represent the horizontal and vertical distances from the center to the ellipse's vertices. Given equation: \(\frac{(x-3)^{2}}{9}+\frac{(y+2)^{2}}{16}=1\) Comparing the equation with the standard form, we can identify the square values of a and b as 9 and 16, respectively. Therefore, \(a^2 = 9\) and \(b^2 = 16\). Taking the square root of these values, we get: a = 3 b = 4

Step 3: Graph the ellipse

To graph the ellipse, start by: 1. Plot the center of the ellipse (h, k) = (3, -2) on the coordinate plane. 2. Move right and left by 'a' units (3 units in this case) from the center point to find the horizontal vertices location. 3. Move up and down by 'b' units (4 units in this case) from the center point to find the vertical vertices location. 4. Sketch the ellipse, passing through the vertices while maintaining an elliptical shape. Following these steps will give you the graph of the ellipse with the equation \(\frac{(x-3)^{2}}{9}+\frac{(y+2)^{2}}{16}=1\), centered at (3, -2) with a horizontal semi-major axis of 3 and a vertical semi-minor axis of 4.

Key Concepts

Graphing Ellipses

When it comes to graphing ellipses, basing your approach on the equation is key to achieving accuracy. Typically, the ellipse equation is given in a standard form, which helps in identifying its center, axis lengths, and orientation on the graph. To begin graphing your ellipse, first locate the center point by referring to the terms within the equation that adjust the x and y positions. Once the center is clear, determine the semi-major and semi-minor axes, which affect how wide and tall the ellipse will appear on your graph, respectively. To graph:

• Identify and plot the center of the ellipse on a coordinate plane.
• Extend outward from the center by the length of the semi-major axis for the horizontal spread, and the semi-minor axis for the vertical spread.
• Connect these end points in a smooth, rounded oval shape, maintaining even curvature around the center.

Breaking down this process ensures the ellipse is plotted correctly, capturing both its position and its characteristic shape.

Conic Sections

Ellipses are part of a fascinating mathematical concept known as conic sections. Conic sections are the curves obtained when a plane intersects a double-napped cone, creating shapes such as circles, ellipses, parabolas, and hyperbolas. Ellipses, specifically, arise from slicing the cone at an angle to its axis but not steep enough to create a parabola. They form closed, symmetric shapes and their unique properties make them important in various fields beyond mathematics, including astronomy and engineering. Understanding conic sections:

• They include four distinct shapes: circles, ellipses, parabolas, and hyperbolas.
• Each shape has a unique equation to represent their curved nature.
• These sections illustrate important geometric concepts such as symmetry and curvature.

Studying conic sections can provide deeper insights into geometry and help explain phenomena in the natural world.

Ellipse Equation

The ellipse equation, crucial in understanding the ellipse geometry, is generally expressed in its standard form: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]In this equation, \((h, k)\) represents the ellipse's center, while \(a\) and \(b\) are the distances from the center to the ellipse's vertices along the x and y axes, respectively. If \(a > b\), the ellipse is elongated horizontally, and if \(b > a\), the elongation is vertical.Key aspects of the ellipse equation:

• \(h\) and \(k\) derive from shifting the ellipse's position horizontally and vertically from the origin.
• The values of \(a\) and \(b\) determine the shape and size of the ellipse.
• This form of the equation always equals 1, ensuring a standard way of expressing ellipses.

Interpreting these components helps in graphing accurate representations of ellipses and lends intuition into their geometric properties.

Algebra

Algebra plays a critical role in analyzing and understanding the properties of ellipses. By manipulating algebraic equations, one can deduce important features of an ellipse such as the center, axes lengths, and general shape. In the context of ellipses, algebra helps to:

• Simplify or rearrange equations to reveal critical parameters.
• Convert from standard forms to general forms and vice versa.
• Apply algebraic techniques to derive specific points or dimensions.

Mastering algebraic manipulation enables learners to tackle complex equations with confidence and visualize the geometric figures they represent effortlessly.