Problem 8 - 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. $$ 3 x^{2}-9 y=0 $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Equation

The given equation of the parabola is \(3x^2 - 9y = 0\). We start by rewriting this in a more standard form: Divide through by 3 to simplify:\[x^2 = 3y\]This is now in the form \(x^2 = 4py\) where \(4p = 3\).

Step 2: Identify the Parabola Parameters

From the equation \(x^2 = 4py\), we can identify that \(4p = 3\). We need to solve for \(p\) to find the focus and directrix parameters:\[ p = \frac{3}{4}\]The parameter \(p\) represents the distance from the vertex to the focus, as well as the distance to the directrix.

Step 3: Find the Coordinates of the Focus

For a parabola in the form \(x^2 = 4py\), the vertex is at the origin \((0,0)\). The focus is \((0, p)\):\[(0, \frac{3}{4})\]Thus, the focus of the parabola is at \((0, \frac{3}{4})\).

Step 4: Determine the Equation of the Directrix

The directrix of the parabola \(x^2 = 4py\) is a horizontal line \(y = -p\). Substitute \(p = \frac{3}{4}\) to find:\[y = -\frac{3}{4}\]Therefore, the equation of the directrix is \(y = -\frac{3}{4}\).

Step 5: Sketch the Parabola

Draw a simple sketch on coordinate axes with vertex at the origin, \((0, 0)\). Mark the focus at \((0, \frac{3}{4})\) and the directrix at \(y = - \frac{3}{4}\). The parabola opens upwards and is symmetric about the y-axis, with the vertex halfway between the focus and directrix.

Key Concepts

Focus of a Parabola

The focus of a parabola is a critical point in the parabola's geometry. In the equation \( x^2 = 4py \), this focus is located at \((0, p)\) when the vertex is at the origin. It plays a central role in defining the parabola's shape and path. The parabola is the set of all points that are equidistant from the focus and a line called the directrix. In this case, the focus is at \((0, \frac{3}{4})\).

• The vertex sits at the halfway point between the focus and the directrix.
• The focus determines how wide or narrow the parabola opens; the smaller the value of \(p\), the sharper the curve.

By pinpointing the focus, you can better understand how the parabola behaves on a graph.

Directrix of a Parabola

The directrix of a parabola is a line that, together with the focus, defines the shape of the parabola. For a parabola represented by \(x^2 = 4py\), the directrix is found at \(y = -p\) when the vertex is at the origin. For the given equation, this leads to the directrix being \(y = -\frac{3}{4}\).

• The directrix is always perpendicular to the axis of symmetry of the parabola.
• Every point on the parabola is equidistant from the focus and any point on the directrix.

Knowing the location of the directrix helps in sketching the parabola and understanding its opening direction.

Graphing Parabolas

Graphing parabolas involves plotting the curve based on the vertex, focus, and directrix. The process is systematic and highlights the symmetrical nature of parabolas. Let's focus on a few steps:

• First, plot the vertex, which in this case is at \((0,0)\).
• Next, plot the focus \((0, \frac{3}{4})\) and draw the directrix line \(y = -\frac{3}{4}\).
• Sketch the parabola as a curve that opens upwards, ensuring it's equidistant from the directrix and focus.

This visual representation is crucial as it brings together all the mathematical discoveries into a precise and comprehensible figure.

Equation Transformation

Equation transformation is vital in working with parabolas as it allows for easier interpretation of their properties. Converting equations like \(3x^2 - 9y = 0\) to a more familiar parabolic form helps reveal information like the parabola's direction and dimensions. After simplifying, we achieve: \[x^2 = 4py\] where \(p = \frac{3}{4}\).

• Dividing through by constant terms simplifies the equation and makes it manageable.
• Transforming the equation shows the relationships between elements like the focus and directrix without altering its properties.

This transformation is a powerful tool in graphing and understanding not just parabolas, but many other functions in algebra.