Problem 19 - Algebra 2

Question

Identify the focus and the directrix of the graph of each equation. $$ x=\frac{1}{2} y^{2} $$

Step-by-Step Solution

Answer

The focus of the parabola is at (1/2, 0) and the directrix is the line x = -1/2.

Step 1: Identifying the value of a

Given the equation of the parabola \(x = \frac{1}{2} y^{2}\), we can compare it with the standard equation for a parabola: \(x = a y^{2}\) we find \(a=\frac{1}{2}\).

Step 2: Finding the value of p

Since a = 1/4p, we can rearrange this to solve for p: \(p = \frac{1}{4a}\). Substituting a=1/2 into this equation gives \(p= \frac{1}{4*\frac{1}{2}} = \frac{1}{2}\).

Step 3: Finding the focus

The parabola opens to the right, so the focus is at (h+p, k), which here is (0+1/2, 0) = (1/2, 0).

Step 4: Finding the directrix

The directrix is a vertical line x = h - p. Substituting the given values gives x = 0 - 1/2 = -1/2.

Key Concepts

Focus of a parabola

The focus of a parabola is a key concept that helps us understand its structure and direction. Think of the focus as a special point inside the parabola, where all the reflections of lines parallel to the direction of the parabola converge. In the equation \( x = \frac{1}{2} y^{2} \), the parabola opens rightward, indicating that its directrix and its focus are positioned horizontally.

• The focus can be found using the formula \((h+p, k)\), where \(p\) is the distance from the vertex to the focus.
• In our example, the vertex is at \((0, 0)\), and we calculate \(p\) as \(\frac{1}{2}\).
• Thus, the focus is at the point \((\frac{1}{2}, 0)\).

This means the focus is \(\frac{1}{2}\) unit to the right of the origin along the x-axis. The focus plays a crucial role in defining the shape and orientation of the parabola.

Directrix of a parabola

Every parabola has a directrix, which is a line located opposite the focus, serving as a reference line for constructing the parabola. For a parabola that opens to the right, like \( x = \frac{1}{2} y^{2} \), the directrix is found behind the vertex.

• The directrix is a vertical line in this case, and its equation is derived from \(x = h - p\).
• Here, \(h = 0\) and \(p = \frac{1}{2}\), giving us a directrix of \(x = -\frac{1}{2}\).

This means the directrix sits \(\frac{1}{2}\) unit to the left of the origin along the x-axis. Consider the directrix as a boundary guideline that complements the focus in controlling the curve of the parabola.

Equation of a parabola

The equation of a parabola can take different forms depending on how it opens. In our specific scenario, the equation is \(x = \frac{1}{2} y^{2}\), which implies several important characteristics:

• This equation is of the form \(x = a y^{2}\), where \(a\) determines the direction and width of the parabola.
• For vertical parabolas, \(x\) and \(y\) are swapped, changing to \(y = ax^{2}\).
• In the equation \(x = \frac{1}{2} y^{2}\), \(a = \frac{1}{2}\), meaning it opens rightward and is relatively wide.
• The vertex, or turning point, of this parabola, is at the origin \((0, 0)\).

Understanding this form helps us locate critical components like the focus and the directrix, guiding us in graphing the parabolic shape accurately. Each variable in the parabola's equation helps us determine its precise geometry.