Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

Vertex: (0, 0); Focus: (0, -3/4); Directrix: y = 3/4.

Step 1: Identify the Form of the Parabola

The equation of the parabola is given as \(x^{2} = -3y\). This resembles the standard form of a vertical parabola, \(x^{2} = -4py\), where the parabola opens downward if \(-4p < 0\). Here, we identify \(4p = 3\), so \(p = \frac{3}{4}\).

Step 2: Find the Vertex of the Parabola

The standard form \(x^2 = -4py\) has its vertex at the origin \((0, 0)\) if it is not shifted. Since \(x^2 = -3y\) is already centered at the origin, the vertex is \((0, 0)\).

Step 3: Determine the Focus of the Parabola

The focus of a parabola in the form \(x^2 = -4py\) is at \((0, -p)\). Using the identified \(p = \frac{3}{4}\), the focus is \((0, -\frac{3}{4})\).

Step 4: Determine the Directrix of the Parabola

The directrix of a vertical parabola \(x^2 = -4py\) is the line \(y = p\). Substituting \(p = \frac{3}{4}\), the directrix is \(y = \frac{3}{4}\).

Step 5: Sketch the Graph

To sketch the parabola, draw the vertex at the origin \((0, 0)\), plot the focus at \((0, -\frac{3}{4})\), and draw the line for the directrix at \(y = \frac{3}{4}\). The parabola opens downward through the vertex, approaching the focus.

Key Concepts

Vertex

The vertex of a parabola is a significant point that is often referred to as the 'tip' or 'turning point' of the curve. It is the point where the parabola changes direction. For the standard vertical parabola equation in the form

• \(x^2 = -4py\)

if the graph has not been shifted, the vertex is located at the origin, which is the point

• \((0, 0)\)

In the example equation

• \(x^2 = -3y\)

the vertex remains at the origin because the graph is centered there. Understanding the location of the vertex is crucial as it gives an important reference point for graphing the parabola and identifying other features like the focus and directrix.

Focus

The focus of a parabola is a point that lies on its axis of symmetry and effectively determines its 'open area'. For a vertical parabola with an equation like

• \(x^2 = -4py\)

the formula to find the focus is

• \((0, -p)\)

where \(p\) is the distance from the vertex to the focus along the y-axis. In our specific case, given

• \(p = \frac{3}{4}\)

the focus is at the point

• \((0, -\frac{3}{4})\)

The focus is significant because the parabola 'wraps' around it, and it is the point toward which the curve "bends" as it extends infinitely in both directions.

Directrix

The directrix of a parabola is a line that is perpendicular to its axis of symmetry. It serves as a boundary that helps define and guide the parabolic shape. The relationship between the focus and any point on the parabola is such that the distance to the focus is equal to the distance to the directrix. For a vertical parabola

• \(x^2 = -4py\)

the directrix is given by the equation

• \(y = p\)

In the given equation

• \(x^2 = -3y\)

with \(p = \frac{3}{4}\), the directrix is

• \(y = \frac{3}{4}\)

The directrix, along with the focus, can help in more accurately sketching the parabola by ensuring that the curve maintains its consistent form.

Graph of a Parabola

Sketching the graph of a parabola involves pinpointing its essential parts, notably the vertex, focus, and directrix. Start with the vertex, which is at

• \((0, 0)\)

In this instance, place the focus at

• \((0, -\frac{3}{4})\)

Draw the line of the directrix at

• \(y = \frac{3}{4}\)

Since the parabola's equation is

• \(x^2 = -3y\)

indicating it opens downward, the curve will spread out and bend towards the focus, moving away from the directrix. A graph that illustrates these features helps in visualizing the structural balance between the vertex, focus, and directrix, providing a comprehensive picture of how parabolas operate geometrically.