Problem 73 - Algebra 2

Question

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. \(y^{2}-3 x+6 y+12=0\)

Step-by-Step Solution

Answer

The equation represents a parabola in standard form: \((y+3)^2 = 3(x-1)\).

Step 1: Move Constants to the Other Side

Start by rearranging the given equation so that the constants are on one side. The equation given is \(y^2 - 3x + 6y + 12 = 0\). Move 12 to the right side to get \(y^2 + 6y - 3x = -12\).

Step 2: Complete the Square for y

To complete the square, focus on the terms involving \(y\): \(y^2 + 6y\). Add and subtract \((\frac{6}{2})^2 = 9\) inside the equation. This forms \((y+3)^2\). The equation becomes \((y+3)^2 - 9 - 3x = -12\).

Step 3: Simplify the Equation

Now simplify by moving \(-9\) across to the right side to obtain \((y+3)^2 - 3x = -3\). Rearranging gives \((y+3)^2 = 3x - 3\).

Step 4: Write in Standard Form

To put the equation in standard form, isolate the square term on one side: \((y+3)^2 = 3(x-1)\). This equation is in the standard form of a parabola.

Step 5: Identify the Type of Conic Section

The equation \((y+3)^2 = 3(x-1)\) represents a parabola due to its structure of having only one squared term.

Step 6: Determine the Vertex and Axis of Symmetry

The vertex of the parabola is at \((1, -3)\) and its axis of symmetry is horizontal (parallel to the x-axis).

Step 7: Graph the Equation

With the vertex at \((1, -3)\), plot this point on the graph. As it is a horizontal parabola, draw the parabola opening to the right following the direction suggested by \(3(x-1)\).

Key Concepts

Standard Form

The standard form of a conic section is a way to express the equation so that it's easier to analyze and graph. For conics like parabolas, the standard form highlights the most important features. In general, the standard form of a parabola is given by

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

Here, \((h, k)\) is the vertex of the parabola.
Converting an equation into standard form often involves completing the square to simplify the expression and bring clarity to characteristics like the vertex and direction of the parabola.

Completing the Square

Completing the square is a technique used to convert quadratic equations into a form that reveals significant attributes of conic sections more clearly. To complete the square:

• Focus on the quadratic term and linear term. If we are dealing with an equation such as \(y^2 + 6y\), the key is to form a perfect square trinomial.
• Add and subtract the value needed to complete the square directly in the equation.

For example, in the transformation of \(y^2 + 6y\), we calculate: \(\left(\frac{6}{2}\right)^2 = 9\).Add and subtract this number in the equation to form \((y+3)^2\).
This method is very useful in identifying and rewriting conic equations in their standard forms, making it easier to graph them.

Parabola

A parabola is a specific type of conic section with a characteristic U-shape. Parabolas can open either upwards, downwards, leftwards, or rightwards.The general structure of a parabola's equation will contain one squared term:

• If the squared term is in \(x\), then the parabola is vertical.
• If the squared term is in \(y\), as seen in \((y+3)^2 = 3(x-1)\), then the parabola is horizontal.

In problems involving conic sections, identifying the conic type is essential. The presence of a single squared variable helps in recognizing the equation as that of a parabola.
Once in the correct form, identifying key features like the vertex and direction becomes straightforward, allowing for accurate graphing.

Vertex

The vertex of a parabola is a point that represents its maximum or minimum value. This is the point where the parabola changes direction.For the equation \((y+3)^2 = 3(x-1)\), the vertex can be identified directly:

• The vertex \((h, k)\) is given by \((1, -3)\).

To find the vertex:

• Look within the completed square terms: \(y+3\) tells us the vertex shifts by \(-3\) in the \(y\)-direction.
• Similarly, \(x-1\) indicates a shift by \(+1\) in the \(x\)-direction.

The vertex is a crucial starting point for graphing the parabola accurately. It not only marks the tip or the lowest/highest point of the parabola but also defines the axis of symmetry.