Problem 20 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Graph the parabola with vertex at (0,0), opening downward, with focus at (0,-2), and directrix y=2.

Step 1: Convert Equation to Vertex Form

The given equation is \(x^2 = -8y\). This is already a standard form for a vertical parabola \(x^2 = 4py\), where \(4p = -8\). So, \(p = -2\). This tells us the parabola opens downwards.

Step 2: Identify Vertex

In the equation \(x^2 = -8y\), there is no \(x\) or \(y\) constant term, so the vertex of the parabola is at the origin \((0, 0)\).

Step 3: Identify the Focus and Directrix

For the parabola of the form \(x^2 = 4py\), the focus is \((0, p)\) and the directrix is \(y = -p\). With \(p = -2\), the focus is at \((0, -2)\) and the directrix is \(y = 2\).

Step 4: Sketch the Parabola

Center your graphing around the vertex at the origin. The parabola opens downward since \(p = -2\) is negative. Plot the focus at \((0, -2)\) and draw the directrix line at \(y = 2\). Use symmetry around the y-axis to help sketch the curve.

Key Concepts

Vertex Form

The vertex form of a parabola is a very useful way to express a quadratic function. It helps to quickly identify the vertex, which is the highest or lowest point on the graph. The general vertex form is \(y = a(x - h)^2 + k\), where

• \((h, k)\) is the vertex of the parabola,
• \(a\) determines the direction and width of the parabola.

In this particular problem, the equation \(x^2 = -8y\) matches the form \(x^2 = 4py\) with no transformation terms. This tells us that the vertex is at the origin \((0, 0)\). This very simple form allows us to quickly conclude that the vertex is pinpointed at zero disruption from the center of the graph.
Moreover, the coefficient \(-8\) in front of \(y\) provides us with the value \(p = -2\), indicating in which direction the parabola opens relative to its vertex.

Focus and Directrix

Understanding the focus and directrix of a parabola is crucial for grasping its geometrical properties. These concepts help define and sketch the parabola correctly. For a vertical parabola represented as \(x^2 = 4py\):

• The **focus** is a fixed point \((0, p)\) that helps "shape" the parabola as all points on the curve are equidistant from this point.
• The **directrix** is a line given by \(y = -p\). It juxtaposes the focus by being a reference the parabola uses to 'open'.

In our example, \(p = -2\), so:

• The focus is located at \((0, -2)\).
• The directrix is the line \(y = 2\).

These elements are key because they explain how and where the parabola extends from its vertex, giving shape and dimension to what could otherwise be an abstract mathematical concept.

Vertical Parabola

The parabola \(x^2 = -8y\) is a perfect example of a vertical parabola. Vertical parabolas are so named because their axis of symmetry is vertical, typically the y-axis. This set-up points the curve upwards or downwards.

• If the parabola equation has \(x^2\) as the main term as opposed to \(y^2\), it represents a vertical parabola.
• In this case, since \(p\) is negative, we know that this particular parabola opens downward.

This visual clue allows us to predict and sketch the general shape without any ambiguity, making vertical parabolas relatively straightforward to manage when plotting graphs.
Mathematically, a negative \(p\) indicates the downward orientation, which can be described as the parabola being inverted over its vertex.

Sketching Graphs

Sketching the graph of a parabola like \(x^2 = -8y\) is simplified by breaking down its components. Start at the vertex, which is at \((0, 0)\) in this instance.

• Since it's a vertical parabola, align your graph symmetrically along the y-axis.
• Plot the focus at \((0, -2)\), which shows where the parabola "reaches" towards.
• Draw the directrix at \(y = 2\); this line provides the open boundary that the parabola should not cross.

The shape and size are defined by the consistency of distance between the curves and these points.
Using symmetry, draw a smooth line curving from the vertex downward, reflecting equally on either side of the y-axis. This technique helps reinforce symmetry in your graph and ensures an accurate visual representation of the parabola.