Problem 39 - Precalculus: Functions and Graphs

Question

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x^{2}+6 x-y^{2}=7$$

Step-by-Step Solution

Answer

The graph is a hyperbola.

Step 1: Recognize Standard Forms

First, recall the standard forms of conics:- Parabola: - Vertical axis: \( (x - h)^2 = 4p(y - k) \) - Horizontal axis: \( (y - k)^2 = 4p(x - h) \)- Circle: \( (x - h)^2 + (y - k)^2 = r^2 \)- Ellipse: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)- Hyperbola: - Horizontal axis: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) - Vertical axis: \( \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \)

Step 2: Analyze the Given Equation

The given equation is \( x^2 + 6x - y^2 = 7 \). Observe the presence and signs of the squared terms. There are two squared terms, \( x^2 \) and \( y^2 \), and they have opposite signs (\( x^2 \) positive, \( -y^2 \) negative). This pattern suggests the graph could be that of a hyperbola.

Step 3: Rearrange the Equation

Rearrange the terms to match the standard form of a hyperbola. Begin by moving all terms to one side: \( x^2 + 6x - y^2 - 7 = 0 \). Set this equal to zero to verify its potential resemblance to the standard hyperbola form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).

Step 4: Identify Conic Section

Since the signs of the squared terms are different and they are not both multiplied by constants that form a perfect sum, the equation \( x^2 + 6x - y^2 = 7 \) resembles that of a hyperbola, given the rearrangement of terms into one equation suggesting subtraction.

Key Concepts

Hyperbola

In the world of conic sections, the hyperbola stands out with its unique shape and properties. Unlike circles and ellipses, which are closed shapes, a hyperbola features two distinct parts, known as branches.
One way to visualize a hyperbola is to imagine two mirrored parabolas separated by clear space. These branches appear to curve towards each other but will never intersect. They open either horizontally or vertically based on their equation.
Key characteristics of a hyperbola include its vertices, which are the closest points of the two branches, and its asymptotes. The asymptotes are straight lines that the branches of the hyperbola approach but never touch. These lines help in sketching a hyperbola and understanding its orientation.

• Vertices: Points where confocal branches are closest.
• Asymptotes: Lines that guide the curvature.

Understanding the properties of a hyperbola allows students to predict how it will appear graphically.

Standard Forms of Conics

Conic sections can be classified into different types with equations that fit certain standard forms. These forms help identify the conic section represented by a given equation.
Each conic has its distinguishing characteristics based on these equations:

• Parabola: One squared term, often taking the form \( (x-h)^2 = 4p(y-k) \) when the axis is vertical.
• Circle: An equation like \( (x-h)^2 + (y-k)^2 = r^2 \) shows that both variables are squared, with equal coefficients.
• Ellipse: Two squared terms with positive signs, e.g., \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).
• Hyperbola: One positive and one negative squared term, such as \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).

Identifying the right form helps distinguish between these conics and understand their geometric properties.

Graphing Equations

Graphing equations of conic sections like hyperbolas involves understanding their standard forms and characteristics. The first step is to rearrange the given equation to match the standard form.
For a hyperbola, after rearranging the equation, one focuses on identifying the center, vertices, foci, and asymptotes. These elements are crucial for accurately sketching the graph:

• Center: The midpoint between the vertices and foci, usually denoted by \( (h, k) \).
• Vertices: Key points directly opposite to the center on the transverse axis.
• Foci: Points inside each branch, crucial for defining the hyperbola.
• Asymptotes: Guiding lines for graph curvature.

Graphing steps often involve moving the graph according to the center, sketching the asymptotes, and then drawing the branches towards these asymptotes, ensuring they curve the right way.

Equation Rearrangement

Rearranging equations is key in identifying and graphing conic sections. To transform an equation to match the standard form, start by grouping and simplifying relevant terms.
In the exercise, the equation was \( x^2 + 6x - y^2 = 7 \). The presence of both \( x^2 \) and \( y^2 \), with one negative and one positive, indicates a hyperbola. Rearranging this equation involves:
1. Moving all terms to one side: Make one side zero by shifting terms to the opposite side, e.g., \( x^2 + 6x - y^2 - 7 = 0 \).
2. Completing the square: Especially if needed, this will simplify the equation and match it with a recognizable standard form.
3. Identifying type: With the correct form, determine the conic type by observing signs and term arrangements.

This process is crucial for recognizing the equation's properties and graphing it accurately. Understanding this rearrangement task helps in transforming complex-looking equations into understandable forms for all types of conic sections.