Problem 17 - A Graphical Approach to College Algebra

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. $$x^{2}-6 x+y=0$$

Step-by-Step Solution

Answer

The equation represents a parabola.

Step 1: Identify the Equation Type

The given equation is:\[ x^2 - 6x + y = 0 \]This equation includes both \(x\) and \(y\), but only \(x\) is squared. This indicates that it is not a circle or an ellipse, which require both \(x\) and \(y\) squared.

Step 2: Rearrange the Terms

Rearrange the terms for better clarity:\[ y = - x^2 + 6x \]This resembles the standard form of a parabola where we have a single squared term in either \(x\) or \(y\).

Step 3: Determine the Conic Section

Since only one variable (\(x\)) is squared and the other (\(y\)) is not squared, and there is no product term \(xy\), the equation represents a parabola. Parabolas are the only conic sections with one variable squared.

Key Concepts

Parabolas

Parabolas are one of the four main types of conic sections. They have a unique shape similar to an arch or a U. Parabolas can open upward, downward, to the left, or to the right, depending on the equation.
In mathematics, parabolas have several interesting properties, such as a focus and a directrix. These are used to define the set of points that make up the curve. The vertex is another important point on a parabola, which is either the highest or lowest point, depending on the direction the parabola opens.
The standard form of a parabola's equation is either:

• Vertical: \( y = ax^2 + bx + c \)
• Horizontal: \( x = ay^2 + by + c \)

Here, the term with the squared variable defines whether the parabola is vertical or horizontal. In our case, since the equation can be rearranged to have \( y = -x^2 + 6x \), it is a vertical parabola. This parabola opens downward because the coefficient of \( x^2 \) is negative.
Understanding the form and orientation of parabolas is crucial when analyzing their graphs and real-life applications, such as the paths of projectiles.

Equation Identification

Identifying the type of conic section just from an equation is a useful skill. Each conic section—circle, ellipse, parabola, and hyperbola—has its distinct characteristics based on its equation.
Here’s how you identify a parabola:

• Check which variable is squared. If only one variable is squared (either \( x^2 \) or \( y^2 \), but not both), the equation is that of a parabola.
• Look for terms without a product of the variables, like \( xy \). If they do not exist alongside a squared term, it usually reassures that the equation is of a parabola in the simplest form.

In our problem, the equation \( x^2 - 6x + y = 0 \) reveals only the \( x \) variable is squared, marking it as a parabola. This clarity simplifies solving and graphing these equations without needing detailed calculations or graphs initially.

Types of Graphs

Conic sections consist of diverse types of graphs, each with its own unique visual form and equation. It is important to distinguish between them based on their algebraic properties.

• Circle: Both \( x \) and \( y \) are squared, with equal coefficients, and no \( xy \) term. Its graph is a perfect round shape.
• Ellipse: Both \( x \) and \( y \) are squared, still no \( xy \) term, but the coefficients are not equal, giving it an oval shape.
• Hyperbola: Both \( x \) and \( y \) are squared and have opposite coefficients, creating two branches that open away from each other.
• Parabola: Only one variable is squared, indicating a U-shaped graph, which could open in any direction.

In terms of identification, spotting whether both, one, or neither of the variables are squared guides you to determine which conic section the graph will represent. Recognizing the phrase "only one squared variable" directly suggests a parabola, the simplest graph to distinguish among conics from algebraic expressions.