Problem 40 - Algebra and Trigonometry

Question

In Exercises 35–42, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$ (y+4)^{2}=12(x+2) $$

Step-by-Step Solution

Answer

The vertex of the parabola is at (-2, -4), the focus is at (-5,-4) and the directrix is at x=1. Plotting these points and lines, we generate the graph of the given parabola.

Step 1: Rewrite the Equation

To rearrange the equation in the standard form, rewrite \((y+4)^2=12(x+2)\) to \(4(3)(y + 4) = (x + 2)\) Now it should be in the form \(4p(y - k) = (x - h)^2\). Therefore, the vertex (h, k) is at (-2, -4) and p equals 3.

Step 2: Find the Vertex, Focus and Directrix

The vertex, as found from the standard form, is at (-2, -4). The value of p is 3, which we obtained from the rearranged equation. Since this is a horizontally opening parabola, the focus, a distance p from the vertex, will be at a point (-2-3, -4) which gives (-5, -4). The directrix, which is also p units from the vertex but on the opposite side, will be the line given by the equation x = -2 + 3, so x = 1.

Step 3: Graph the Parabola

First position the vertex at (-2, -4). Then draw the directrix at x = 1. The focus point is at (-5, -4). Now, generate the parabola opening horizontally to the left, with the given focus point inside the parabola and the given directrix line outside the parabola on the right.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is the point where it changes direction. This is either the maximum or minimum point, depending on how the parabola opens. In our example,

• the given equation is \( (y+4)^2 = 12(x+2) \).
• Rewriting it in the standard form helps identify the vertex.
• Here, the vertex is \( (h, k) = (-2, -4) \).

The vertex is important because it's essentially the 'center' of our parabola, giving us a key reference point for plotting and understanding the structure.
Once you've found the vertex, it acts as a pivotal point for locating other elements like the focus and the directrix.

Focus of a Parabola

The focus of a parabola is a special point that helps define its shape. For a parabola that opens left or right:

• The focus lies a distance \( p \) from the vertex.
• In our case, \( p = 3 \).
• Since the parabola opens horizontally to the left, the focus is at \( (-5, -4) \).

The focus is crucial because every point on the parabola is equidistant from this point and the directrix. It's a guiding beacon for the shape, helping understand how the curve is formed.
Visualizing the focus inside the parabola aids in grasping the geometry's symmetry and direction.

Directrix of a Parabola

The directrix of a parabola is a line that, together with the focus, defines the parabola. It is equidistant from any point on the parabola as the focus is:

• For our parabola, \( p = 3 \).
• The directrix is a vertical line: \( x = 1 \).

Think of the directrix as a boundary that helps establish the parabola's open end. While the focus sits inside the curve, the directrix balances it from the outside.
This balance ensures the reflective property of parabolas, where paths to the focus are symmetric across the curve.

Graphing Parabolas

Graphing a parabola involves plotting its vertex, focus, and directrix for precise visualization.

• We place the vertex at \((-2, -4)\).
• Plot the focus at \((-5, -4)\).
• Draw the directrix as a vertical line \(x = 1\).

From here, sketch the parabola as a smooth curve opening horizontally to the left. The curve should always pass through points that are equidistant from the focus and directrix.
Graphing visually connects all these elements, showing how the vertex, focus, and directrix interact to form the parabolic shape. Engaging in this task solidifies understanding of how each part influences the overall figure.