Problem 26 - Precalculus: Mathematics for Calculus

Question

Use a graphing device to graph the parabola. $$x^{2}=-8 y$$

Step-by-Step Solution

Answer

The parabola \(x^2 = -8y\) opens downward with vertex \((0, 0)\), focus \((0, -2)\), and directrix \(y = 2\).

Step 1: Identify the Standard Form

The given equation is \(x^2 = -8y\). This is almost in vertex form for a vertical parabola, \(x^2 = 4p(y - k)\), with \(h = 0, k = 0\), and \(4p = -8\).

Step 2: Determine the Value of \(p\)

From the equation \(x^2 = -8y\), we have \(4p = -8\). Solving for \(p\) gives \(p = -2\). This means the parabola opens downward with the vertex at the origin \((0, 0)\).

Step 3: Find the Focus and Directrix

The focus of the parabola is located \(p\) units from the vertex. Since \(p = -2\), the focus is at \((0, -2)\). The directrix is a line \(-p\) units from the vertex in the opposite direction, so it is \(y = 2\).

Step 4: Graph the Parabola

Use a graphing device to plot the parabola. Start by marking the vertex at the origin \((0, 0)\), the focus at \((0, -2)\), and draw the directrix line \(y = 2\). Sketch the parabola opening downward, ensuring it is symmetric with respect to the x-axis.

Key Concepts

Graphing Parabolas

Parabolas are U-shaped curves that can open either upwards, downwards, left, or right. When graphing a parabola, it's crucial to identify the vertex, which is the highest or lowest point of the curve, depending on its orientation. The example equation given here is \(x^2 = -8y\).
To graph a parabola effectively:

• Start by plotting the vertex. For this equation, the vertex is located at the origin \((0, 0)\).
• Determine the direction in which the parabola opens by looking at the sign and position of the squared term. Since \(x^2\) implies a vertical parabola and the coefficient is negative, this parabola opens downward.
• Plot additional key points, such as the focus, and draw the directrix to guide the shape of the parabola.
• Make sure the parabola is symmetric about its axis.

Graphing a parabola may initially seem like a complex task, but by breaking it down into these simple steps, it becomes straightforward. Remember to always revisit your plotted points and lines for symmetry and accuracy.

Vertex Form

The vertex form of a parabola is a useful way to express its equation. This form gives you easy access to the vertex's coordinates directly from the equation, and it appears as \(y = a(x - h)^2 + k\) for vertical parabolas, where \((h, k)\) is the vertex.
For equations in the format \(x^2 = 4p(y - k)\), like \(x^2 = -8y\), you can quickly identify the vertex as \((h,k)=(0,0)\). The number \(4p\) provides information about the parabola's width and the direction it opens.
Using the vertex form not only helps in identifying the vertex but also assists in determining whether the parabola opens upwards or downwards:

• If \(a > 0\) or \(4p > 0\), the parabola opens upwards.
• If \(a < 0\) or \(4p < 0\), the parabola opens downwards.

Understanding and rewriting the equation in vertex form can greatly simplify graphing and solving problems involving parabolas.

Focus and Directrix

Every parabola has a focus and a directrix, which are crucial features for understanding its geometric properties. The focus is a point inside the parabola, and the directrix is a line outside it.
For the parabola \(x^2 = -8y\), the formula \(x^2 = 4p(y - k)\) was used to identify that \(p = -2\). This indicates the focus is located at \((0, -2)\), meaning it's "below" the vertex when the parabola opens downward.

• The focus helps in determining the set of points that make up the parabola, as they are equidistant from this point and the directrix.
• The directrix in this example is the horizontal line \(y = 2\).

The focus and directrix together describe the parabola's shape, size, and orientation. Graphically, these elements play a key role because every point on the parabola is equidistant from both the focus and the directrix.