Problem 4 - A Graphical Approach to Precalculus with Limits

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. $$x=3 y^{2}+5 y-6$$

Step-by-Step Solution

Answer

The equation represents a horizontally oriented parabola.

Step 1: Recognize the Equation Type

The given equation is \( x = 3y^2 + 5y - 6 \). This equation expresses \( x \) as a function of \( y \), rather than the common \( y \) in terms of \( x \). This indicates the equation might define a conic section rotated or oriented differently on the coordinate plane.

Step 2: Identify the Conic Section

Observe the term \( 3y^2 \) in the equation. This represents a quadratic term, suggesting that this conic will have terms raised to the second power. Because the equation has one squared term (\( y^2 \)) and no \( xy \) term, it fits the pattern of a parabola. A single squared variable indicates a parabola.

Step 3: Conclusion on Orientation

Since the equation expresses \( x \) in terms of \( y^2 \), and \( x \) is linear, the parabola is oriented such that it opens horizontally (either left or right) rather than the usual vertical opening. It is rotated compared to a standard \( y = ax^2 + bx + c \) parabola.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two. Typically, it is written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. Key characteristics of quadratic equations include:

• The highest power of the variable is 2.
• It can include a linear term (\( bx \)) and a constant term (\( c \)).
• It can be factored, solved using the quadratic formula, or completed through the square.

In the context of conic sections, identifying the presence of a single squared term can help determine the type of conic. If you observe one squared term, like \( x = 3y^2 + 5y - 6 \), it's indicative of a parabola in conic sections, rather than a circle or ellipse where both variables might be squared.

Parabola

A parabola is a specific type of conic section defined by a quadratic equation where only one of the variables is squared. It appears as a U-shaped curve in traditional graphs when expressed as \( y = ax^2 + bx + c \), opening upwards or downwards depending on the sign of \( a \). However, in the equation \( x = 3y^2 + 5y - 6 \), the variable \( y \) is squared. This indicates an alternate orientation. This change affects the direction in which the parabola opens.Characteristics of a parabola:

• A vertex, which is the highest or lowest point depending on the parabola's orientation.
• An axis of symmetry that passes through the vertex, dividing the parabola into two mirror-image halves.
• The direction they open (either vertically or horizontally) can be determined by the position of the squared term.

Understanding these features helps in visualizing and sketching the parabola correctly based on the given quadratic form.

Graph Orientation

Understanding graph orientation in the context of quadratic equations is crucial for correctly interpreting conic sections. For standard quadratic equations \( y = ax^2 + bx + c \):- The graph opens upwards if \( a > 0 \) and downwards if \( a < 0 \).However, when the quadratic equation is in the form of \( x = ay^2 + by + c \), like our original equation \( x = 3y^2 + 5y - 6 \):

• The graph opens horizontally (right if \( a > 0 \), left if \( a < 0 \)).
• The orientation is affected because the variable associated with the squared term indicates how the parabola is aligned on the plane.

Recognizing the orientation of the parabola helps to predict the graph's behavior without plotting each point, allowing us to visualize its shape and direction based on the equation's structure.