Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) y^{2}-5 x-4 y-6=0

Step-by-Step Solution

Answer

Parabola.

Step 1: Rearrange the equation

Start by moving the constant term to the other side of the equation. The given equation is \( y^2 - 5x - 4y - 6 = 0 \). Add 6 to both sides to get \( y^2 - 5x - 4y = 6 \).

Step 2: Group variable terms

To complete the square, group the \( y \) terms together. Your equation now is: \( y^2 - 4y = 5x + 6 \).

Step 3: Complete the square for the \( y \)-terms

Take the coefficient of \( y \), which is \(-4\), take half of it to get \(-2\), and then square it to get \(4\). Add and subtract \(4\) inside the equation to complete the square: \( (y^2 - 4y + 4) - 4 = 5x + 6 \).

Step 4: Simplify the completed square

The left side of the equation can be rewritten as a squared term: \( (y - 2)^2 - 4 = 5x + 6 \).

Step 5: Isolate the squared term

Move \( -4 \) to the other side of the equation: \( (y - 2)^2 = 5x + 10 \).

Step 6: Write in standard form

Divide everything by \(5\) to match the standard form of a parabola: \( \left( y - 2 \right)^2 = 5(x + 2) \).

Step 7: Identify the conic section

The given equation is in the standard form of a parabola \( (y - k)^2 = 4p(x - h) \). Therefore, the conic section represented by this equation is a parabola.

Key Concepts

Completing the Square

Completing the square is a technique used to simplify quadratic expressions, making them easier to solve or manipulate. In this case, it's used to rewrite an equation into a recognizable form, such as the standard form of a parabola.The process involves creating a perfect square trinomial from a quadratic expression. Here's how it works step by step:

• Identify the quadratic and linear terms you need to complete the square. For instance, if you start with something like \(y^2 - 4y\), your goal is to turn this into a perfect square trinomial.
• Take the coefficient of the linear term, halve it, and square the result. With \(-4\) as the coefficient, the calculation becomes \((-4/2)^2 = 4\).
• Add and subtract this square within the expression. What this does is create an expression like \((y - 2)^2 - 4\), which is a perfect square minus a constant.

Completing the square transforms equations into forms that are easier to solve or match with common equations in geometry, such as those for conic sections.

Parabolas

Parabolas are a type of conic section characterized by a distinct U-shaped curve. They have a few key properties and elements like the vertex, the axis of symmetry, and the focus.In this exercise, the equation is manipulated to reveal that it represents a parabola. Here's what makes an equation a parabola and the basic elements to pay attention to:

• The equation of a parabola can often be written in the form \((y-k)^2 = 4p(x-h)\) or \((x-h)^2 = 4p(y-k)\). This shows if it opens horizontally or vertically.
• The parameters \(h\) and \(k\) represent the coordinates of the vertex, the highest or lowest point (for vertical parabolas), or the furthest left or right point (for horizontal parabolas).
• The distance \(p\) is crucial as it indicates how far the focus and directrix are from the vertex.

Recognizing the structure of a parabola's equation allows for easy identification of these feature points, making graphing and understanding the parabola's properties much simpler.

Equation Rearrangement

Rearranging equations is a fundamental skill in solving and understanding algebraic expressions. It involves manipulating an equation to achieve a desired form or isolate particular variables.Here’s how and why rearranging an equation works:

• This process often begins by moving constants and like terms to one side of the equation. With the original problem \(y^2 - 5x - 4y - 6 = 0\), moving \(-6\) to the right-hand side helped simplify the equation to \(y^2 - 4y = 5x + 6\).
• Grouping like terms is crucial for processes like completing the square or factoring. For example, we need the \(y\) terms together when completing the square.
• Equations are often rearranged into standard forms to easily identify what conic section or geometric shape they represent. In this context, rearranging identified the equation as a parabola.

Understanding how to rearrange equations can simplify solving them and aid in identifying their structure and the concepts they represent, such as conic sections.