Problem 18 - 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. $$11-3 x=2 y^{2}-8 y$$

Step-by-Step Solution

Answer

The equation represents a parabola.

Step 1: Rewrite the Equation

First, rewrite the given equation in a more recognizable form: \( 2y^2 - 8y + 3x = 11 \). We notice that this equation involves terms in \( y^2 \) and \( y \), and a term in \( x \).

Step 2: Identify the Conic Section Standard Form

For conic sections, the general equation in the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \) helps identify the type of conic section. Here, \( A = 0 \), \( B = 0 \), \( C = 2 \), \( D = 3 \), \( E = -8 \), and \( F = -11 \).

Step 3: Determine the Type Based on Coefficients

In the equation, there is a square term \( y^2 \) but no \( x^2 \) term. Conic sections without \( x^2 \) terms or with only \( y^2 \) terms are typically parabolas.

Step 4: Conclusion on Conic Type

Since the equation lacks an \( x^2 \) term while possessing a \( y^2 \) term and \( C eq 0 \), this fits the profile of a parabola, specifically oriented along the x-direction.

Key Concepts

Parabola

A parabola is one of the simplest and most common types of conic sections. Think of a parabola as a U-shaped curve. It can open upwards, downwards, to the right, or to the left. Parabolas are defined by their vertex, which is the highest or lowest point of the curve, and their axis of symmetry, a vertical or horizontal line that passes through the vertex dividing the parabola into two mirror-image halves.

In terms of algebra, a parabola can be recognized by the form of its equation. A key feature of parabola equations is that they include a squared term, such as \( y^2 \) or \( x^2 \), but not both at the same time in their basic form. For example, if there is a \( y^2 \) term without an \( x^2 \) term in the form of the equation, it indicates that the parabola opens sideways—either to the right or the left.

Parabolas are common in real-life applications too! They appear in the path of projectiles in physics and are also used in the design of car headlights and satellite dishes to direct signals efficiently.

Conic Section Identification

Identifying the type of conic section represented by an equation starts with recognizing the specific structure of the equation. Conic sections include four different types of curves: circles, ellipses, parabolas, and hyperbolas. Each type has distinctive characteristics based on the coefficients of the squared terms in their algebraic representation.

• Circles have equal coefficients on both \( x^2 \) and \( y^2 \) terms, and they are always positive.
• Ellipses also feature \( x^2 \) and \( y^2 \) terms with the same sign but usually with different coefficients.
• Parabolas contain only one squared term, either \( x^2 \) or \( y^2 \), which helps identify the parabola's direction.
• Hyperbolas have \( x^2 \) and \( y^2 \) terms with opposite signs.

Using these patterns, you can easily identify which type of conic section the equation is describing without needing to graph it. The equation \( 2y^2 - 8y + 3x = 11 \) is identified as a parabola because of the presence of a \( y^2 \) term and absence of \( x^2 \).

Equation Standard Form

The concept of standard form is essential when working with conic sections because it helps in easily identifying the type of conic and its properties.

The general standard form for conic sections can be expressed as \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). The values of \( A \), \( B \), and \( C \) determine the type of conic:

• If \( C eq 0 \) and \( A = B = 0 \), it usually forms a parabola.
• If \( A = C eq 0 \) and both are positive, the equation represents a circle.
• If \( A eq C \) and both have the same sign, the conic is an ellipse.
• If \( A \) and \( C \) have opposite signs, the conic is identified as a hyperbola.

By comparing the given equation to this standard form, you can swiftly deduce the nature of the conic. For example, in our situation, where \( A = 0 \), \( B = 0 \), and \( C = 2 \), it aligns with the description of a parabola.