Problem 56 - College Algebra with Modeling and Visualization

Question

Graph the parabola. Label the vertex, focus, and directrix. $$ x=2(y-2)^{2}-1 $$

Step-by-Step Solution

Answer

Vertex at \((-1, 2)\), focus at \((-\frac{7}{8}, 2)\), directrix is \(x = -\frac{9}{8}\).

Step 1: Identify the Standard Form

The given equation is \( x = 2(y-2)^2 - 1 \), which is in the form of a parabola opening horizontally, \( x = a(y-k)^2 + h \). Here, \( a = 2 \), \( h = -1 \), and \( k = 2 \). This identifies the vertex form for horizontal parabolas.

Step 2: Locate the Vertex

The vertex \( (h, k) \) of the parabola is found directly from the equation. Thus, the vertex is \( (-1, 2) \). This is the point where the parabola is at its widest, opening in the positive x-direction because \( a > 0 \).

Step 3: Determine the Axis of Symmetry

Since this is a horizontally opening parabola, the axis of symmetry is a horizontal line through the y-coordinate of the vertex, \( y = 2 \).

Step 4: Find the Focus

The focus of the parabola can be calculated using the formula for distance from vertex: \( \rac{1}{4a} \). Here \( a = 2 \), so the distance from the vertex to the focus along the axis of symmetry is \( \rac{1}{8} \). Therefore, the focus is at \( \igg( -1 + \rac{1}{8}, 2 \igg) = \igg( -\frac{7}{8}, 2 \igg) \).

Step 5: Identify the Directrix

The directrix of a parabola is a line at the same distance from the vertex as the focus, but in the opposite direction. Therefore, the directrix is at \( x = -1 - \rac{1}{8} = -\frac{9}{8} \).

Step 6: Sketch the Graph

To graph the parabola, plot the vertex at \((-1, 2)\), the focus at \(-\frac{7}{8}, 2\), and draw the directrix line \(x = -\frac{9}{8}\). Since \(a = 2\) is positive, draw the parabola opening to the right, symmetrical about \(y = 2\).

Key Concepts

Vertex

The vertex of a parabola is a crucial point as it represents the peak or the lowest point when the parabola opens vertically, or the widest part when it opens horizontally.
The equation \( x = 2(y - 2)^2 - 1 \) gives us a horizontally opening parabola, making its vertex \((h, k)\) easy to locate directly from the equation.
In this case, \( h = -1 \) and \( k = 2 \), so the vertex is at the point \((-1, 2)\).
This point is pivotal as it helps to determine the direction in which the parabola opens.

• If \( a > 0 \), as it is here, the parabola opens in the positive x-direction.
• Conversely, if \( a < 0 \), the parabola would open in the negative x-direction.

Understanding the vertex also aids in graphing the parabola accurately.

Focus

The focus of a parabola is not on its graph, but a significant point in the plane.
The focus is used to define the parabola as the set of all points equidistant from the focus and the directrix.
In horizontally opening parabolas, the focus sits on the axis of symmetry.
To find the focus, we calculate the distance from the vertex using \( \frac{1}{4a} \).
For our equation, \( a = 2 \), making this distance \( \frac{1}{8} \).
With the vertex at \((-1, 2)\), the focus ends up at \( \left( -1 + \frac{1}{8}, 2 \right) \), which simplifies to \( \left( -\frac{7}{8}, 2 \right) \).

• The focus lies on the same horizontal line as the vertex, yet inside the parabola's opening.
• This helps in providing insight into the parabola's "steepness" and structure.

By identifying the focus, students can better understand the shaping behavior of parabolas.

Directrix

The directrix of a parabola is a line that helps to define the shape of the parabola in relation to its focus.
This line is "imaginary," meaning it isn't a line on the parabola itself, but it influences the curvature and is perpendicular to the axis of symmetry.
For a parabola like the one from our equation, opening horizontally, the directrix is a vertical line.

• It is placed at the same distance from the vertex as the focus, but opposite in direction.

Since the focus is at \( \left( -\frac{7}{8}, 2 \right) \), the directrix is at \( x = -1 - \frac{1}{8} \), which equates to \( x = -\frac{9}{8} \).
This setup ensures the balance needed to maintain the parabolic shape as dictated by its definition.

Axis of Symmetry

The axis of symmetry in a parabola provides a line over which the parabola is a mirror image of itself.
In a horizontally opening parabola like our equation describes, the axis of symmetry runs parallel to the x-axis.

• The axis goes through the vertex, sharing the same y-coordinate as the vertex.
• Thus for \( x = 2(y-2)^2 - 1 \), the axis of symmetry is \( y = 2 \).

This helps keep the focus and directrix in symmetrical positions about the parabola.
The axis of symmetry assists in placing the focus and directrix and guides students to graph a more accurate parabola.
By reflecting one side of the parabola onto the other over this axis, students can check the correctness of their sketches.