Problem 23 - Precalculus with Calculus Previews

Question

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(6 y^{2}-12 y-24 x-42=0\)

Step-by-Step Solution

Answer

Vertex: (-2, 1); Focus: (-1, 1); Directrix: x = -3; Axis of symmetry: y = 1.

Step 1: Rewriting the Equation

Start by rearranging and organizing the given equation: \[ 6y^2 - 12y = 24x + 42. \]Divide both sides by 6 to simplify:\[ y^2 - 2y = 4x + 7. \]

Step 2: Completing the Square

To complete the square on the left side, take the coefficient of \( y \), which is \(-2\), divide it by 2 to get \(-1\), and square it to get 1. Add and subtract this square inside the equation:\[ y^2 - 2y + 1 - 1 = 4x + 7. \]This transforms into:\[ (y-1)^2 = 4x + 7 + 1. \]Thus, the equation becomes:\[ (y-1)^2 = 4x + 8. \]

Step 3: Putting Equation into Standard Form

Further, simplify the equation by transforming it:\[ (y-1)^2 = 4(x + 2). \]This is now in the standard form \((y-k)^2=4p(x-h)\), indicating a parabola that opens rightward.

Step 4: Finding the Vertex

From the standard form \((y-k)^2 = 4p(x - h)\), identify the vertex as (h, k). Here, \( h = -2 \) and \( k = 1 \), so the vertex is:\[ (-2, 1). \]

Step 5: Identifying the Focus

Given the form \((y-k)^2 = 4p(x-h)\), calculate \(p\) from \(4p = 4\), so \(p = 1\). Thus, the focus, located \(p\) units to the right of the vertex, is:\[ (-1, 1). \]

Step 6: Determining the Directrix

The directrix is a vertical line \(p\) units to the left of the vertex. This line is given by the equation:\[ x = -3. \]

Step 7: Finding the Axis of the Parabola

For a horizontal parabola of the form \((y-k)^2 = 4p(x-h)\), the axis is horizontal through the vertex. Hence, the axis of symmetry is:\[ y = 1. \]

Step 8: Graphing the Parabola

Plot the vertex at \((-2, 1)\), the focus at \((-1, 1)\), and the directrix at \(x=-3\) on a coordinate plane. Draw the axis of symmetry as a horizontal line \(y=1\) and sketch the parabola opening to the right, through the focus and away from the directrix.

Key Concepts

Vertex of Parabola

The vertex of a parabola is the point where the curve changes direction. It is the "turning point" of the parabola. For the equation in the standard form \((y-k)^2=4p(x-h)\), the vertex can be easily found as \((h, k)\). In our example, the equation \((y-1)^2 = 4(x + 2)\) reveals that the vertex is at \((-2, 1)\).
This means that the parabola is horizontally aligned. In practical terms, the vertex acts as a reference point from which other features of the parabola, such as the focus and directrix, are determined.

Focus of Parabola

The focus is a special point within the parabola that can be used to define the curve. For the parabola in standard form \((y-k)^2 = 4p(x-h)\), \(p\) is the distance from the vertex to the focus. Here, \(p = 1\), and the parabola opens to the right, so we move one unit rightwards from the vertex.
Therefore, the focus is located at \((-1, 1)\). This point is critical in determining how "wide" or "narrow" the parabola is around its vertex.

• The focus sits inside the parabola.
• The vertex and axis of symmetry point directly toward it.

Directrix of Parabola

The directrix is a line perpendicular to the axis of symmetry of the parabola. It provides a geometric boundary for the parabola and is found opposite the focus.
Given the horizontal layout, it lies \(p\) units to the left of the vertex for our example. Therefore, the directrix has the equation \(x = -3\).

• Directrix is as far from the vertex as the focus, only in the opposite direction.
• It helps in defining the parabola comprehensively, balancing the focus's position.

Axis of Symmetry

The axis of symmetry is a crucial line which acts as a mirror, splitting the parabola into two symmetrical halves. For a parabola in the form \((y-k)^2=4p(x-h)\), this line is horizontal. In our case, aligning vertically through the vertex, the axis of symmetry is at \(y=1\).

• It represents the direction along which the parabola opens.
• The axis helps in plotting the parabola accurately using points like vertex and focus.

Knowing the axis of symmetry simplifies understanding the graph's structure as it provides an essential guideline while plotting or graphing the parabola.