Problem 72 - Precalculus: Functions and Graphs

Question

Exer. \(69-72\) : Graph the ellipses on the same coordinate plane, and estimate their points of intersection. $$\frac{x^{2}}{3.1}+\frac{(y-0.2)^{2}}{2.8}=1 ; \quad \frac{(x+0.23)^{2}}{1.8}+\frac{y^{2}}{4.2}=1$$

Step-by-Step Solution

Answer

Solve equations graphically or algebraically for intersection points.

Step 1: Identify key features

Determine intercepts, asymptotes, critical points, and end behavior.

Step 2: Plot the graph

Plot key features and connect them to form the complete graph.

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental concept in geometry, providing a two-dimensional surface where equations of ellipses and other geometric shapes are plotted. It consists of two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin point (0,0). The coordinate plane allows us to precisely locate points using ordered pairs \((x, y)\), enabling us to graph equations and visualize their relationships.When graphing ellipses, the coordinate plane facilitates understanding their geometric properties, such as orientation and intersections with other shapes. It is essential to accurately translate the equations of ellipses onto this plane to analyze their dimensions and positions, which informs the procedure for estimating points of intersection.

Semi-Major Axis

The semi-major axis is one of the most important elements of an ellipse. It represents half of the longest diameter through the center of the ellipse. When analyzing an ellipse's equation in the form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), the value \(a\) typically denotes the semi-major axis if \(a > b\). Consequently, this axis outlines the ellipse's maximum width.In practical terms, the semi-major axis determines how stretched the ellipse is along the longer dimension. By understanding and identifying the semi-major axis while working with ellipses, you get a clearer picture of the shape's orientation and potential intersection points with other ellipses or lines on the coordinate plane.

Semi-Minor Axis

The semi-minor axis complements the semi-major axis by representing the shortest radius from the center to the boundary of the ellipse. In the standard ellipse equation \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), \(b\) is the semi-minor axis when \(a > b\), marking the shorter distance across the ellipse.The semi-minor axis is crucial for understanding the degree of the ellipse's "squash." When combined with the semi-major axis, it helps define the overall shape and aspect ratio of the ellipse. Recognizing the semi-minor axis helps in determining the ellipse's placement and structure on the coordinate plane, particularly when observing overlaps or intersections with other shapes.

Equation of an Ellipse

An equation of an ellipse provides a mathematical description and allows us to sketch its shape in the coordinate plane. The standard form, \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), signifies an ellipse centered at the point \((h, k)\). Here, the denominators \(a^2\) and \(b^2\) represent the squares of the semi-major and semi-minor axes, respectively.Understanding the equation's structure enables us to readily identify features such as:

• Center: The point \((h, k)\) around which the ellipse is symmetrically drawn.
• Axes: Determines the lengths and orientations of the semi-major and semi-minor axes.
• Shape: Whether the ellipse is more elongated or more circular based on the ratio of \(a\) to \(b\).

Detecting these characteristics from the equation helps us systematically graph the ellipse and comprehend its geometric properties on the coordinate plane. By using the given equations, students can visualize ellipses and manage to locate intersections between different ellipses easily.