Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$y^{2}+3 x=0$$

Step-by-Step Solution

Answer

Focus: \((-\frac{3}{4}, 0)\), Directrix: \(x = \frac{3}{4}\).

Step 1: Identify the type of parabola

The given equation of the parabola is \(y^2 + 3x = 0\). Notice that it is of the form \(y^2 = -4px\), which is a standard equation for a parabola that opens to the left or right.

Step 2: Rearrange the equation

Rewrite the equation \(y^2 + 3x = 0\) in the form \(y^2 = -4px\). We get \(y^2 = -3x\). Here, we compare this with \(y^2 = -4px\) to find \(-4p = -3\), giving \(p = \frac{3}{4}\).

Step 3: Determine the orientation of the parabola

Since the equation \(y^2 = -3x\) fits the form \(y^2 = -4px\) and \(-4p\) is negative, the parabola opens to the left.

Step 4: Find the focus of the parabola

For a parabola \(y^2 = -4px\), the focus is located at \((-p, 0)\) if it opens to the left. Substituting the value of \(p\), the focus is at \(\left(-\frac{3}{4}, 0\right)\).

Step 5: Find the directrix of the parabola

The equation of the directrix for a parabola \(y^2 = -4px\) is \(x = p\). Thus, the directrix is \(x = \frac{3}{4}\).

Step 6: Sketch the parabola, focus, and directrix

Draw the parabola on the Cartesian plane with its vertex at the origin \((0, 0)\), opening to the left. Place the focus at \((-\frac{3}{4}, 0)\) and draw the line for the directrix at \(x = \frac{3}{4}\). The vertex, focus, and directrix all align on the horizontal axis.

Key Concepts

Focus and Directrix

In parabolas, the concepts of "focus" and "directrix" are pivotal to understanding their properties. The focus is a fixed point inside the parabola, and every point on the parabola is equidistant from both the focus and the directrix, which is a fixed line. This relationship is crucial in describing how parabolas curve.

• The focus acts like a guiding point that the parabola wraps around.
• The directrix serves as a boundary on the opposite side of the curve from the focus.

In the case of the equation provided, the focus is calculated using the derived value of \(p\) from the rearranged equation \(y^2 = -4px\). For our parabola, the focus is placed at \((-\frac{3}{4}, 0)\). The directrix, being a vertical line, is given by the equation \(x = \frac{3}{4}\).These elements are essential when visually sketching the parabola or predicting its behavior.

Parabola Orientation

Understanding parabola orientation is necessary to graph its curriculum accurately. The orientation describes the direction in which the parabola opens. This is determined by the sign and placement of terms in its equation.

• If the term \(y^2\) appears, the parabola opens horizontally either to the left or right.
• If the term \(x^2\) appears, it opens vertically, either upwards or downwards.
• The direction can further be determined by the sign:
  • A positive term like \(y^2 = 4px\) opens to the right.
  • A negative term like \(y^2 = -4px\) opens to the left.

In our given equation \(y^2 = -3x\), the negative sign indicates a horizontal parabola opening to the left. Recognizing this pattern is vital when dealing with transformed equations and assists in the accurate sketching.

Parabola Sketching

Sketching a parabola involves illustrating its key elements correctly on a coordinate plane. Start by identifying the vertex, focus, and directrix.

• Vertex: This is typically located at the origin \((0, 0)\) for standard equations unless shifted by additional transformations.
• Focus: Place it as calculated, here at \((-\frac{3}{4}, 0)\), guiding the curve of the parabola.
• Directrix: Draw it at \(x = \frac{3}{4}\), ensuring it is perpendicular to the axis through the vertex.

To accurately sketch, remember that the parabola should symmetrically pass through a series of points equidistant to the focus and the directrix. Plot several critical points to ensure the curve is smooth and correctly proportioned. This method ensures craftiness and precision in graphing mathematical figures.

Parabola Equations

Equations of parabolas are mathematical representations that describe their curve. The standard form of a parabola equation varies depending upon its orientation.

• Vertical Parabolas: Typically use the form \((x - h)^2 = 4p(y - k)\), where \((h, k)\) is the vertex.
• Horizontal Parabolas: Use the form \((y - k)^2 = 4p(x - h)\).

Our equation \(y^2 + 3x = 0\) was rearranged to \(y^2 = -3x\), fitting the horizontal format \((y - k)^2 = -4p(x - h)\). Here, we aligned it to identify that the parabola opens horizontally. Parameters like \(p\) help find essential features, including focus and directrix. Mastering these forms allows the navigation of transformations and modifications in complex problems with ease.