Problem 63 - Precalculus with Limits

Question

In Exercises 57-72, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(y^2+12x+4y+28=0\)

Step-by-Step Solution

Answer

The graph of the equation \(y^2+12x+4y+28=0\) is a parabola that opens to the left.

Step 1: Simplify the given equation

The first step is to simplify the given equation and group similar terms together. Let's rewrite the equation to do this: \(y^2 + 4y + 12x + 28 = 0\). This can be further simplified as: \(y^2 + 4y = -12x - 28\).

Step 2: Complete the square for y terms

To complete the square on the left-hand side of the equation, an appropriate number must be calculated and added to both sides to produce a perfect square trinomial. Half the coefficient of 'y' is 2. Squaring 2 gives us 4. Hence we add 4 to both sides: \(y^2 + 4y + 4 = -12x - 28 + 4\). This simplifies to: \((y + 2)^2 = -12x - 24\).

Step 3: Classify the graph

Our equation now takes the form \((y + 2)^2 = -12x - 24\) or equivalently \(y = (-6\sqrt{x + 2})^2 -2)\). This is the standard form of a parabola that opens to the left or right. Since the coefficient of x is negative, thus we know that the parabola opens towards the left.

Key Concepts

Parabola

A parabola is a unique type of curve often seen in algebra. It is defined as a set of all points that are equidistant from a fixed point called the focus and a fixed line called the directrix.
In the context of graphs, a parabola can open upwards, downwards, left, or right, depending on its equation. For a standard parabolic equation in the form (ax^2 + bx + c = y), the parabola typically opens vertically (up or down). However, when the equation is structured like (y^2 + bx = c), the parabola opens horizontally (left or right).
In the given exercise, the equation was restructured to ((y + 2)^2 = -12x - 24). With y squared and equal to an expression in x, it indicates a horizontally oriented parabola. Specifically, the negative coefficient before x confirms that this parabola opens to the left.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This is useful for rewriting equations in a standard form that is easier to analyze graphically.
The process involves finding a specific constant to add to both sides of the equation to complete the square. For the given quadratic equation in the y terms, (y^2 + 4y), we first identify the coefficient of the linear term (which is 4 in this case).
We take half of this coefficient, giving us 2, and then square it, resulting in 4. By adding 4 to both sides of the equation, the left side becomes a perfect square, ((y + 2)^2). This modification simplifies solving and graphing the equation.

Graph Classification

Graph classification involves determining the type of conic section an equation represents. Conic sections include parabolas, circles, ellipses, and hyperbolas, each with its own unique characteristics and standard forms.
To classify the equation ((y + 2)^2 = -12x - 24), examining its structure is crucial. In any conic classification, the relationship between the variables and the coefficients provides critical insight.
Parabolas are identified when the equation can be reduced to show one variable is proportional to the square of another, as we see with (y terms squared equaling expressions containing x). Here, having only one squared term confirms it is a parabola. The negative coefficient of x in the equivalently transformed equation highlights that the graph opens sideways, specifically to the left, clearly classifying it as a horizontal parabola.