Problem 25 - College Algebra with Modeling and Visualization

Question

Graph the parabola. Label the vertex, focus, and directrix. $$ 2 y^{2}=-8 x $$

Step-by-Step Solution

Answer

Vertex: (0, 0), Focus: (-1, 0), Directrix: x = 1.

Step 1: Rewrite the Equation in Standard Form

The given equation is \( 2y^2 = -8x \). To express it in standard form, divide both sides by 2 to isolate the \( y^2 \) term: \[ y^2 = -4x. \] This equation represents a parabola that opens to the left.

Step 2: Identify the Vertex

The standard form for a horizontally oriented parabola is \( (y-k)^2 = 4p(x-h) \). Our equation \( y^2 = -4x \) can be seen as \( (y-0)^2 = 4(-1)(x-0) \), which means the vertex \((h, k)\) is at \((0, 0)\).

Step 3: Identify the Focus and Directrix

For the equation \( y^2 = -4x \), the value \( 4p = -4 \) gives \( p = -1 \). Since the parabola opens to the left, the focus, which is \( p \) units from the vertex towards the direction of opening, is at \((-1, 0)\). The directrix is a vertical line \( x = 1 \), found by moving \( p \) units to the right from the vertex.

Step 4: Draw the Graph

Plot the vertex \((0, 0)\) on the coordinate plane. Mark the focus at \((-1, 0)\) and draw the directrix as the vertical line \( x = 1 \). Sketch the parabola opening leftward through the vertex point, equidistant from the focus and the directrix.

Key Concepts

Vertex

The vertex of a parabola is a fundamental point that defines its shape and position. Imagine a smile or a sad face: the vertex is the tip of the curve, its highest or lowest point depending on the opening direction. For the equation \( y^2 = -4x \), the vertex is neatly centered at \((0, 0)\). Imagine plotting this on a graph where the vertical \(y\)-axis and the horizontal \(x\)-axis meet right at this vertex point. It serves as a central point from which the parabola "grows". In the standard form of a parabola that opens horizontally, \((y-k)^2 = 4p(x-h)\), the vertex \((h, k)\) defines its precise location on the grid.
For our specific example, the fact that both \(h\) and \(k\) are zero simplifies things a lot; you simply start plotting right at the origin of the coordinate plane. Understanding the vertex also helps you know how the parabola expands on either side, bounded by both the focus and the directrix.

• Think of the vertex as the balancing point of the parabola.
• It's the starting line for plotting both the focus and directrix.
• Because this parabola opens to the left, it "mirrors" horizontally from the vertex.

Focus

The focus of a parabola is an intriguing point with fascinating properties. It's not part of the parabola itself but plays a crucial role in its formation. For our parabola given by \( y^2 = -4x \), the focus is at \((-1, 0)\).
Here's how to think about it:

• The focus is always found "inside" the parabola. For left-opening curves like ours, it's placed to the left of the vertex.
• Mathematically, you can find the focus by moving \(p\) units from the vertex toward the parabola's opening; here, \(p = -1\).
• This focus is your guide to how the parabola curves. Every point on the parabola is equidistant from the focus and the directrix.

Imagine the focus as a beacon: as the parabola curves outwards, each point on the curve strives to be equidistant to this focal point. This unique feature defines how the parabola hugs around the focus.In essence: The focus can be seen as that secret point the curve "focuses" on, affecting the depth and direction of its curve.

Directrix

Understanding the directrix is essential to fully grasping a parabola's structure. Imagine it as an invisible line that commands the shape of the parabola. For our example equation \( y^2 = -4x \), the directrix is the vertical line \(x = 1\).
Here's how it works:

• The directrix is always positioned opposite to the focus, on the other side of the vertex.
• For left-opening parabolas like ours, it lies to the right of the vertex.
• It forms an integral part of the geometric definition: every point on the parabola is equidistant from both the directrix and the focus.

In practical terms, the directrix doesn't appear on the parabola itself but helps contour its boundary. While the focus "pulls" the curve inwards, the directrix "pushes" it outward, ensuring the parabolic shape. Think of the directrix as a kind of leash guiding the distance every point on the parabola maintains from it, striking a fascinating balance.To visualize:- Sketch the directrix as a straight vertical line distinct from your parabola.- Notice how each point on the parabola is an equal distance from both the focus and this line.