Problem 50 - Algebra for College Students

Question

Vertex \((6,-4)\), symmetric with respect to the line \(y=-4\), and contains the point \((8,-3)\) $$ 2 y^{2}+16 y-x+38=0 $$

Step-by-Step Solution

Answer

The directrix of the given parabola is the line \(y = -4\), and the given point \((8, -3)\) lies on the parabola.

Step 1: Determine Directrix Equation

Since the parabola is symmetric with respect to the line \(y = -4\), it implies that the line is a horizontal axis of symmetry for a vertical parabola. The vertex of the parabola is also given as \((6, -4)\). Therefore, the line \(y = -4\) also represents the directrix of the parabola.

Step 2: Rewrite Parabolic Equation in Standard Form

Given the equation \(2y^2 + 16y - x + 38 = 0\), first solve for \(x\) to express it in the form \(x = ay^2 + by + c\). Simplifying, we have: \[x = 2y^2 + 16y + 38\]

Step 3: Identify Parabola Orientation and Vertex

For a parabola of the form \(x = ay^2 + by + c\), check the orientation. It opens sideways (horizontally) as \(x\) is expressed in terms of \(y\). The vertex formula \((h, k)\) for such a parabola can be used to find the vertex already given as \((6, -4)\).

Step 4: Use the Vertex to Inform Axis of Symmetry

The symmetry of the parabola means it mirrors over \(y = -4\). Hence, the vertex \((h, k) = (6, -4)\) confirms this line and aligns with the expression derived in Step 1.

Step 5: Verify by Plugging a Point

Use the point \((8, -3)\) known to lie on the parabola to verify:Substitute \(y = -3\) into the equation derived in Step 2: \[x = 2(-3)^2 + 16(-3) + 38\] \[x = 2(9) - 48 + 38 = 18 - 48 + 38 = 8\] which confirms the point \((8, -3)\) lies on the parabola.

Key Concepts

Vertex

In the world of parabolas, the vertex is a crucial point. It is the peak or the trough of a parabola, depending on its orientation. Think of it as the central point where everything balances out. In our example, the vertex is given as \((6, -4)\). This special point is not just any point; it serves as the turning point or the "hinge" of the parabola.
To put it simply, the vertex is the point where the parabola makes its sharpest turn. If the parabola opens upwards or downwards, the vertex will be the lowest or highest point, respectively. In our case, the equation is written in terms of \(x\). Therefore, the parabola opens sideways (either left or right), and the vertex is at the middle of this arc-like shape.
Being aware of the vertex's position is key to understanding the graph's overall shape. It can be found in the equation when it's presented in the vertex form of a parabola, given by \((h, k)\). Here, \(h\) and \(k\) are simply the \(x\) and \(y\) coordinates of the vertex.

Directrix

The directrix is another fascinating aspect of a parabola. It's a fixed line, and every point on the parabola is equidistant from the directrix and the focus, another crucial point but often less discussed in basic exercises. In our problem, the line \(y = -4\) is not just the axis of symmetry but also the directrix.
In our scenario, since the vertex \((6, -4)\) lies on the directrix, this gives us a straightforward horizontal line for the directrix as well, \(y = -4\). This aligns with the symmetry line and underlines the fact that for vertical parabolas, the directrix is a horizontal line. The relation of the vertex to the directrix helps in determining the parabola's openness and direction for a sketch or further analysis.

Axis of Symmetry

The axis of symmetry is a line that divides the parabola into two identical halves. Imagine drawing a line through the vertex that perfectly divides the parabola into mirrored segments. In this particular exercise, the line of symmetry is also given by \(y = -4\).
This axis of symmetry can help you find other points on the parabola by reflecting known points across the line. This characteristic is crucial because it simplifies the graphing process for parabolas. For instance, if you know one point on one side of the axis of symmetry, its mirror image will be on the other side. The vertex lies directly on this line, making it the center of the parabola's mirror image.

Standard Form of a Parabola

The standard form of a parabola is a way to write the parabola's equation that reveals significant information about its shape and orientation. In our example, the equation \(2y^2 + 16y - x + 38 = 0\) needed to be rearranged.
By manipulating this equation, we can express it as \(x = 2y^2 + 16y + 38\). Notice that the equation is in terms of \(x\), suggesting a horizontally opening parabola. This form is useful as it lets us plug in values of \(y\) to find corresponding \(x\) values, or vice versa, depending on the situation.
The standard form helps to identify whether the parabola opens vertically or horizontally, and if it opens upwards, downwards, left, or right. Knowing this allows you to easily sketch the parabola and understand the problem better.