Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given equation. \(\frac{(x+3)^{2}}{4}+\frac{(y+2)^{2}}{16}=1\)

Step-by-Step Solution

Answer

The center of the ellipse is at (-3, -2), with vertices at (-3, 2), (-3, -6), (-1, -2), and (-5, -2).

Step 1: Identify the Conic Section

The given equation \( \frac{(x+3)^2}{4} + \frac{(y+2)^2}{16} = 1 \) is in the form of \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), which represents the equation of an ellipse. The center of the ellipse is at \((h, k)\).

Step 2: Determine the Center of the Ellipse

From the equation, we identify \(h = -3\) and \(k = -2\). Therefore, the center of the ellipse is \((-3, -2)\).

Step 3: Identify the Major and Minor Axes Lengths

The values of \(a^2 = 4\) and \(b^2 = 16\) imply that \(a = 2\) and \(b = 4\). Since \(b > a\), the major axis is vertical, and the minor axis is horizontal.

Step 4: Calculate the Vertices

The vertices along the major axis, which is along the y-direction, will be at \((-3, -2 + 4)\) and \((-3, -2 - 4)\), i.e., \((-3, 2)\) and \((-3, -6)\). For the minor axis along the x-direction, the vertices will be \((-3 + 2, -2)\) and \((-3 - 2, -2)\), i.e., \((-1, -2)\) and \((-5, -2)\).

Step 5: Sketch the Ellipse

Using the center \((-3, -2)\), plot the vertices calculated in Step 4. Draw a smooth curve connecting these points, forming an ellipse. The major axis is vertical and the minor axis is horizontal.

Key Concepts

Ellipse

An ellipse is a shape that resembles a stretched circle. It is one of the fundamental conic sections, along with parabolas and hyperbolas.
In mathematics, an ellipse is defined by its equation in the standard form: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]Here, \((h, k)\) denotes the center of the ellipse, while the values \(a\) and \(b\) represent the distances from the center to the ends of the major and minor axes, respectively.

• If \(a > b\), the ellipse is elongated along the x-axis, making it horizontal.
• Conversely, if \(b > a\), as in this exercise, the ellipse elongates along the y-axis, resulting in a vertical orientation.

Understanding the equation helps in determining the shape and orientation of the ellipse. Each point on the ellipse satisfies the equation, maintaining a balanced geometric shape.

Graphing

Graphing is an essential technique in mathematics that helps visualize equations. By plotting points on a coordinate plane, students can better understand the relationships expressed algebraically.
For graphing an ellipse, follow these steps:

• Identify the center of the ellipse, which in this case is \((-3, -2)\), to serve as a reference point.
• Locate the vertices using the given values of \(a\) and \(b\).
• Mark the major axis by adding and subtracting \(b\) along the y-axis, and the minor axis by adding and subtracting \(a\) along the x-axis.
• Connect these points with a smooth curve to shape the ellipse.

Graphing not only aids in solving problems but also helps verify solutions by providing a visual representation of the geometrical figure.

Mathematics Education

Mathematics education emphasizes understanding concepts through multiple representations - symbolic, graphical, and numerical.
Learning to work with conic sections like ellipses provides students with the chance to apply algebraic skills in geometry.

• Graphical representation helps in grasping abstract algebraic ideas by making them visible and tangible.
• Practice with graphing promotes comprehension of spatial properties and relationships.
• Engaging in exercises encourages learners to interpret real-world scenarios through mathematical lenses.

By encompassing these different methods, educators can cater to diverse learning styles, thereby reinforcing students' understanding of geometry and algebra.

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space.
Understanding geometric concepts such as ellipses involve recognizing their attributes and knowing how to calculate parameters like axes, area, and vertices.

• The major and minor axes are essential components that describe the shape of the ellipse.
• Vertices mark the extent of the ellipse along each axis.
• Geometric properties allow for describing objects and phenomena in various fields such as physics, engineering, and art.

Studying geometry enhances critical thinking by enabling students to solve problems involving shape and dimension, essential skills in real-world applications.