Problem 20 - 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. $$ (x+1)^{2}=2(y+4) $$

Step-by-Step Solution

Answer

Vertex: (-1, -4), Focus: (-1, -7/2), Directrix: y = -9/2.

Step 1: Rewrite the Equation

The given equation is \( (x+1)^2 = 2(y + 4) \) Start by expanding the equation to find the standard form of a vertical parabola: \( (x - h)^2 = 4p (y - k) \) where \((h, k)\) is the vertex.

Step 2: Identify the Vertex

Rewrite the given equation to match the standard form: \( (x + 1)^2 = 2(y + 4) \) can be rewritten as \( (x + 1)^2 = 2[y - (-4)] \) Now we can identify \(h = -1\), \(k = -4\). Therefore, the vertex \(V\) is at the point \( (-1, -4) \).

Step 3: Determine p

To find \(p\), compare the coefficient of \(y\) from the standard form with the given equation. In the given equation, \(4p = 2\) Solving for \(p\), we find \( p = \frac{1}{2} \).

Step 4: Calculate Focus

The focus \((F)\) is at a distance \(p\) from the vertex along the axis of symmetry. For a vertical parabola, the axis is vertical. The vertex is \((-1, -4)\), so the focus is at \((-1, -4 + \frac{1}{2}) = (-1, -\frac{7}{2})\).

Step 5: Find Directrix

The directrix \((d)\) is a line that is \(p\) units from the vertex on the opposite side of the focus. The equation of the directrix is \(y = -4 - \frac{1}{2} = -\frac{9}{2} \).

Key Concepts

Parabola Vertex

To understand the vertex of a parabola, think of it as the most prominent turning point of the curve. For parabolas, the standard form of the equation is \((x - h)^2 = 4p(y - k)\) for a vertically oriented parabola, where \((h, k)\) stands for the vertex. In the given equation, \((x + 1)^2 = 2(y + 4)\), we observe that it can be rearranged to match the standard form as \((x + 1)^2 = 2(y - (-4))\), indicating that \(h = -1\) and \(k = -4\). Therefore, the vertex \(V\) is located at \((-1, -4)\). This vertex is the point where the parabola changes direction. It acts as the connection point between the two arms of the parabola and provides insight into the orientation and position of the curve in relation to the coordinate plane.

Parabola Focus

The focus of a parabola is a special point that, together with the directrix, helps define the shape of the curve. For a vertical parabola like in our example, the focus lies along the vertical axis, directly above or below the vertex, depending on the direction the parabola opens. The distance from the vertex to the focus is referred to as \(p\). This can be calculated from the equation \(4p = 2\), giving \(p = \frac{1}{2}\). Therefore, starting from the vertex at \((-1, -4)\), we move up \(\frac{1}{2}\) unit to reach the focus, located at \((-1, -\frac{7}{2})\). The importance of the focus cannot be understated: it acts as the point toward which all points on the parabola reflect equal distances from the focus and the directrix.

Parabola Directrix

The directrix of a parabola is a straight line that plays a pivotal role in shaping the curve along with the focus. For a vertical parabola, it is horizontal and located such that it is the same distance \(p\) from the vertex as the focus, but in the opposite direction.Reiterating from our example, the directrix is below the vertex by \(\frac{1}{2}\) unit, resulting from the calculation \(y = -4 - \frac{1}{2} = -\frac{9}{2}\). The directrix serves as a crucial component in understanding the definition of a parabola: every point on the parabola is equidistant to the focus and the directrix. This geometric property underlies how a parabola behaves and how it is constructed, creating an elegant balance between the parabola's arms.