Problem 4 - Intermediate Algebra

Question

The graph of each equation is a parabola. Determine whether the parabola opens upward, downward, to the left, or to the right. Do not graph. $$x=3 y^{2}+2 y-5$$

Step-by-Step Solution

Answer

The parabola opens to the right.

Step 1: Determine the Form

The given equation is \( x = 3y^2 + 2y - 5 \). This equation is in the form \( x = ay^2 + by + c \), which is characteristic of a parabola that opens horizontally (either to the left or to the right).

Step 2: Identify the Direction

For a parabola in the form \( x = ay^2 + by + c \), the coefficient \( a \) determines the direction of the opening. If \( a > 0 \), the parabola opens to the right. If \( a < 0 \), it opens to the left.

Step 3: Apply the Rule

In the equation \( x = 3y^2 + 2y - 5 \), the coefficient of \( y^2 \) is \( 3 \), which is greater than 0. According to the rule, since \( a = 3 > 0 \), the parabola opens to the right.

Key Concepts

Quadratic Equations

Quadratic equations are central in mathematics, representing functions where the highest degree term is squared. Generally, a quadratic equation is written in the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( x \) represents an unknown variable. What makes these equations special is their parabolic graph, forming a smooth, U-shaped curve called a parabola.
\( a \), the leading coefficient, plays a major role in defining the quadratic's properties:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.

However, not all quadratic equations involve \( x \) as the variable to be squared. Sometimes, \( y \) can be squared, leading to parabolas that open horizontally, which is crucial to understand when dealing with different orientations.

Horizontal Parabolas

Horizontal parabolas appear when the equation is structured such that \( y \), instead of \( x \), is squared. This means you would generally have an equation like \( x = ay^2 + by + c \).
Unlike the more familiar vertical equations, here the orientation on the plane changes. The parabola opens either to the left or to the right, depending on the leading coefficient \( a \):

• If \( a > 0 \), the parabola opens to the right.
• If \( a < 0 \), it opens to the left.

This distinction is key not only in solving mathematical problems but also in understanding their geometric implications, such as how objects might move or project in physical space. In real-world contexts like projectile motion, recognizing the type of parabola is vital for accurate model predictions.

Parabolic Graph Orientation

Understanding the orientation of a parabolic graph helps in predicting how the graph behaves without plotting it. Orientation, in simple terms, is the direction in which the parabola opens. Vertically oriented parabolas open upwards or downwards, while horizontally oriented parabolas open leftwards or rightwards.
Key to determining orientation is recognizing the form of the equation:

• Standard form \( y = ax^2 + bx + c \), results in a vertical parabola.
• Alternative form \( x = ay^2 + by + c \), results in a horizontal parabola.

Having clear guidelines based on the coefficient \( a \) allows you to predict behavior without graphing: upwards, downwards, right, or left. Understanding orientation not only aids in visualizing the graphs but is also essential for interpreting solutions to real-world scenarios, like structural engineering and physics.