Problem 26 - College Algebra

Question

For the following exercises, rewrite the given equation in standard form, and then determine the vertex \((V),\) focus \((F),\) and directrix \((d)\) of the parabola. $$ y^{2}-24 x+4 y-68=0 $$

Step-by-Step Solution

Answer

Vertex: \((-3, -2)\), Focus: \((3, -2)\), Directrix: \(x = -9\).

Step 1: Rewrite the equation by organizing terms

Start by grouping the equation in terms of \(x\) and \(y\). Move the \(x\)-terms to the other side of the equation:\[ y^2 + 4y = 24x + 68 \]

Step 2: Complete the square for the y-terms

To complete the square for \(y\), take half of the coefficient of \(y\) in \(y^2 + 4y\), square it, and add it to both sides. The coefficient is 4, half of it is 2, and squaring it gives 4:\[ y^2 + 4y + 4 = 24x + 68 + 4 \]Simplifying gives:\[ (y + 2)^2 = 24x + 72 \]

Step 3: Write the equation in standard form

The equation now can be written as a completed square on one side with \(x\) linear on the other:\[ (y + 2)^2 = 24(x + 3) \]This represents the standard form \((y - k)^2 = 4p(x - h)\), where \(h = -3\), \(k = -2\), and \(4p = 24\).

Step 4: Identify vertex, focus, and directrix

From the standard form \((y - k)^2 = 4p(x - h)\), identify \(p = 6\), since \(4p = 24\). The vertex \(V\) is \((-3, -2)\). The focus \(F\) is \(6\) units right of the vertex at \((3, -2)\). The directrix \(d\) is \(x = h - p = -3 - 6 = -9\).

Key Concepts

Completing the Square

Completing the square is a technique to transform a quadratic equation into a perfect square trinomial. This process makes it easier to rewrite the quadratic in different forms, such as the vertex form. To complete the square, follow these steps:

• Re-organize the terms by moving constants to the other side of the equation.
• Focus on the variable terms. In our case, we worked with the terms involving the variable \(y\).
• Take half of the coefficient of the linear term \(y\), square it, and add to both sides. For instance, with \(y^2 + 4y\), here, the coefficient of \(y\) is 4. Half of 4 is 2, and squaring it gives 4.
• This adjustment gives a perfect square trinomial, \((y + 2)^2\), equal to the adjusted terms on the other side of the equation.

This method not only facilitates a transition into standard and vertex forms but also enhances understanding of the parabola's geometric properties.

Vertex Form

The vertex form of a parabola's equation reveals key information about its graph, specifically the vertex, which is the highest or lowest point of the parabola. The equation is written as:
\[ (y - k)^2 = 4p(x - h) \]
Here, \( (h, k) \) is the vertex of the parabola.
For this exercise, after converting the equation by completing the square, we have:
\[ (y + 2)^2 = 24(x + 3) \]
We can compare this to the general vertex form, identifying the vertex as \((-3, -2)\).Having the equation in vertex form is immensely helpful. It tells us:

• The direction and width of the parabola by the term \(4p\), where \(p\) is the distance from the vertex to the focus or directrix.
• The precise location of the vertex, helping to sketch the graph accurately.

Focus and Directrix

The focus and directrix are key features defining the shape and orientation of a parabola. They are pivotal in understanding the geometrical properties:

• The focus is a point inside the parabola from which distances to any point on the parabola are equidistant to the directrix.
• The directrix is a line that divides the parabola symmetrically and is used to define its shape.

In our equation, \[ (y + 2)^2 = 24(x + 3) \]identifies \(4p = 24\). Hence, \(p = 6\). This shows:

• The focus is found \(p\) units to the right of the vertex for this horizontal parabola. So, the focus is \((3, -2)\).
• The directrix lies \(p\) units to the left of the vertex at \(x = -9\).

These points help in graphing and understanding the orientation and dimensions of the parabola.

Standard Form of a Parabola

The standard form of a parabola is crucial as it provides a simple way to analyze and graph the parabola's characteristics:
For parabolas that open left or right, the form is:\[ (y - k)^2 = 4p(x - h) \]
For those that open up or down:\[ (x - h)^2 = 4p(y - k) \]
The provided equation was transformed into the standard form:\[ (y + 2)^2 = 24(x + 3) \]This implies a horizontal orientation with vertex at \((-3, -2)\). This compact expression allows easy identification of:

• The vertex \((h, k)\) which remains constant regardless of its precise form.
• The factor \(4p\) determines the focus and directrix position relative to this vertex.

Remember, the standard form offers a straightforward way to see the orientation and general geometry of the parabola, whether you're solving problems analytically or graphing them visually.