Problem 36 - Precalculus Mathematics for Calculus

Question

Use a graphing device to graph the conic. $$4 x^{2}+9 y^{2}-36 y=0$$

Step-by-Step Solution

Answer

Graph it as an ellipse centered at (0, 2) with axes lengths 6 and 4.

Step 1: Identify the Conic Equation

The given equation is \(4x^2 + 9y^2 - 36y = 0\). This equation can be identified as a conic section because it involves both \(x^2\) and \(y^2\) terms, suggesting it is either a circle, ellipse, hyperbola, or parabola.

Step 2: Rearrange the Equation

First, rearrange the terms of the equation to group \(y\) terms together. The equation can be rewritten as \(4x^2 + 9(y^2 - 4y) = 0\).

Step 3: Complete the Square

To convert the \(y\) terms into a perfect square, complete the square in \(y^2 - 4y\). \(y^2 - 4y\) can be rewritten as \((y-2)^2 - 4\). Thus, the equation becomes \(4x^2 + 9((y-2)^2 - 4) = 0\).

Step 4: Simplify the Equation

Distribute the \(9\) into the completed square form:\(4x^2 + 9(y-2)^2 - 36 = 0\). Then add \(36\) to both sides to simplify:\(4x^2 + 9(y-2)^2 = 36\).

Step 5: Divide by 36

Divide the entire equation by 36 to get it into standard form:\(\frac{x^2}{9} + \frac{(y-2)^2}{4} = 1\). This is the equation of an ellipse.

Step 6: Identify the Ellipse's Properties

The equation \(\frac{x^2}{9} + \frac{(y-2)^2}{4} = 1\) represents an ellipse centered at (0, 2). It has a semi-major axis length of 3 (since \(\sqrt{9} = 3\)), aligned with the x-axis, and a semi-minor axis length of 2 (since \(\sqrt{4} = 2\)), along the y-axis.

Step 7: Graph the Ellipse

Use a graphing device to plot the ellipse. Draw the ellipse centered at (0, 2) with a width along the x-axis of 6 units (2 × 3) and height along the y-axis of 4 units (2 × 2). Ensure proper labeling of axes and key points such as the center.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They have four distinct types: circles, ellipses, parabolas, and hyperbolas, each with unique characteristics.

• A circle is formed when the intersecting plane is perpendicular to the cone's axis and does not pass through the apex. It appears as a perfectly round shape.
• An ellipse occurs when the intersecting plane is tilted, but not enough to become parallel to the edge of the cone. This results in an elongated circle.
• A parabola is created when the intersecting plane is parallel to the edge of the cone, forming a U-shaped curve.
• A hyperbola emerges when the intersecting plane intersects both nappes of the cone, resulting in two separate curves.

Understanding the properties of conic sections helps in classifying and analyzing these curves. They appear in various natural and human-made structures, lending beauty and functionality to architecture and design. In this exercise, we identify an ellipse as the conic section based on the original equation.

Completing the Square

Completing the square is a technique used to simplify quadratic expressions and equations, making them easier to solve or graph. This process transforms expressions into a perfect square trinomial, which is easier to manipulate. In our exercise, we focus on completing the square for the terms involving y:

• Start with the expression \(y^2 - 4y\).
• To complete the square, we add and subtract the square of half the coefficient of y inside the expression. This results in \((y - 2)^2 - 4\).

This transformation allows us to rewrite the conic equation in a form that makes it easier to graph. Completing the square is a handy tool, especially when working with conic sections, as it often reveals insights into the geometric properties of the curve, like its center or vertices.

Graphing Conics

Graphing curves derived from conic sections, like ellipses, involves understanding their equations and properties. Ellipses have distinctive features that are critical for accurate graphing:

• The center of the ellipse provides a reference point for positioning the curve on a coordinate plane.
• Axes lengths—semi-major and semi-minor—determine the shape and orientation of the ellipse.
• The semi-major axis is the longest radius and the semi-minor axis is the shortest radii of the ellipse.

In our example, the equation \(\frac{x^2}{9} + \frac{(y-2)^2}{4} = 1\) represents an ellipse. Here:

• The center is at \((0, 2)\).
• The semi-major axis is aligned along the x-axis with a length of 3 (\(\sqrt{9} = 3\)).
• The semi-minor axis lies along the y-axis with a length of 2 (\(\sqrt{4} = 2\)).

Using this information, you can draw the ellipse centered at \((0, 2)\) with precise dimensions and orientation.

Standard Form of Ellipse

The standard form of an ellipse's equation is useful for identifying its key attributes easily. The standard form is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where

• \((h, k)\) are the coordinates of the center.
• \(a^2\) and \(b^2\) represent the squares of the semi-major and semi-minor axes lengths, respectively.

In the exercise, our equation \(\frac{x^2}{9} + \frac{(y-2)^2}{4} = 1\) follows this form. The center is \((0, 2)\). The semi-major axis length is 3, derived from \(x^2/9\), while the semi-minor axis is 2, from \((y-2)^2/4\). Observing these parameters helps in sketching the ellipse with accuracy.Understanding the equation in its standard form allows you to quickly interpret an ellipse's dimensions and position, streamlining both theoretical understanding and practical application in graphing.