Problem 45 - Precalculus: Functions and Graphs

Question

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x^{2}+3 x=3 y-6$$

Step-by-Step Solution

Answer

The graph of the equation is a parabola with a vertical axis.

Step 1: Rewrite the Equation

First, rewrite the equation to standard form by rearranging terms. The given equation is \(x^2 + 3x = 3y - 6\). Rearrange it to: \(x^2 + 3x - 3y = -6\).

Step 2: Move Terms with Same Variable

Separate the terms with \(y\) to one side and constant terms on the other. Here, rearrange to: \(x^2 + 3x + 6 = 3y\), making it easier to identify the type of conic.

Step 3: Complete the Square

To reveal the structure more clearly, complete the square for the \(x\) terms. The expression \(x^2 + 3x\) can be completed as follows: \((x^2 + 3x + \frac{9}{4}) - \frac{9}{4}\), which becomes \((x + \frac{3}{2})^2 - \frac{9}{4}\).

Step 4: Substitute Back Into Equation

Substitute the completed square back: \((x + \frac{3}{2})^2 - \frac{9}{4} + 6 = 3y\). This simplifies to \((x + \frac{3}{2})^2 = 3y - \frac{15}{4}\).

Step 5: Identify the Type of Conic Section

The equation \((x + \frac{3}{2})^2 = 3(y - \frac{5}{4})\) is in the form \((x - h)^2 = 4p(y - k)\). This is the standard form of a parabola with a vertical axis.

Key Concepts

Parabola

One of the four types of conic sections, a parabola is typically formed by cutting a cone with a plane that is parallel to one side of the cone. A parabola has a unique U-shape and can open either upwards, downwards, to the left, or to the right.

Parabolas have a point called the vertex, which is the point where the curve is at its maximum or minimum, and an axis of symmetry that divides the parabola into two congruent halves. The vertex form of a parabola helps easily determine its direction and supports further analysis, such as its reflection properties, which are widely used in satellite dishes and automobile headlights.

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This is particularly useful for simplifying and solving equations. For instance, if you start with an expression like \(x^2 + bx\), you want to add and subtract the square of half the coefficient of \(x\).

• Find half of the coefficient of \(x\), which is \(\frac{b}{2}\).
• Square it: \((\frac{b}{2})^2\).
• Add and subtract this value within the expression to form a complete square: \(x^2 + bx + (\frac{b}{2})^2 - (\frac{b}{2})^2\).

This results in \((x + \frac{b}{2})^2\), making the expression easier to manage. Completing the square is a valuable method not only in algebra but also in calculus, where it aids in integrations and finding maximum or minimum values of functions.

Equation Rearrangement

Rearranging an equation involves changing its format or structure to reveal more meaningful insights. In the context of conic sections like parabolas, rearranging an equation helps in identifying the exact form the equation takes, which can be pivotal in graphing or further analysis.

For instance, taking the equation \(x^2 + 3x = 3y - 6\), you can rearrange it by gathering like terms and positioning them to better suit completing the square. This step is conducted to expose the equation's deeper geometric meaning and connection to standard forms, often guiding subsequent steps like graph plotting or vertex identification.

Standard Form of a Parabola

The standard form of a parabola with a vertical axis is \((x - h)^2 = 4p(y - k)\), where \((h, k)\) is the vertex of the parabola. This equation shows the symmetrical nature of the parabola, its direction (up or down for vertical), and how "wide" or "narrow" it appears.

• The parameter \(p\) indicates the distance from the vertex to the focus, which also equals the distance from the vertex to the directrix, a fixed line used to define the parabola.
• A positive \(p\) means the parabola opens upwards, and a negative \(p\) means it opens downwards.

Knowing the standard form simplifies the process of graphing parabolas and provides a quick snapshot of its essential properties. Understanding this form is integral in various fields, including physics, where parabolic paths are common, such as the trajectory of projectiles.