Problem 35 - A Graphical Approach to College Algebra

Question

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$4 x^{2}-8 x+9 y^{2}-36 y=-4$$

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Rewrite the Equation in Standard Form

Start by rewriting the given equation to reveal its structure. The equation is \( 4x^{2} - 8x + 9y^{2} - 36y = -4 \). We'll rearrange terms: \( 4x^2 - 8x = -9y^2 + 36y - 4 \). Factor and complete the square for both \(x\) and \(y\) terms separately.

Step 2: Factor and Complete the Square for x

For the \(x\) terms: \(4x^2 - 8x\). Factor out the 4: \(4(x^2 - 2x)\). Complete the square inside the parenthesis by adding and subtracting 1: \(4((x - 1)^2 - 1) = 4(x - 1)^2 - 4\).

Step 3: Factor and Complete the Square for y

Now for the \(y\) terms: \(9y^2 - 36y\). Factor out the 9: \(9(y^2 - 4y)\). Complete the square inside the parenthesis by adding and subtracting 4: \(9((y - 2)^2 - 4) = 9(y - 2)^2 - 36\).

Step 4: Rewrite the Equation With Completed Squares

Substitute the completed squares back into the equation: \(4(x - 1)^2 - 4 + 9(y - 2)^2 - 36 = -4\). Simplify by combining constants: \(4(x - 1)^2 + 9(y - 2)^2 = 36\).

Step 5: Identify the Conic Section Type

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

Key Concepts

Completing the Square

Completing the square is a method used to transform quadratic expressions into a perfect square trinomial. This technique is helpful when working with conic sections, as it allows us to rewrite the equation in a recognizable standard form.

To begin, look at a quadratic expression in the form of \( ax^2 + bx \). The goal is to add and subtract the same value to create a square of a binomial. Here’s how you do it:

• First, factor out any common factor from the quadratic terms \((ax^2\) if necessary).
• Take half of the linear coefficient \(b\), square it, and add and subtract this square within the expression.
• Reorganize to form \((x + h)^2 - k\).

For example, to complete the square of \(x^2 - 2x\):- Factor to get \((x^2 - 2x)\).- Take half of \(-2\), which is \(-1\), then square it to get 1.- Add and subtract 1 to form \((x-1)^2 - 1\).

This process converts quadratic terms into a neat square, simplifying graphing and identifying the conic section type.

Ellipse

An ellipse is a set of points in a plane where the sum of the distances from two fixed points (called foci) to any point on the ellipse is constant. Understanding the properties of an ellipse is essential when working with its equation.

The general equation for an ellipse is of the form:\[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]

• \((h, k)\) represent the center of the ellipse.
• \(a^2\) and \(b^2\) are the denominators that denote the square of the semi-major and semi-minor axes, respectively.
• If \(a > b\), the ellipse is stretched wider horizontally. If \(b > a\), it’s taller vertically.

To determine the orientation and shape of an ellipse graphically:- Identify the center point \((h, k)\).- Calculate the distances for major and minor axes using \(a\) and \(b\).- Remember that the total constant in the general equation should be 1 to recognize its form.

An ellipse is oval and smooth, making it distinct from other conic sections like circles, parabolas, and hyperbolas.

Standard Form of Conic Sections

The standard form of conic sections provides a comprehensive way to express different types such as circles, ellipses, parabolas, and hyperbolas using recognizable equations. Understanding these forms is critical for identifying and graphing conic sections.

For a conic section to be in standard form, we need to simplify its equation into a format that reveals its type clearly. Here's how different conic sections typically look:

• **Circle**: \((x - h)^2 + (y - k)^2 = r^2\)
• **Ellipse**: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)
• **Parabola**: \((y-k) = a(x-h)^2\) or \((x-h) = a(y-k)^2\)
• **Hyperbola**: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\)

In our original exercise, the equation simplifies to an ellipse through completing the square and dividing by 36 to normalize the equation. Thus, achieving the standard form allows us to quickly recognize we are dealing with an ellipse by observing the format of its equation.

Revisiting the importance of completing the square, it acts as a bridge to transform complex quadratic equations into a more accessible standard conic section form. This makes graphing and further analysis straightforward, especially in identifying specific features such as axes and focal points.