Problem 22 - A Graphical Approach to Precalculus with Limits

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$x^{2}+2 x=-4 y$$

Step-by-Step Solution

Answer

The equation represents a parabola.

Step 1: Rearrange the Equation

The given equation is \( x^2 + 2x = -4y \). To identify the conic section, rearrange it into the standard form. Add 4y to both sides to get \( x^2 + 2x + 4y = 0 \), which is needed for further analysis.

Step 2: Complete the Square

To understand the nature of the equation, complete the square for the \(x\) terms. Start with \(x^2 + 2x\). Take half of the coefficient of \(x\), which is 1, and square it to get 1. Add and subtract 1 inside the equation: \( (x^2 + 2x + 1) - 1 + 4y = 0 \). This can be rewritten as \( (x+1)^2 - 1 + 4y = 0 \).

Step 3: Simplify the Equation

Simplify the equation by moving terms around: \((x+1)^2 = 1 - 4y \). Then, rearrange to isolate \(y\): \( 4y = 1 - (x+1)^2 \), leading to \( y = \frac{1 - (x+1)^2}{4} \).

Step 4: Identify the Conic Section

Notice the equation \( y = \frac{1 - (x+1)^2}{4} \) is in the form of a parabola. This matches the general form of a vertical parabola \( y = ax^2 + bx + c \) after completing the square. Therefore, the equation represents a parabola.

Key Concepts

Parabolas

A parabola is a U-shaped curve that can open either up, down, left, or right. The general equation of a parabola in standard form is either \( y = ax^2 + bx + c \) or \( x = ay^2 + by + c \). Parabolas have a distinct feature called the vertex, which is the highest or lowest point on the graph, depending on the direction it opens.
Parabolas also have an axis of symmetry, which is a vertical or horizontal line that divides the parabola into two mirror-image halves. In addition, when identifying parabolas, focus on:

• The coefficient of the squared term, \( a \).
• The direction of the opening (up or down for \( y \), left or right for \( x \)).
• Whether the vertex lies at the origin or if it has been shifted.

In this exercise, after rearranging and simplifying the given equation into standard form, it was identified as a parabola.

Completing the Square

Completing the square is a method used to simplify quadratic equations, which helps in transforming equations into a recognizable standard form. This process is essential when identifying the shape and position of conic sections, such as parabolas.
Here’s a simple way to complete the square:

• Start with the quadratic expression, like \( x^2 + bx \).
• Find half of the coefficient of \( x \), which is \( \frac{b}{2} \), and square it to form \( \left(\frac{b}{2}\right)^2 \).
• Add and subtract this square inside the equation.
• Rewrite the expression as a perfect square trinomial: \( (x + \frac{b}{2})^2 \).

Applying this to the exercise, by completing the square on \( x^2 + 2x \), we convert it into the form \( (x + 1)^2 - 1 \), which is crucial for identifying the parabola.

Equation Identification

Equation identification in conic sections involves recognizing the standard forms of various conic shapes: parabolas, circles, ellipses, and hyperbolas. Each conic section has a distinct set of properties and characteristics reflected in its equation.
When you come across an equation, like in this exercise, focus on these steps to identify the type of conic section:

• Rearrange the equation to collect similar terms and simplify.
• Consider completing the square or factoring to transform the equation into a recognizable standard form.
• Match the modified equation with standard forms:\( y = ax^2 + bx + c \) for parabolas, \( (x-h)^2 + (y-k)^2 = r^2 \) for circles, etc.

By transforming the given equation \( x^2 + 2x = -4y \) into \( y = \frac{1 - (x+1)^2}{4} \), it was identified as a parabola, showcasing the importance of understanding these steps.