Question

Sketch the graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. $$x=-y^{2} / 16$$

Short answer

Question: Sketch the graph of the parabola with the equation x = -1/16 * y^2 and verify the sketch using a graphing utility. Identify the vertex, focus, and equation of the directrix.

Step-by-step solution

Identify the Vertex

The vertex for the equation \(x = -\frac{1}{4p}y^2\) is at the origin (0,0). So the vertex of the given parabola is (0,0).

Determine the Value of p

In our equation, \(x = -\frac{1}{4p}y^2\), we can identify the value of \(- \frac{1}{4p}\) as -1/16. That gives us \(- \frac{1}{4p} = -\frac{1}{16}\). Solving for p, we find that p = 4.

Focus and Directrix

Since the parabola opens to the left, the focus will be (vertex_x - p, vertex_y) and the equation for the directrix will be x = vertex_x + p. By plugging in our values, we get: Focus: (0 - 4, 0) = (-4, 0) Directrix: x = 0 + 4, which simplifies to x=4

Sketch the Graph

To sketch the graph: 1. Plot the vertex at the origin (0,0) 2. Indicate the focus at (-4,0) 3. Draw the directrix with a vertical line at x = 4 4. Draw the parabola opening to the left, with the vertex at the origin, and passing through the focus.

Use a Graphing Utility to Check our Work

To confirm our sketch is accurate, use a graphing utility (such as Desmos or a graphing calculator) to plot the parabola with the equation $$x = -\frac{1}{16}y^2$$. Check the graph's vertex, focus, and directrix against our solutions from Steps 1-3.

Key concepts

Focus of a Parabola

When it comes to parabolas, understanding the focus is crucial. The focus is a special point. It's located inside the parabola. Imagine it as the 'eye' of the parabola that attracts its entire structure. In the equation provided, which is rearranged into the form \( x = -\frac{1}{4p}y^2 \), the focus helps define how the parabola curves.

From the given equation \( x = -\frac{1}{16}y^2 \), we derived that \( p = 4 \). This tells us the distance from the vertex to the focus. Since the parabola opens to the left, it means the focus will be 4 units to the left of the vertex.

• Vertex is at the origin: (0,0)
• Distance to reach the focus: 4 units left

Thus, for this parabola, the focus point becomes \((-4, 0)\). Remember, this point is where all the lines reflecting off the parabola will converge, a unique characteristic of parabolas.

Directrix of a Parabola

The directrix is another crucial component when studying the geometry of a parabola. Unlike the focus, the directrix is a line, not a point. It lies outside the parabola. If the focus 'pulls' the parabola towards it, the directrix helps push the parabola away, working as a balance.

For the parabolic equation \( x = -\frac{1}{16}y^2 \), we already found the distance \( p = 4 \). Thus, with the vertex at \((0,0)\) and the parabola opening leftward, the directrix is positioned 4 units to the right of the vertex.

• From vertex (0,0) move 4 units right to find the directrix

Hence, the equation of the directrix for this particular parabola becomes \( x = 4 \). This vertical line represents a boundary that creates the unique 'arch' of the parabola we observe.

Using Graphing Utilities

A graphing utility can be your best friend when visualizing parabolas. These tools offer an easy way to understand complex equations by turning numbers into graphs. For our specific example, plotting the equation \( x = -\frac{1}{16}y^2 \) can help verify our manual calculations.

Here's how a graphing utility enhances learning:

• It provides a visual confirmation of the parabola shape.
• Verifies the location of key points like the vertex and focus.
• Shows the parabola in relation to the directrix line.

By using platforms such as Desmos or graphing calculators, you can plot this parabola and see it with your own eyes. This interactive method makes abstract concepts concrete, reinforcing your understanding efficiently.