Problem 40 - Precalculus with Limits

Question

In Exercises 33-46, find the vertex, focus, and directrix of the parabola, and sketch its graph. \((x+5) + 8(y-2)^2 = 0\)

Step-by-Step Solution

Answer

The vertex of the given parabola is \((-5, 2)\), the focus is \((-7, 2)\), and the directrix is the line \(x = -3\).

Step 1: Write the Equation in Standard Form

Firstly, rewrite the given equation in standard form. This initially involves isolating \(y\) in terms of \(x\). The equation \((x+5) + 8(y-2)^2 = 0\) can be rearranged to \((y-2)^2 = - (x+5) / 8\).

Step 2: Identify the Properties of the Parabola

Next, properties of the parabola can be inferred from the standard form. The vertex of the parabola is given by the point (h, k), and comparing our equation with the standard form we find \(h = -5\) and \(k = 2\). Therefore the vertex of the parabola is \(-5, 2\).For a parabola in this form, the value of \(p\) is equal to \(-1 / 4a\), and comparing our equation with the standard form we find \(a = -1/8\). Therefore, \(p = 2\).Since our equation involves a term \((y-k)^2\), the parabola opens horizontally, and since \(a<0\), it opens to the left.

Step 3: Find the Focus and Directrix of the Parabola

For a parabola that opens horizontally, the focus is at the point \((h-p, k)\) and the directrix is the vertical line at \(x = h+p\).From step 2, we know \(h = -5\) and \(p = 2\). Therefore, the focus of the parabola is at the point \((-5-2, 2) = (-7, 2)\) and the directrix is the vertical lineme \(x = -5+2 = -3\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a fundamental point that determines the position and direction of the parabola. It is the point where the parabola changes direction and is often thought of as the "tip" or "turning point" of the parabola.
To find the vertex of a parabola given in the form \((y-k)^2 = 4a(x-h)\), you identify \(h\) and \(k\) within the equation.

• The standard form of our specific problem, \((y-2)^2 = - (x+5) / 8\), reveals \(h = -5\) and \(k = 2\).
• Therefore, the vertex is at the point \((-5, 2)\).

This is the starting point from which the parabola opens horizontally. Knowing the vertex is crucial for drawing and understanding the parabola's orientation and shape.

Focus of a Parabola

The focus of a parabola is a special point located along the axis of symmetry inside the parabola. It is important as it determines the shape and direction in which the parabola opens. In parabolas that open horizontally, finding the focus means finding the distance \(p\) from the vertex.
To calculate the focus for our problem:

• First, note from the equation that \(a = -\frac{1}{8}\).
• The formula for \(p\) is \(p = \frac{1}{4a}\), which equals \(2\).
• The focus is then \((h-p, k) = (-5-2, 2) = (-7, 2)\).

The parabola will "aim" towards the focus, pointing in its direction as it opens leftwards, which highlights why the focus is integral to understanding the parabola's structure.

Directrix of a Parabola

The directrix of a parabola is a straight line that, together with the focus, defines the set of points that form the parabola. Each point on the parabola is equidistant from both the focus and the directrix.
For a parabola that opens horizontally, the directrix is a vertical line:

• The position of the directrix is derived as \(x = h+p\).
• For our equation, with \(h = -5\) and \(p = 2\), the directrix is the line \(x = -3\).

The directrix offers an important boundary, providing great insight into the geometry of a parabola and how the points align relative to the vertex and the focus.

Graphing Parabolas

Graphing a parabola involves plotting its key points and structure. With the vertex, focus, and directrix calculated, we can sketch the parabola confidently.
Consider the characteristics derived:

• The parabola has its vertex at \((-5, 2)\).
• It opens horizontally to the left (since \(a < 0\)).
• The focus, positioned at \((-7, 2)\), lies within the parabolic curve.
• The directrix, \(x = -3\), runs parallel to the direction in which the parabola is open.

Plotting these elements will illustrate a complete graph of the parabola, showcasing its symmetry and shape. Understanding these components ensures accurate graphing and offers a deeper comprehension of the parabola's nature.