Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. \(y^{2}+12 x-6 y-51=0\)

Step-by-Step Solution

Answer

Yes, it is a parabola. Standard form: \((y - 3)^2 = -12(x - 5)\).

Step 1: Identify the Type of Equation

Identify the form of the given equation. The equation provided is: \(y^{2} + 12x - 6y - 51 = 0\). We notice that this equation is a quadratic because of the \(y^2\) term. A parabola in standard form typically involves one squared term, either \(x^2\) or \(y^2\). Hence, this equation could represent a parabola.

Step 2: Rearrange the Equation

Group the terms involving \(y\) together: \(y^2 - 6y + 12x - 51 = 0\). To rewrite in standard form, we aim to complete the square for the \(y\) terms.

Step 3: Complete the Square for the y Terms

To complete the square for \(y^2 - 6y\), take half of the coefficient of \(y\) which is \(-6\), giving \(-3\), and square it to get \(9\). Add and subtract \(9\) to the equation: \(y^2 - 6y + 9 + 12x - 51 - 9 = 0\).

Step 4: Simplify the Equation

The equation now is \((y - 3)^2 + 12x - 60 = 0\). Simplify this to \((y - 3)^2 = -12x + 60\).

Step 5: Express in Standard Form

Rewrite the equation into the standard form of a parabola, which looks like \((y - k)^2 = 4p(x - h)\). Here, move all terms except \((y - 3)^2\) to one side: \((y - 3)^2 = -12(x - 5)\).

Step 6: Identify Parameters and Confirm

Compare \((y - 3)^2 = -12(x - 5)\) with \((y - k)^2 = 4p(x - h)\). Here, the vertex is at \((h, k) = (5, 3)\) and \(4p = -12\), giving \(p = -3\). This equation confirms it is a parabola that opens to the left.

Key Concepts

Quadratic Equation

In the world of algebra, quadratic equations hold a special place. They are expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).

These equations can appear in different forms and reveal a variety of solutions. Typically, they include a squared term, making them non-linear.

Importantly, not all quadratic equations strictly involve \(x\); some, as in our case, involve \(y^2\).

• If a squared term is \(x^2\), it describes vertical parabolas.
• If it is \(y^2\), it describes horizontal parabolas.

Quadratic equations form the foundational structure from which parabolas are defined.

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a form that is easier to work with.

In our exercise, we needed to arrange the equation \(y^2 - 6y + 12x - 51 = 0\) to complete the square for \(y\).

Here's how it works:

• Take the coefficient of \(y\), which is \(-6\) in this case.
• Halve it to get \(-3\), then square \(-3\) to get \(9\).
• Add and subtract this square inside the equation: \(y^2 - 6y + 9 - 9 = 0\).

This adjustment creates a perfect square trinomial \((y-3)^2\), allowing you to express the quadratic in a more manageable form for analysis and calculation. It's like polishing a rough gemstone into a shining jewel in mathematical analysis.

Standard Form of a Parabola

The standard form of a parabola helps us quickly identify key features like its vertex and the direction it opens.

Typically, for parabolas opening horizontally, the standard form is \((y-k)^2 = 4p(x-h)\). In our case, the rearranged equation \((y - 3)^2 = -12(x - 5)\) fits this model.

The key components in this form include:

• \(h\) and \(k\), which specify the vertex of the parabola.
• \(4p\), which helps determine the parabola's direction and focal width.

Using standard form, it's straightforward to decipher much about the parabola from just a glance—be it finding the vertex, understanding its symmetry, or grasping how wide or steep it appears.

Vertex of a Parabola

The vertex of a parabola is a point of utmost importance. It serves as the peak or the lowest point of the curve, depending on its orientation.

In standard form \((y-k)^2 = 4p(x-h)\), the vertex is located at \(h, k\). For our equation \((y - 3)^2 = -12(x - 5)\), the vertex is at \(5, 3\).

The vertex not only indicates a pivotal point in the parabola's path but also symmetrically divides the parabola.

• It helps in understanding the graph's layout.
• It provides a quick reference as to how the parabola behaves.

Identifying and using the vertex allows for complete insight into graphed quadratic relationships, making it a crucial aspect of parabola analysis.