Problem 33
Question
Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$\frac{1}{3}(x+2)=y^{2}$$
Step-by-Step Solution
Verified Answer
The graph of the equation is a parabola with a horizontal axis.
1Step 1: Rewrite the Equation in Standard Form
The given equation is \( \frac{1}{3}(x+2) = y^2 \). Multiply every term by 3 to eliminate the fraction, giving us \( x + 2 = 3y^2 \). Then, modify the equation to obtain \( x = 3y^2 - 2 \). Now identify the standard form, which resembles \( x = ay^2 + by + c \).
2Step 2: Determine the Form of the Conic Section
The equation \( x = 3y^2 - 2 \) corresponds to the standard form of a parabola with a horizontal axis, represented by \( x = ay^2 + by + c \), where the squared term is \( y^2 \). In this case, the coefficient of \( y^2 \) is 3, confirming it is a parabola opening in the horizontal direction.
Key Concepts
ParabolasGraphing EquationsStandard Form of Conic Sections
Parabolas
A parabola is a special type of curve on a graph, defined by its symmetrical U-shape. It can open either upwards, downwards, or sideways. Parabolas are one of the simplest forms of conic sections, and they appear frequently in mathematics due to their unique properties.
An important feature of parabolas is their vertex, which is the highest or lowest point of the curve depending on its orientation. Parabolas also have axes of symmetry, which help determine their orientation—either vertical or horizontal. The direction in which a parabola opens is determined by the squared component in its equation.
For example, if the variable $x$ is squared, the parabola opens vertically. Conversely, if the variable $y$ is squared, the parabola opens horizontally. These characteristics are pivotal when identifying a graph's orientation after converting an equation into standard form.
An important feature of parabolas is their vertex, which is the highest or lowest point of the curve depending on its orientation. Parabolas also have axes of symmetry, which help determine their orientation—either vertical or horizontal. The direction in which a parabola opens is determined by the squared component in its equation.
For example, if the variable $x$ is squared, the parabola opens vertically. Conversely, if the variable $y$ is squared, the parabola opens horizontally. These characteristics are pivotal when identifying a graph's orientation after converting an equation into standard form.
Graphing Equations
Graphing equations involves plotting points on a coordinate plane to represent solutions to those equations. Understanding how different types of equations influence the shape and position of their graphs is essential for visual interpretation and analysis.
When graphing a parabola, particular attention is given to identifying the main components, such as the vertex, axis of symmetry, and direction of opening—using the equation's coefficients. It's important to rewrite equations in a user-friendly form to make graphing easier.
For the equation \(x = 3y^2 - 2\), the graphing process involves:
When graphing a parabola, particular attention is given to identifying the main components, such as the vertex, axis of symmetry, and direction of opening—using the equation's coefficients. It's important to rewrite equations in a user-friendly form to make graphing easier.
For the equation \(x = 3y^2 - 2\), the graphing process involves:
- Identifying it as a parabola because the \(y^2\) term is isolated.
- Noting the horizontal orientation as the \(y\) term is squared.
- Using parameters such as the coefficient (3) to determine how "wide" or "narrow" the parabola will appear.
Standard Form of Conic Sections
The standard form of conic sections provides a blueprint for recognizing and analyzing the different types of curves, such as parabolas, circles, ellipses, and hyperbolas. Each conic section has a specific standard form depending on its characteristics.
For parabolas, the standard forms are \(y = ax^2 + bx + c\) for vertical parabolas and \(x = ay^2 + by + c\) for horizontal parabolas. These forms indicate which variable is squared and the direction of the parabola.
In our example, \(x = 3y^2 - 2\) resembles the standard form of a horizontally oriented parabola. Examining this form helps quickly identify key aspects like axis orientation and opening direction.
For parabolas, the standard forms are \(y = ax^2 + bx + c\) for vertical parabolas and \(x = ay^2 + by + c\) for horizontal parabolas. These forms indicate which variable is squared and the direction of the parabola.
In our example, \(x = 3y^2 - 2\) resembles the standard form of a horizontally oriented parabola. Examining this form helps quickly identify key aspects like axis orientation and opening direction.
- The presence of \(y^2\) indicates a horizontal parabola.
- The absence of linear terms (terms with single power of variables) besides the squared term confirms the parabola's symmetry about a specific axis.
Other exercises in this chapter
Problem 32
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Exer. 31-34: Find an equation for the set of points in an \(x y\)-plane that are equidistant from the point \(P\) and the line \(l\). $$ P(-6,3) ; \quad l: x=-2
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