Problem 35 - Algebra and Trigonometry with Analytic Geometry

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 equation represents a hyperbola with a horizontal axis.

Step 1: Identify the General Equation Form

The given equation is \(x^2 + 6x - y^2 = 7\). We need to determine which conic section form it represents by rearranging if necessary. The equation involves \(x^2\) and \(y^2\) with coefficients indicating it is of the form \(Ax^2 + By^2 + Cx + Dy + E = 0\), which is used for identifying conic sections.

Step 2: Analyze the Coefficients

Compare the given equation with the standard form: \(Ax^2 + By^2 + Cx + Dy + E = 0\). Here, \(A = 1\), \(B = -1\), \(C = 6\), \(D = 0\), and \(E = -7\). Note that \(A\) and \(B\) have opposite signs, indicating this is the equation of a hyperbola.

Step 3: Verify Axis Orientation

In a hyperbola equation \(Ax^2 + By^2 = C\), if \(A > 0\) and \(B < 0\), as in our equation, the hyperbola opens horizontally (along the x-axis). Conversely, if \(A < 0\) and \(B > 0\), it opens vertically. Our equation satisfies the condition for a horizontal hyperbola.

Key Concepts

Hyperbola

A hyperbola is a type of conic section that you can recognize by its equation involving both squared terms, like \(x^2\) and \(y^2\), with opposite signs. Unlike ellipses and circles, hyperbolas have a distinct property where they open up either horizontally or vertically, forming two separate curves. This shape looks a bit like two opposite-facing parabolas.
Features of a hyperbola include:

• Two asymptotes that intersect at the hyperbola's center.
• A transverse axis that determines its main orientation.
• Conjugate axis, perpendicular to the transverse axis.

Together, these characteristics help decide if the graph of a conic section is a hyperbola. Understanding these features can make it easier to graph and analyze them effectively.

Equation Analysis

Analyzing an equation helps us determine the type of conic section it represents. In the case of a hyperbola, when you look at an equation like \(x^2 + 6x - y^2 = 7\), you can identify it by looking at the coefficients of \(x^2\) and \(y^2\). The equation matches the general conic form \(Ax^2 + By^2 + Cx + Dy + E = 0\).
In hyperbolas:

• Coefficients of \(x^2\) and \(y^2\) have opposite signs.
• This sign difference makes it a hyperbola since this reflects their characteristic diverging curves.

By replacing \(A\) and \(B\) with these coefficients, you determine the equation type. Checking these signs and calculating coefficients is the first step in equation analysis when deciding the graph type.
Once you understand these components, you can easily apply them to other conic equations too!

Axis Orientation

The axis orientation of a hyperbola refers to the direction it opens. This is impacted by the coefficients \(A\) and \(B\) in the conic section equation. In a hyperbola, the signs of these coefficients give you the clue about the axis orientation.
Here’s how you can determine the orientation:

• If \(A > 0\) and \(B < 0\), the hyperbola opens horizontally, or along the x-axis.
• If \(A < 0\) and \(B > 0\), the hyperbola opens vertically, or along the y-axis.

In our example equation \(x^2 + 6x - y^2 = 7\), \(A\) is positive and \(B\) is negative, indicating a horizontal axis orientation.
Knowing the orientation helps you understand the graph's structure and predict its shape. It's a simple tip that provides valuable insight into how the hyperbola will look when plotted. Keeping these rules in mind can strengthen your graphing skills significantly.