Problem 36 - College Algebra

Question

For the following exercises, graph the parabola, labeling the focus and the directrix $$ -5(x+5)^{2}=4(y+5) $$

Step-by-Step Solution

Answer

The parabola has vertex (-5, -5), opens downwards, focus (-5, -6), and directrix y = -4.

Step 1: Recognize the Parabola Form

The equation given \[-5(x+5)^{2}=4(y+5)\]is of the form \[a(x-h)^{2}=4p(y-k)\]which represents a vertical parabola centered at \((h, k)\). In this equation, \[a = -5, \quad h = -5, \quad k = -5.\]

Step 2: Determine the Orientation and Parameter

Since the coefficient of \((x+5)^{2}\)is negative, the parabola opens downwards. Comparing with the standard form, we have \[4p = 4.\]From this, solve for \(p\):\[p = \frac{1}{1} = 1.\]

Step 3: Identify the Vertex

The vertex \((h, k)\)is given by \((-5, -5)\).

Step 4: Find the Focus

The focus of a parabola is located a distance \(p\) from the vertex along the axis of symmetry. Since the parabola is vertical and opens downward, compute the focus \((h, k-p)\):\((-5, -5 -1) = (-5, -6).\)

Step 5: Determine the Directrix

The directrix of a vertical parabola is a horizontal line located at \(k+p\) from the vertex. Since \(k = -5\) and \(p=1\), the equation of the directrix is \(y = -5 + 1 = -4.\)

Step 6: Graph the Parabola

Plot the vertex \((-5, -5)\), focus \((-5, -6)\), draw the directrix \(y = -4\), and sketch the downward-opening parabola with its axis of symmetry along the line \(x = -5\).

Key Concepts

Graphing Parabolas

Graphing a parabola can seem daunting at first, but breaking it down step by step makes it much simpler. When you're given a quadratic equation, the first step is to identify its type and orientation. In the case of \[-5(x+5)^{2}=4(y+5),\]we see this is a vertical parabola because it follows the format \[a(x-h)^{2}=4p(y-k).\] To graph this parabola:

• Start by finding the key features like the vertex, focus, and directrix.
• Determine the orientation of the parabola; here, it opens downwards due to the negative coefficient of \((x+5)^{2}\).
• Plot these points and lines on a graph to help visualize the parabola's path.

Once you have these features outlined, it gets easier to sketch the parabola accurately.

Focus and Directrix

The focus and directrix are two critical components in defining the shape and position of a parabola. They are used to derive the set of points that make up the parabola. Here's how to find them for a vertical parabola like ours:

• Focus: The focus is a unique point inside the parabola that is the same fixed distance from any parabola point as the directrix. For \[-5(x+5)^{2}=4(y+5)\], calculate the focus by moving \(p\) units vertically from the vertex. Since the parabola opens downwards, the focus is at \((-5, -6)\).
• Directrix: This is a horizontal line located at a distance \(p\) from the vertex in the opposite direction of the parabola's opening. For this equation, the directrix is \(y = -4\).

Knowing these helps in understanding the parabolic trajectory and its precise positioning.

Vertex of a Parabola

The vertex is perhaps the most vital part of a parabola as it indicates the maximum or minimum point, from which the parabola curves away. In the equation\[-5(x+5)^{2}=4(y+5),\]the vertex is found at \((h, k) = (-5, -5).\) This spot is crucial as it serves as the reference point when determining the focus and directrix.
Always remember:

• The vertex is the middle point of the parabola's curve.
• The axis of symmetry goes through the vertex; for vertical parabolas, it's a vertical line \(x = h.\)
• It helps to divide the parabola into symmetrical halves.

Being the central part, graphing the vertex correctly guides the rest of the parabola's silhouette.

Orientation of Parabolas

The orientation of the parabola tells you the direction in which it opens. For the equation \[-5(x+5)^{2}=4(y+5),\]we know the parabola opens downwards due to the negative coefficient \(-5\) appearing before the squared term. Understanding the orientation is essential because:

• It affects the direction of opening; positive leads upwards, negative leads downwards.
• Helps in correctly positioning the focus and directrix.
• Affects how you draw points and the overall parabola curve.

When graphing, recognizing that downward orientation means the focus will be below the vertex, and the directrix above, guides you in plotting these components accurately.