Problem 31 - Algebra 2

Question

Identify the vertex, the focus, and the directrix of each graph. Then sketch the graph. $$ -8 x=y^{2} $$

Step-by-Step Solution

Answer

The vertex is at (0,0), the focus is at (-2, 0), and the directrix is the line \( x = 2 \).

Step 1: Understand Parabolic Form

A horizontal parabola equation is in the form \( 4p(x - h) = (y - k)^2 \), where \( (h, k) \) is the vertex. In this case, we can rewrite our equation as \( x = -1/8 * y^2 \)

Step 2: Identify the Vertex

In our equation, \( h = 0 \) and \( k = 0 \). Thus, the vertex of the parabola is at the origin, (0,0).

Step 3: Identify the Focus

The distance from vertex to the focus is given by \( p \) in \( 4p = -8 \), so \( p = -2 \). Since this is a parabola opening to the left, we subtract \( p \) from the \( x \) coordinate of the vertex to get the focus which is (-2, 0).

Step 4: Identify the Directrix

The directrix line equation for a leftward opening parabola is \( x = h + p \). Replacing \( h = 0 \) and \( p = -2 \), we get the directrix equation as \( x = 2 \).

Step 5: Sketch the Graph

Once the vertex, focus, and directrix are determined, graph the parabola with these elements by plotting the vertex first at point (0,0), add the focus at (-2,0) and draw the directrix line at \( x = 2 \). The parabola opens to the left, with the focus being inside the parabola and directrix line being on the right.

Key Concepts

Vertex

When dealing with parabolas, the vertex is a crucial point. It's where the parabola either reaches its maximum or minimum value, tipping directions. For a parabola like the one given in the equation \[-8x = y^2\], it's expressed in the form \[4p(x - h) = (y - k)^2\].Here, \((h, k)\) represents the vertex of the parabola.

In this specific case, the equation can be rewritten to match the standard form, revealing that both \(h\) and \(k\) are zero. This means the vertex is right at the origin, \((0,0)\).

• The position of the vertex helps in determining the placement of the rest of the parabola.
• Knowing the vertex is vital as it's the starting point for graphing the rest of the elements.

Focus

The focus is another essential feature of a parabola. It's a point inside the parabola that, along with the directrix, defines its shape.To find the focus, first identify the value \(p\) in the equation \[4p = -8\], so \(p = -2\). Since the parabola opens to the left, this affects where the focus is located.

The focus for a leftward-facing parabola is shy away in the same direction as the opening.For this equation, the focus is shifted negatively along the x-axis from the vertex, resulting in the point \((-2,0)\).

• Determining \(p\) is crucial in ensuring the focus is in the correct position relative to the vertex.
• The focus is internal to the parabola, helping to define its directional opening.

Directrix

The directrix is a straight line that complements the focus to help shape a parabola. It's not a part of the parabola but plays a magic role alongside the focus to guide the structure.To find the directrix for the parabola, use the formula \[x = h + p\]. With \(h = 0\) and \(p = -2\), calculate it as \[x = 2\], placing it on the positive side of the x-axis.

• The directrix is parallel to the y-axis in this orientation, enforcing the parabolic shape.
• Ensures a consistent distance from the vertex and focus with respect to the parabola's structure.

It's the invisible architect, making sure that every point on the parabola is equidistant from both the focus and the directrix.