Problem 61 - Precalculus: Mathematics for Calculus

Question

Use a graphing device to graph the conic. $$9 x^{2}+36=y^{2}+36 x+6 y$$

Step-by-Step Solution

Answer

The conic is a parabola centered at (2, -3).

Step 1: Simplify the Equation

Start by simplifying the given equation:\[ 9x^2 + 36 = y^2 + 36x + 6y \]Subtract 36 from both sides to eliminate the constant:\[ 9x^2 = y^2 + 36x + 6y \]

Step 2: Rearrange Terms

Reorder the terms to group all terms related to x and y together:\[ 9x^2 - 36x = y^2 + 6y \]

Step 3: Complete the Square for x-Terms

To complete the square for the x-terms:\[ 9(x^2 - 4x) \]Factor out the 9:\[ 9((x - 2)^2 - 4) \]This simplifies to:\[ 9(x - 2)^2 - 36 \]

Step 4: Complete the Square for y-Terms

To complete the square for the y-terms:\[ y^2 + 6y \]Add and subtract 9 inside the square to complete it:\[ (y + 3)^2 - 9 \]

Step 5: Substitute Back and Simplify

Substitute the completed squares into the equation:\[ 9(x - 2)^2 - 36 = (y + 3)^2 - 9 \]Simplify and rearrange:\[ 9(x - 2)^2 = (y + 3)^2 + 27 \]

Step 6: Divide to Recognize the Form

Divide the entire equation by 9 to match standard forms:\[ (x - 2)^2 = \frac{1}{9}(y + 3)^2 + 3 \]This equation represents a parabola.

Step 7: Graph the Conic Section

Use a graphing calculator or software to visualize the parabola described by the equation \((x - 2)^2 = \frac{1}{9}(y + 3)^2 + 3\). Set the vertex at (2, -3) and adjust the scale accordingly to reflect the transformation.

Key Concepts

Parabola

A parabola is a special curve, shaped like an open bowl or a U. It is one of the four types of conic sections and is defined as the set of all points equidistant from a single point, called the focus, and a straight line, called the directrix. Parabolas have different orientations; they can open up, down, left, or right.

In the equation form, a parabola centered at the origin can be given by either \( y = ax^2 \) or \( x = ay^2 \). However, the given exercise involved transforming the original equation into a recognizable parabola form. The shape and orientation of a parabola depend on several factors:

• The direction it opens is determined by the variable written first (in our case, the equation ends up suggesting it opens left/right).
• The vertex form of a parabola is \( (x-h)^2 = 4p(y-k) \) for vertical parabolas, and \( (y-k)^2 = 4p(x-h) \) for horizontal parabolas.
• The vertex's coordinates are ( \(h,k\)) where the parabola turns.

Understanding a parabola's structure helps in graphing or solving these conic sections.

Completing the Square

Completing the square is a method used to manipulate quadratic expressions into a perfect square trinomial, a form that makes solving equations or graphing them easier. This method is indispensable when dealing with conic sections, like our parabola. Let's explore the steps involved:

1. **Identify the quadratic terms:** For both x and y, look for an expression in the form of \( ax^2 + bx \) and \( ay^2 + by \).
2. **Separate the linear coefficient and constant:** You aim to adjust \( bx \) and \( by \) parts.
3. **Create a perfect square trinomial:** - For x, for example, focus on \( x^2 - 4x \). Take half of -4 (which is -2) and square it, giving you 4. Add and subtract this square within the expression to balance the equation. - This transforms \( x^2 - 4x \) into \( (x-2)^2 - 4 \).
4. **Apply the method to both x and y terms separately:** In our solution, completing the square formed \( 9(x - 2)^2 = (y + 3)^2 + 27 \), a form made easier to manage.
By mastering this, graphing or manipulating quadratic equations becomes straightforward, aiding in tasks like identifying the vertex.

Graphing

Graphing is a visual way to represent equations, and it's particularly useful for understanding the properties of conic sections like parabolas. After simplifying and transforming our original equation into \( (x - 2)^2 = \frac{1}{9}(y + 3)^2 + 3 \), we can better visualize the parabola. Here's how:

1. **Identify the vertex and direction:** From the equation \( (x - 2)^2 = \frac{1}{9}(y + 3)^2 + 3 \), we deduce the vertex at (2, -3).
2. **Understand the direction and scale:** Since \( x \) is squared, the parabola opens horizontally (left/right opened).
- The \( rac{1}{9} \) factor compresses the parabola, making it wider than a standard parabola.
3. **Plot points around the vertex:** Start at the vertex and compute for symmetrical points around it, ensuring the graph accurately reflects the parabola's openness and direction.
4. **Draw the parabola:** Use software or graphing devices for accurate depiction, especially in more complex transformations.

Graphing provides insight into the behavior and properties of the parabola, confirming the algebraic manipulations done previously. It solidifies understanding of the overall shape and structure of the conic section being studied.