Problem 64 - College Algebra

Question

Graph each semi ellipse. $$y=-\sqrt{4-4 x^{2}}$$

Step-by-Step Solution

Answer

The graph of the given function will be a vertical semi-ellipse in the lower half of xy plane, with its major axis along the x-axis from -2 to 2, minor axis along the y-axis from 0 to -2, and center at origin.

Step 1: Identifying General Form of Ellipse

The given equation can be rewritten as \(y^2=4-4x^2\) which is in the form of \(-y^2/a^2 + x^2/b^2 = 1\) where \(a=2\) and \(b=\sqrt{4}=2\). The negative sign in front of \(y^2\), indicates that the semi-ellipse is a vertical one in the lower half of the xy-plane.

Step 2: Sketching the semi-ellipse

On the cartesian plane, start at origin. The semi major axis b is the representation of the distance from origin to a point on the x-axis. It can be plotted along the horizontal axis with coordinates \((2, 0)\) and \((-2, 0)\). The y-coordinates for the same x-coordinates can be calculated by substituting x-values into the given equation. For \(x = 2\), \(y = -\sqrt{4 - 4 (2)^2} = 0\). Similarly for \(x = -2\), we have the same y value. Hence, creating a semi-ellipse in the lower half of the xy-plane.

Step 3: Finalizing the Graph

Now, we just need to connect the points smoothly to form a semi-ellipse. The curve should be symmetrical around the x-axis, and it should go downward as the negative sign in front of the square root indicates. Our semi-ellipse will range from \(-2\) to \(2\) on the x-axis, and from \(0\) to \(-2\) on the y-axis.

Key Concepts

College Algebra

College algebra is a foundational course that explores various types of functions, their properties, and their graphs. Understanding the concepts in college algebra is crucial for students, as it serves as a bridge to more advanced topics in mathematics and science. When studying conic sections, such as ellipses, you learn both their algebraic representation and their geometric interpretation.

This synergy between algebra and geometry helps to solidify your knowledge and aids in visualizing and solving real-world problems. A semi-ellipse, like the one presented in the exercise, showcases this relationship. Although it's only a portion of an ellipse, understanding how to graph it still relies on the principles of algebra, such as solving equations and inequalities.

Ellipse Equation

The general equation of an ellipse on the Cartesian plane is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) represent the lengths of the semi-major and semi-minor axes, respectively. However, our exercise deals with a semi-ellipse, represented by a modified version of this equation, which is \( y = -\sqrt{4-4x^2} \).

This modified form arises from the manipulation of the general equation to solve for \( y \) and the introduction of the square root, which leads to only one half of the ellipse being depicted. The presence of a negative sign before the square root further indicates that the semi-ellipse is oriented downwards, which is confirmed when graphing it. By understanding these modifications, students can learn how the equation’s form affects the shape and position of an ellipse on a graph.

Cartesian Plane

The Cartesian plane is a two-dimensional plane created by two perpendicular number lines intersecting at a point called the origin. The intersection creates a coordinate system where every point is defined by an ordered pair \( (x, y) \). The ability to plot equations on the Cartesian plane is a powerful tool for visualizing functions and their behaviors.

The Cartesian plane also allows us to draw and understand conic sections like ellipses. In the context of graphing a semi-ellipse, you begin at the origin and, by accounting for the positive and negative values of \( x \) and \( y \) within your equation, plot the corresponding points to create the curve. The semi-major and semi-minor axes lie along the axes of the Cartesian plane, guiding where your semi-ellipse will be located. Properly graphing a semi-ellipse involves plotting points, considering symmetry, and smoothly connecting these points to reveal the shape outlined by the equation.