Problem 2 - Algebra for College Students

Question

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ y^{2}=-4 x $$

Step-by-Step Solution

Answer

The vertex is (0,0), focus is (-1,0), and the directrix is x=1.

Step 1: Identify the Form of the Parabola

The given equation is \( y^2 = -4x \). This matches the general form of a sideways parabola \( (y-k)^2 = -4p(x-h) \), where the parabola opens left or right depending on the sign of \( p \). Here, \( p \) is negative, indicating the parabola opens left.

Step 2: Determine the Vertex

In the equation \( y^2 = -4x \), the terms \( \pm k \) and \( \pm h \) are not present, which means \( h = 0 \) and \( k = 0 \). Therefore, the vertex of the parabola is \( (0, 0) \).

Step 3: Calculate the Focus

The standard form \( (y - k)^2 = -4p(x - h) \) allows us to identify \( 4p = 4 \), thus \( p = 1 \). Since the parabola opens left, the focus is one unit to the left of the vertex. The focus is \((-1, 0)\).

Step 4: Find the Directrix

The directrix of a parabola \( (y - k)^2 = -4p(x - h) \) is a vertical line \( x = h + p \). Since \( h = 0 \) and \( p = 1 \), the directrix is \( x = 1 \).

Step 5: Sketch the Graph

The graph of the parabola \( y^2 = -4x \) is a sideways parabola opening to the left and centered at the origin (0,0). Plot the vertex at (0,0), the focus at (-1,0), and draw the directrix as the line \( x = 1 \). The parabola opens towards negative x and lies between the focus and directrix.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key point where the curve changes direction. It is essentially the "turning point" of the parabola. For the equation \( y^2 = -4x \), the vertex can be identified by examining its form. The general form for a sideways parabola is \( (y-k)^2 = -4p(x-h) \), where \((h, k)\) represents the vertex.
The given equation lacks \(h\) and \(k\) terms outside the square, implying they equal zero. Thus, for \( y^2 = -4x \), the vertex is at \((0, 0)\).

• The vertex marks the point where the direction changes.
• In this sideways parabola, the vertex is at the origin, \((0,0)\).
• The vertex is crucial for sketching the parabola as it sets the central reference point.

Understanding where the vertex lies helps in plotting the basic structure of the parabola accurately. Once located, it aids in determining how the parabola opens and in identifying other features like the focus and directrix.

Focus of a Parabola

The focus of a parabola is a vital point that dictates the parabolic shape. For the parabola \( y^2 = -4x \), the focus is determined using the standard equation \( (y-k)^2 = -4p(x-h) \). In this case, we find \(p\) by setting \(4p = 4\), yielding \(p = 1\).
Given that the parabola opens to the left (since \(-4\) is negative), the focus is located one unit left of the vertex. Therefore, it is situated at \((-1, 0)\).

• The focus is a point inside the parabola, crucial for defining its curvature.
• The parabola's symmetry ensures that any point on the curve is equidistant from both the focus and the directrix.
• Understanding the focus allows us to grasp how parabolas direct light, sound, or any reflections towards this point.

Hence, realizing where the focus lies enhances our ability to sketch and comprehend the parabolic curve's properties.

Directrix of a Parabola

The directrix of a parabola is a straight line that helps define the parabola's geometric shape along with the focus. It is the line that maintains the equidistant property of every point on the parabola from both the focus and itself.
For the parabola represented by \( y^2 = -4x \), we compute the directrix by using the coordinates of the vertex \((h, k)\) and the value of \(p\). The directrix equation, \(x = h + p\), simplifies to \(x = 1\) since \(h = 0\) and \(p = 1\).

• The directrix for this particular parabola is a vertical line located at \(x = 1\).
• Understanding the directrix allows for the visualization of the parabola's width and how it spreads.
• The line acts as a boundary opposite to the direction of the parabola’s opening.

Knowing the position of the directrix provides better insight into the parabola's size and symmetry, which is essential for accurately sketching it.