Problem 35 - Precalculus

Question

Find the vertex, focus, and directrix of the parabola. Then sketch the parabola. $$(x-1)^{2}+8(y+2)=0$$

Step-by-Step Solution

Answer

The vertex of the parabola is at the point (1, -2), the focus is at the point (1, -4), and the directrix is the line \(y = 0\). The parabola opens downwards.

Step 1: Identifying the Vertex

From the given equation \((x-1)^2 + 8(y+2) = 0\), there are no terms of y on the other side of the equation. So, convert the equation into the standard parabola form. \((x-1)^2 = -8(y+2)\). So, h=1 and k=-2. Therefore, the vertex of the parabola is at the point (1, -2).

Step 2: Finding the focus of the parabola

The value of 4p is -8 from the standard form of the parabola equation. From this, the value of 'p' can be calculated as p = -2. The focus is the point (h, k+p) = (1, -2 - 2) = (1, -4)

Step 3: Calculating the Directrix of the Parabola

The equation of the directrix is \(y = k - p = -2 - (-2) = 0\).

Step 4: Sketching the Parabola

Finally, based off of all these calculations, a sketch of the parabola can be created. Mark the vertex at the point (1, -2), the focus at the point (1, -4), and draw the directrix line at \(y=0\). The parabola opens downwards because the 'p' is negative.

Key Concepts

Vertex

The vertex of a parabola is a significant point that helps in understanding its shape and orientation. It acts like a turning point and is typically represented by the coordinates \((h, k)\). To find the vertex from the equation \((x-1)^2 = -8(y+2)\), we note the standard form \((x-h)^2 = 4p(y-k)\). Here, \(h = 1\) and \(k = -2\).
Thus, the vertex is located at \((1, -2)\).

• It represents the highest or lowest point of the parabola, depending on its orientation.
• The vertex provides a reference for other elements of the parabola such as the focus and the directrix.

Focus

The focus of a parabola is a point that, together with the directrix, defines the parabola's set of points. Every point on the parabola is equidistant from the focus and the directrix.
In our problem, we determine the focus using the formula \(h, k+p\). Using the fact that \(p = -2\), which is derived from \(4p = -8\), the focus is located at \((1, -4)\).

• The focus is located inside the parabola, along the axis of symmetry.
• Since \(p\) is negative in this example, the parabola opens downward, positioning the focus below the vertex.

Directrix

A parabola’s directrix is a line that, much like the focus, helps form the parabolic shape. It's perpendicular to the axis of symmetry and located on the opposite side of the vertex from the focus.
The equation for the directrix, given by \(y = k - p\), leads us to \(y = 0\) in this scenario.

• The directrix provides a boundary over which the parabola never crosses.
• Every point on the parabola is at an equal distance to both the focus and this line.

Parabola Sketching

Sketching a parabola is the culmination of identifying its vertex, focus, and directrix. This process helps visualize the curve's orientation and position.
For our equation, we mark the vertex at \((1, -2)\) and the focus at \((1, -4)\). The directrix at \(y = 0\) provides a baseline reference line above the vertex.
When sketching:

• Draw the vertex and use it as a central point.
• Ensure the focus is marked below the vertex since it opens downwards (as seen from the negative \(p\)).
• The directrix is a straight line above vertex, showing where the parabola would reflect points if folded.
• The parabola's curve will open towards the focus, bending away from the vertex and getting closer to the directrix without crossing it.

This visualization approach helps in appreciating the symmetrical nature and better understanding properties of parabolas.