Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ (x+2)^{2}=-8(y-1) $$

Step-by-Step Solution

Answer

Vertex: (-2, 1), Focus: (-2, -1), Directrix: y = 3; the parabola opens downward.

Step 1: Identify the Form of the Parabola Equation

The provided equation is \((x+2)^2 = -8(y-1)\). This is in the form \((x-h)^2 = 4p(y-k)\), which represents a vertical parabola that opens either upwards or downwards, depending on the sign of \(p\). Here, \(-8 = 4p\), so solve for \(p\).

Step 2: Find the Value of p

Set \(4p = -8\) and solve for \(p\): \[ p = \frac{-8}{4} = -2 \] This indicates that the parabola opens downwards with vertex \((h, k)\) and \(p = -2\).

Step 3: Identify the Vertex

The standard form \((x-h)^2 = 4p(y-k)\) shows that the vertex of the parabola is at \((h, k)\). Thus, from \((x+2)^2 = -8(y-1)\), the vertex is \((-2, 1)\).

Step 4: Determine the Focus

For the vertical parabola form \((x-h)^2 = 4p(y-k)\), the focus is at \((h, k+p)\). We have \(h = -2, k = 1, p = -2\). Thus, the focus is \((-2, 1 - 2) = (-2, -1)\).

Step 5: Find the Directrix

The directrix is given by the line \(y = k - p\). With \(k = 1\) and \(p = -2\), \[ y = 1 - (-2) = 1 + 2 = 3 \] Therefore, the directrix is the line \(y = 3\).

Step 6: Sketch the Graph

To sketch:1. Plot the vertex at \((-2, 1)\).2. Plot the focus at \((-2, -1)\).3. Draw the directrix line \(y = 3\).4. Since \(p < 0\), the parabola opens downward from the vertex toward the focus.

Key Concepts

Vertex

In the context of a parabola, the vertex is an important point that signifies the peak or the lowest point of the curve, depending on its orientation. In a standard parabolic equation like \[(x-h)^2 = 4p(y-k)\]the vertex is signified by the coordinates \((h, k)\). This form helps identify the vertex quickly because it shows how the parabola is shifted from the origin.
In our example equation, \[(x+2)^2 = -8(y-1)\]the vertex can be identified by comparing it to the form \((x-h)^2 = 4p(y-k)\). Here, \(h = -2\) and \(k = 1\), so the vertex of the parabola is at \((-2, 1)\).

• The vertex is the point where the parabola changes direction.
• The vertex determines if the parabola opens upwards or downwards.

Keep in mind, while the vertex gives a central clue about the direction and position of the parabola, it does not fully describe the entire curve. However, knowing this point allows for easier graphing and understanding of the parabola's behavior.

Focus

The focus of a parabola is another critical concept in understanding its geometry. It is a specific point located inside the parabola that helps to define its shape. In a vertically oriented parabola, such as \[(x-h)^2 = 4p(y-k)\],the focus is located at \((h, k+p)\).The focus can be thought of as a light source that reflects off the parabolic shape.
In our specific equation, \[(x+2)^2 = -8(y-1)\],we identify \(p\) as \(-2\). Hence, the focus, using \(h = -2\) and \(k = 1\), is found at \((-2, -1)\).

• The focus is used, along with the directrix, to maintain the reflective property of parabolas.
• All points on the parabola are equidistant from the focus and the directrix.

Understanding the focus helps in applications such as satellite dishes and telescopes, where precise alignment is crucial.

Directrix

The directrix of a parabola is a straight line that sits opposite the parabola’s opening from the focus. This line, along with the focus, plays a vital role in defining the locus of all points on the parabola. In the equation \[(x-h)^2 = 4p(y-k)\],the directrix is located at the equation \(y = k - p\).
With our equation, \[(x+2)^2 = -8(y-1)\],we have determined \(k = 1\) and \(p = -2\). Therefore, the directrix is a line parallel to the x-axis, described by \(y = 3\).

• The directrix helps maintain the parabola’s definition such that each point is equidistant from the focus and the directrix.
• It is often used to understand how the parabola can be replicated or adjusted for various designs and functions.

The presence of the directrix ensures a consistent relationship between all elements of the parabola, making it a fundamental concept in various practical and theoretical applications.