Problem 45 - Precalculus

Question

Find the vertex, focus, and directrix of the parabola. Use a graphing utility to graph the parabola. $$y^{2}+x+y=0$$

Step-by-Step Solution

Answer

Vertex: \((-0.25, -0.5)\), Focus: \((0.75, -0.5)\), Directrix: \(x = -1.25\).

Step 1: Rewrite the Equation in Standard Form

The given equation is \(y^{2}+x+y=0\). Rearranging this to match the standard form of a parabola with a horizontal axis of symmetry \((x-h)^2 = 4p(y-k)\), the equation becomes \((y+0.5)^2=-1*(x+0.25)\) after completing the square in y terms.

Step 2: Determine the Vertex

The vertex of the parabola is given by \((h,k)\). From the rewritten equation, it can be determined that \(h = -0.25\) and \(k = -0.5\). Therefore, the vertex of the given parabola is \((-0.25, -0.5)\).

Step 3: Find the Focus

The value of \(p\) can be determined by looking at the coefficient of \(x\), which is \(-1\). The focus of the parabola can subsequently be found by taking the point \((h,k)\) and subtracting \(p\) from the \(x\)-coordinate. Therefore, the focus of the parabola is \((-0.25-(-1), -0.5)\), or \((0.75, -0.5)\).

Step 4: Calculate the Directrix

The directrix of a parabola with a horizontal axis of symmetry can be found with the equation \(x = h+p\). With \(h = -0.25\) and \(p = -1\), the equation of directrix is \(x = -1.25\).

Step 5: Graph the Parabola

Once the key features have been found, the given parabola can be graphed. This can be done with a graphing utility by inputting the standard form equation found in Step 1, then marking the vertex, focus, and directrix as calculated in Steps 2 - 4.

Key Concepts

Vertex

The vertex of a parabola is a crucial point that represents its peak or lowest point, depending on its orientation. For the equation provided, we start by transforming it into the standard form:

• Original equation: \(y^2 + x + y = 0\)
• Standard form: \((y + 0.5)^2 = -1(x + 0.25)\)

This form helps us easily identify the vertex.
The vertex is given by the coordinates \((h, k)\) in this equation.
For our transformed equation, \(h = -0.25\) and \(k = -0.5\), making the vertex \((-0.25, -0.5)\). This point is significant because it helps define the parabola's axis of symmetry and gives a benchmark from which other features, like the focus and directrix, are derived.

Focus

The focus of a parabola is a special point that helps in understanding the shape and direction of the parabola.
It lies on the parabola's axis of symmetry and plays a role in the reflective property characteristic of parabolas.
For the given equation, we find the focus by using the parameter \(4p\) from the standard form equation. Here, \(p\) helps to determine the distance from the vertex to the focus:

• In the equation \((y + 0.5)^2 = -1(x + 0.25)\), \(p = -0.25\).
• We find the focus by adjusting the \(x\)-coordinate of the vertex by \(p\).
• Focus: \((-0.25 - (-1), -0.5) = (0.75, -0.5)\).

This focus point indicates the direction the parabola opens towards and contributes to its reflective properties.

Directrix

The directrix is a line that serves as a reference for constructing a parabola.
Together with the focus, it defines the geometric property that each point on the parabola is equidistant from the focus and the directrix.
To find the directrix of the parabola with a horizontal axis of symmetry:

• We use the calculated values from earlier: \(h = -0.25\) and \(p = -1\).
• The equation for the directrix becomes: \(x = h + p\).
• In this case, the directrix is \(x = -1.25\).

This line is essential for understanding how the parabola is structured and how it relates spatially to both the focus and vertex.