Problem 37 - Precalculus: Functions and Graphs

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

Answer

The graph is a parabola with a horizontal axis.

Step 1: Identify the General Equation Form

The given equation is \( \frac{1}{3}(x+2) = y^2 \). To identify the type of conic section, we need to compare this equation with the standard forms of conic sections' equations. We start by multiplying both sides by 3 to make the equation easier to recognize: \((x + 2) = 3y^2 \). Now, the equation looks similar to \( (x-h) = 4p(y-k)^2 \), which is the form of a parabola with a horizontal axis.

Step 2: Determine the Orientation of the Parabola

The standard form of a parabola with a horizontal axis is \((x-h) = 4p(y-k)^2\). In our equation, \((x+2) = 3y^2\), it can be rewritten as \((x-(-2)) = 3(y-0)^2\). Here, \(h = -2\), \(k = 0\), and \(4p = 3\). The presence of \(y^2\) confirms that the parabola opens horizontally.

Step 3: Confirm the Type of Conic

To conclude, the form \((x-h) = 4p(y-k)^2\) with \(y^2\) and no \(x^2\) indicates it is a parabola. Other conic sections, such as circles, ellipses, and hyperbolas, involve both \(x^2\) and \(y^2\) terms in their quadratic forms. Therefore, this conic section is a parabola.

Key Concepts

Parabola

A parabola is one of the fascinating shapes in geometry known as conic sections. It has a unique, U-shaped curve that is symmetrical along its axis. Parabolas have distinct properties that make them stand out, such as reflective symmetry and the property that any path taken by a projectile under uniform gravity and no other forces is a parabola. There are two primary types of parabolas based on their orientation:

• Vertical Parabola: Opens up or down.
• Horizontal Parabola: Opens left or right.

In a coordinate plane, parabolas can be conveniently identified using their equations, which vary depending on their orientation.A vertical parabola has the form \( (y - k)^2 = 4p(x - h) \), and a horizontal parabola has the form \( (x - h)^2 = 4p(y - k) \). Notice that in a vertical parabola, the \(x\) term is squared, while in a horizontal, the \(y\) term is squared. This distinction is key in identifying the direction in which a parabola opens.

Equation of Parabola

The equation of a parabola is essential to understanding its characteristics and orientation. It provides critical information about how the curve behaves on a graph. Standard parabolic equations have specific forms that directly tell you whether the parabola opens vertically or horizontally. The general equations are as follows:

• Vertical: \( (x - h)^2 = 4p(y - k) \)
• Horizontal: \( (y - k)^2 = 4p(x - h) \)

Here, \((h, k)\) represents the vertex of the parabola, which is the point where it changes direction. The parameter \(p\) indicates the distance between the vertex and the focus, which is a critical point that helps describe the parabola's steepness and shape.
In the horizontal equation, if you see a \(y^2\) term and no \(x^2\) term, you know immediately that the parabola must be horizontal, as noted in the example equation \((x+2) = 3y^2\). Here, \(h = -2\), \(k = 0\), and \(4p = 3\), indicating the parabola opens to the side horizontally.

Conic Section Identification

Conic sections are the curves formed by the intersection of a plane and a double-napped cone. These curves, namely circles, ellipses, hyperbolas, and parabolas, have distinct equations and characteristics that differentiate them.To identify a conic section from its equation, it's crucial to look at the form of the equation and the types of terms involved. For instance:

• Circle: Both \(x^2\) and \(y^2\) terms have the same coefficient.
• Ellipse: Both \(x^2\) and \(y^2\) terms appear but with different coefficients.
• Hyperbola: \(x^2\) and \(y^2\) terms have opposite signs.
• Parabola: Only one of the variables, either \(x\) or \(y\), is squared.

In the given equation \( (x+2)=3y^2 \), there is no \(x^2\) term, but the \(y^2\) term is present. This absence of \(x^2\) helps identify it as a parabola, particularly a horizontal one, leading to a straightforward identification among conic sections.