Problem 22 - 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. $$x^{2}+2 x=-4 y$$

Step-by-Step Solution

Answer

The equation represents a parabola.

Step 1: Write the Equation in Standard Form

Start by moving all terms involving x and y to one side of the equation to identify the type of conic section. The given equation is \( x^2 + 2x = -4y \). Rearranging terms, we have \( x^2 + 2x + 4y = 0 \).

Step 2: Compare with Standard Conic Sections

The general form for any conic section is \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). For our equation \( x^2 + 2x + 4y = 0 \), the coefficients are \( A = 1, B = 0, C = 0, D = 2, E = 4, F = 0 \), and no second power term for y, indicating \( C = 0 \).

Step 3: Determine Type Using Condition Criteria

Check if the given equation matches one of the criteria for conic sections:- Circle if \( A = C \) and \( B = 0 \) - Ellipse if \( A \, \&\, C \) are both positive or both negative - Parabola if either \( A \) or \( C \) is zero - Hyperbola if \( A \cdot C < 0 \)Since \( C = 0 \), and only \( x^2 \) is present, this matches the form of a parabola.

Step 4: Identify the Conic Section

Since the condition for a parabola is that either \( A = 0 \) or \( C = 0 \), and in our case \( C = 0 \), this equation represents a parabola.

Key Concepts

Parabola

A parabola is a unique shape that forms a U-like curve, which is defined scientifically as a plane curve that is mirror-symmetrical. It is a result of the intersection of a plane with a cone, parallel to one of its sides. Parabolas appear not only in mathematics but in many aspects of daily life, such as in satellite dishes and car headlights.

When it comes to the standard form of its equation, a parabola can be defined in the forms such as:

• Vertical Parabolas: The equation is generally written as \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients that determine the parabola's shape and position.
• Horizontal Parabolas: These are expressed in the form of \( x = ay^2 + by + c \).

One of the key characteristics of a parabola is its vertex. This is the point where the parabola changes direction, which is also the minimum or maximum point of the curve. Additionally, parabolas have an axis of symmetry which passes through the vertex, dividing the curve into two congruent parts.

Equation of Conic Sections

Conic sections are various shapes that can arise from slicing a cone with a plane. They include parabolas, ellipses, circles, and hyperbolas. Each type has a unique set of properties and equations associated with it.

The general equation of a conic section in its broadest form can be represented as:

• \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)

Here, \( A \), \( B \), \( C \), \( D \), \( E \), and \( F \) are constants that determine the shape and position of the conic.

• Circle: Occurs when \( A = C \), \( B = 0 \).
• Ellipse: Formed when \( A eq C \), \( A \, C > 0 \).
• Parabola: Appears when either \( A \) or \( C \) is zero.
• Hyperbola: Defined when \( A \cdot C < 0 \).

Standard Form of Conic Sections

To make the identification and graphing of conic sections easier, their equations are often converted into a standard form. This process involves completing the square, a technique used to simplify quadratic forms.

Among the standard forms, each conic section has its own specific equation:

• Circle: \( (x - h)^2 + (y - k)^2 = r^2 \)
• Ellipse: \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \)
• Parabola: \( (x-h)^2 = 4p(y-k) \) for a vertical parabola, or \( (y-k)^2 = 4p(x-h) \) for a horizontal parabola.
• Hyperbola: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \)

Completing the square is key especially for transforming equations into these forms. In doing so, graphing, modeling, and interpreting are simplified, providing a clearer understanding of each shape's geometrical properties and symmetries.