Problem 19 - Algebra for College Students

Question

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ (x-2)^{2}=-4(y+2) $$

Step-by-Step Solution

Answer

Vertex: (2, -2), Focus: (2, -3), Directrix: y = -1.

Step 1: Identify the Form of the Parabola Equation

The given equation \((x-2)^2 = -4(y+2)\) can be compared to the standard form of a vertical parabola, which is \((x-h)^2 = 4p(y-k)\). In this comparison, \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus and directrix.

Step 2: Determine the Vertex

The vertex of the parabola is \((h, k)\). From the equation \((x-2)^2 = -4(y+2)\), we can see that \(h = 2\) and \(k = -2\). Thus, the vertex is \((2, -2)\).

Step 3: Calculate the Value of p

In the equation \((x-2)^2 = -4(y+2)\), compare it to the form \((x-h)^2 = 4p(y-k)\). We find \(-4 = 4p\), so \(p = -1\). The negative sign indicates that the parabola opens downwards.

Step 4: Determine the Focus

The focus of a parabola is located at \((h, k + p)\). Since the vertex is \((2, -2)\) and \(p = -1\), the focus is at \((2, -2 + (-1)) = (2, -3)\).

Step 5: Find the Equation of the Directrix

The directrix of a parabola is a line at \(y = k - p\). With the vertex at \((2, -2)\) and \(p = -1\), the directrix is \(y = -2 - (-1) = y = -1\).

Key Concepts

Vertex

In the study of parabolas, the vertex plays a crucial role, as it represents the turning point of the curve. For the equation \[(x-2)^2 = -4(y+2)\], the vertex can be derived from the form \[(x-h)^2 = 4p(y-k)\].

• Here, \(h\) and \(k\) represent the coordinates of the vertex.

By comparing these equations, one can identify that \(h = 2\) and \(k = -2\). Thus, the vertex is located at \((2, -2)\). This point is crucial as it's symmetrically centered along the axis of symmetry of the parabola, serving as a reference point for determining the shape and direction of the curve.
Understanding the location of the vertex helps provide insight into other features of the parabola, such as its focus and directrix.

Focus

The focus of a parabola lies along its axis of symmetry, and it plays an integral part in defining the shape and properties of the curve. Using the standard parabola equation form, \((x-h)^2 = 4p(y-k)\), the focus is determined by the coordinates \((h, k + p)\).
In our example \[(x-2)^2 = -4(y+2)\], the vertex is \((2, -2)\) and \(p\) is calculated as \(-1\).

• This gives the focus at \((2, -3)\).

The presence of the focus is vital because every point on the parabola is equidistant from the focus and the directrix, maintaining this edge in perfect balance. Being aware of where the focus is located allows one to sketch an accurate representation of the parabola, showing its downward opening direction due to the negative \(p\) value.

Directrix

The directrix of a parabola is a fixed straight line that provides a reference when plotting the curve. For a vertical parabola as seen in the studied equation \[(x-2)^2 = -4(y+2)\], the directrix is parallel to the width of the opening of the parabola. Positioned at \(y = k - p\), it forms a simplistic way to define the curvature.

• In this scenario, the advanced calculation provides the directrix at \(y = -1\).

The utility of the directrix comes in handy as it helps structure the parabola's form by working hand in hand with the focus. Points on the parabola maintain equidistance from the focus and directrix. Thus, in any graphical representation, the directrix serves as a contrast balance to the focal point, together defining the elegant U-shape that characterizes parabolas.