Problem 17 - Algebra for College Students

Question

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ y^{2}-12 x+24=0 $$

Step-by-Step Solution

Answer

Vertex at (2,0), focus at (5,0), directrix x=-1.

Step 1: Rearrange the Equation

First, rearrange the given equation to make it easier to complete the square. Start with the given equation: \(y^2 - 12x + 24 = 0\). Move the constant to the other side of the equation to isolate the \(x\)-related terms: \(y^2 = 12x - 24\).

Step 2: Complete the Square on Y-side

Since the equation involves \(y^2\), we suspect the parabola opens horizontally. To complete the square involving \(y\), rewrite it as: \((y - 0)^2\). This simplifies our equation to: \((y - 0)^2 = 12x - 24\).

Step 3: Rewrite the Equation in Standard Form

After completing the square, the equation should resemble the standard form of a horizontal parabola: \((y - k)^2 = 4p(x - h)\). Factor out the 12 from the right side: \(12(x - 2)\). Thus, the equation becomes \((y - 0)^2 = 12(x - 2)\).

Step 4: Identify the Vertex

In the standard form \((y - k)^2 = 4p(x - h)\), the vertex is \((h, k)\). For \((y - 0)^2 = 12(x - 2)\), the vertex \((h, k)\) is \((2, 0)\).

Step 5: Find the Focus

Determine \(4p\) from the equation \((y - 0)^2 = 12(x - 2)\). This gives \(4p = 12\), so \(p = 3\). The focus for a horizontal parabola at vertex \((h, k) = (2, 0)\) is \((h + p, k)\), which is \((5, 0)\).

Step 6: Determine the Directrix

The directrix is a vertical line \(x = h - p\). Using the vertex \((h, k) = (2, 0)\) and \(p = 3\), the directrix is \(x = 2 - 3 = -1\).

Step 7: Sketch the Parabola

Plot the vertex \((2, 0)\), the focus \((5, 0)\), and the directrix line \(x = -1\). Draw the parabola opening to the right as it is a horizontal parabola with the given equation, symmetrically aligned relative to the vertex.

Key Concepts

Vertex

The vertex of a parabola is an important point where the curve changes direction. For parabolas, the vertex can be thought of as the "peak" or the "lowest point", but it depends on the curve's orientation.
In our equation, \((y - 0)^2 = 12(x - 2)\), the parabola is horizontal. Therefore, the vertex is given by the point \((h, k)\), which is clearly noted as \((2, 0)\) from our standard form.
The vertex serves as a point of symmetry:

• For vertical parabolas, it determines the minimum or maximum value of the parabola.
• For horizontal parabolas, it helps us understand the leftmost or rightmost point.

Knowing the vertex is crucial since it provides a starting point for graph sketching and further analysis.

Focus

The focus of a parabola is a point from which distances to any point on the curve are equidistant when compared to the directrix. It is essential in understanding the parabola's width and how "open" it appears.
The focus is determined by the \(p\) value from the equation. In our exercise, \((y - 0)^2 = 12(x - 2)\), we find that \(4p = 12\), which gives \(p = 3\). This means the focus is located at \((h + p, k)\), yielding the coordinates \((5, 0)\).
The focus tells us:

• How "deep" the parabola opens in the direction of the focus.
• It acts as a reference point for defining the curved path of the parabola.

Understanding the focus allows us to construct more precise and accurate graphs.

Directrix

The directrix of a parabola is a line that helps us define its shape, much like the focus. It is always perpendicular to the axis of symmetry of the parabola.
In this scenario, for our parabola \((y - 0)^2 = 12(x - 2)\), the directrix is calculated as \(x = h - p\). With our values \(h = 2\) and \(p = 3\), the directrix is the vertical line \(x = -1\).
What to know about the directrix:

• It helps maintain the balance between the focus and any point on the parabola.
• It gives us a boundary for defining the parabola's curve.

Combining the concept of the focus and directrix ensures you can fully specify a parabolic curve.

Graph Sketching

Sketching the graph of a parabola involves plotting points like the vertex, focus, and directrix, allowing you to visualize the graph more intuitively.
For this parabola \((y - 0)^2 = 12(x - 2)\), you'll start with:

• Plotting the vertex at \((2, 0)\).
• Adding the focus point at \((5, 0)\).
• Drawing the directrix as a vertical line \(x = -1\).

With these points and line in place, you'll have a clearer picture of how the parabola should be shaped as it opens to the right.Remember these tips when sketching:

• The parabola is symmetrical around its vertex.
• The direction it opens (left, right, up, down) relies on its orientation in the equation.

Graph sketching combines mathematical precision with visual interpretation, creating a complete understanding of the parabola.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a form where the parabola's characteristics like the vertex become more apparent.
In our exercise with \(y^2 - 12x + 24 = 0\), completing the square helps in defining the proper form of \((y - k)^2 = 4p(x - h)\).
The steps are:

• Rearrange the equation to isolate terms involving \(y\) (or \(x\) for vertical parabolas).
• Factor out any coefficients from the squared term variable.

Applying these steps yields \((y - 0)^2 = 12(x - 2)\), making it easy to find the vertex, focus, and directrix.
This technique is vital for:

• Converting any standard quadratic equation into a more usable form.
• Understanding the inherent properties of parabolic curves clearly.

Mastering completing the square can simplify solving quadratic problems and aid in graphical analysis.