Problem 65 - Precalculus

Question

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$x^{2}-6 x-2 y+7=0$$

Step-by-Step Solution

Answer

The graph of the equation is a parabola.

Step 1: Re-arrange the equation

The given equation is \(x^2 - 6x - 2y + 7 = 0\). To begin to identify the type of graph this equation represents, it's helpful to rewrite the equation into one of the standard forms listed above. In order to do this, the terms containing x and y should be grouped together and the constant should be isolated on the right side of the equation. Doing this gives us: \(x^2 - 6x = 2y - 7\).

Step 2: Identify the graph type

Looking at this equation, we see that y is isolated on one side. The other side of the equation, \(x^2 - 6x\), is a quadratic in terms of x. While there is currently no constant term in the quadratic, one could potentially be added or subtracted without changing the overall equation. These characteristics suggest that the graph would be of a parabola.

Step 3: Confirm the graph type

To confirm that the graph is indeed a parabola, one method is to complete the square for the quadratic term. Doing so will result in an equation that matches the standard form of a parabolic equation. First, the coefficient of the linear term in the quadratic (here, -6) is halved and squared, giving 9, which is then added to and subtracted from the equation. Doing this transforms the equation to \(x^2 - 6x + 9 = 2y - 7 + 9\), which simplifies to \((x - 3)^2 = 2(y + 1)\). In general terms, this equation represents a parabola that opens upwards with a vertex at (3, -1).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental to understanding a wide range of mathematical and real-world concepts. These equations are typically in the form of \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). In the context of graphing, parabolas are the geometric shapes that result from these equations.

Here are some key characteristics of quadratic equations:

• They involve the highest power of \( x \) being 2, hence the name quadratic.
• The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the coefficient \( a \).
• Quadratic equations can be solved by various methods such as factoring, using the quadratic formula, or graphing.

Understanding the basic structure of quadratic equations is crucial when dealing with conic sections, particularly when determining whether an equation represents a parabola or another shape.

Completing the Square

Completing the square is an essential algebraic technique used to transform quadratic equations into a form that is easier to work with, particularly when identifying the vertex form of a parabola. This technique is particularly useful when graphing parabolas as it helps determine the vertex and direction of the parabola.

Here's a concise guide to completing the square:

• Start with a quadratic equation, typically in the form \( x^2 + bx \).
• Take the coefficient of \( x \) (\( b \)), halve it, and then square the result.
• Add and subtract this square within the equation to form a perfect square trinomial.
• Rewrite the perfect square trinomial as \((x - h)^2\) where \( h \) is the halved coefficient of \( x \).

Using completing the square, you can quickly rewrite equations and match them to the standard forms of parabolas, such as \((x - h)^2 = a(y - k)\), where \((h, k)\) is the vertex of the parabola. This simplification plays a pivotal role in classifying conic sections in equations.

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. They represent many forms such as circles, parabolas, ellipses, and hyperbolas. Understanding these shapes and their equations is crucial in graphing and determining the type of curve a particular equation represents.

Here is a quick overview of conic sections:

• Circle: Created when a plane intersects a cone parallel to its base. It is characterized by the equation \( (x-h)^2 + (y-k)^2 = r^2 \) where \((h, k)\) is the center and \(r\) is the radius.
• Ellipse: Formed when a plane cuts through the cone at an angle, resulting in the shape of an oval. Its general equation is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).
• Parabola: Occurs when a plane is parallel to the side of the cone. Its equation can be written in vertex form as \( (x-h)^2 = 4p(y-k) \).
• Hyperbola: Formed when a plane cuts through both halves of the cone. Its equation is expressed as \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).

By recognizing these forms, you can accurately classify the type of conic section represented by a given equation, providing a visual and analytical understanding of its graph.