Problem 3 - Precalculus: Functions and Graphs

Question

Exer. \(1-12\) : Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$2 y^{2}=-3 x$$

Step-by-Step Solution

Answer

Vertex at (0, 0), focus at \((-\frac{3}{8}, 0)\), directrix \(x = \frac{3}{8}\).

Step 1: Rewrite the Equation

Start with the given equation of the parabola: \(2y^2 = -3x\). To find the vertex, focus, and directrix, we need to rewrite this equation in the standard form of a parabola that opens horizontally, \((y-k)^2 = 4p(x-h)\). Divide the entire equation by 2 to simplify: \(y^2 = -\frac{3}{2}x\).

Step 2: Determine the Vertex

The vertex form of a horizontal parabola is \((y-k)^2 = 4p(x-h)\). Here, \(y^2 = -\frac{3}{2}x\) can be rewritten as \((y-0)^2 = -\frac{3}{2}(x-0)\), meaning the vertex \((h, k)\) is \((0, 0)\).

Step 3: Find the Focus

The value \(4p\) in the standard form indicates \(-\frac{3}{2} = 4p\), hence \(p = -\frac{3}{8}\). For a horizontal parabola opening to the left, the focus \(\left(h + p, k\right)\) is at \(\left(0 - \frac{3}{8}, 0\right)\) or \(-\frac{3}{8}, 0\).

Step 4: Determine the Directrix

The directrix of a horizontal parabola \(x = h - p\) is given by \(x = 0 + \frac{3}{8}\) or \(x = \frac{3}{8}\).

Step 5: Sketch the Graph

Draw the parabola on a coordinate plane with vertex at \((0, 0)\), the focus at \((-\frac{3}{8}, 0)\), and the directrix as the vertical line \(x = \frac{3}{8}\). The parabola should open to the left.

Key Concepts

Vertex of Parabola

The vertex of a parabola is a crucial point that lies exactly at the tip of the parabola's curve. In essence, it acts like the 'origin' of the parabola in its respective direction, kind of like the center point of the parabola. The standard form of a parabola makes it easy to identify the vertex. For horizontal parabolas, the equation generally looks like

• \((y-k)^2 = 4p(x-h)\)

Here,

• \(h\) and \(k\) are the x and y coordinates of the vertex.

For the given parabola, \(2y^2 = -3x\), we rewrote it to fit this format as

• \((y-0)^2 = -\frac{3}{2}(x-0)\)

Therefore, the vertex of this particular parabola is at the origin

• \((0, 0)\)

The position of the vertex gives insight into the direction in which the parabola opens, the width, and the placement relative to its graph. A proper grasp of the vertex helps you predict and sketch the graph accurately.

Focus of Parabola

The focus of a parabola provides valuable information about the parabola's shape and direction. It's a fixed point located within the curve, and the parabola itself is always oriented around this point. The focus and directrix work in harmony and define the parabola's reflective properties. For horizontal parabolas, the focus is located at:

• \((h + p, k)\)

where \(p\) is the distance from the vertex to the focus. In our case:

• \((0 + (-\frac{3}{8}), 0)\)
• meaning at \((-\frac{3}{8}, 0)\)

This negative value of \(p\) (\(-\frac{3}{8}\)) indicates that the parabola opens to the left, as the focus is to the left of the vertex at \((0, 0)\). Understanding the focus helps one ensure that the parabola is drawn accurately. The parabola's defining property is that all points are equidistant to the focus and the directrix; knowing where the focus is, therefore, gives us the ability to describe each point on the curve.

Directrix of Parabola

A parabola's directrix is a straight line that, alongside the focus, dictates its shape. It is perpendicular to the axis of symmetry of the parabola. For a horizontal parabola, the equation for the directrix is:

• \(x = h - p\)

Providing balance to the focus, the directrix ensures that every point on the parabola shares an equal distance from both the directrix and focus. In this specific example with an equation \(2y^2 = -3x\), the directrix becomes:

• \(x = 0 + \frac{3}{8}\)
• or simply \(x = \frac{3}{8}\)

The directrix is a vertical line positioned at \(x = \frac{3}{8}\), right of the vertex. Directrix adjustments affect the parabola's width and how 'stretched' it appears on a graph. Recognizing the center balance between focus and directrix helps sketch a more precise parabola, revealing the parabola's properties just from these two features.