Problem 11 - Algebra 2

Question

Write an equation of a parabola with a vertex at the origin and the given focus. focus at \((2,0)\)

Step-by-Step Solution

Answer

The equation of a parabola with vertex at the origin and focus at \((2,0)\) is \(y^2 = 8x\).

Step 1: Identify the parameters of a standard form parabola equation

The standard form of a parabola that opens to the right or left is \(y^2 = 4px\), where 'p' is the distance from the vertex to the focus. By the definition of a parabola, 'p' equals to the distance from the vertex to the focus which is \(2,0\).

Step 2: Calculate the p-value

Since the vertex is at the origin, the value of 'p' can be found by just the x-coordinate of the focus. So, 'p' equals 2.

Step 3: Substitute the p-value into the standard form

Substituting 'p' = 2 into the equation \(y^2 = 4px\), The final equation is \(y^2 = 8x\).

Key Concepts

Vertex at Origin

When a parabola has its vertex at the origin, it significantly simplifies the equation and the graph of the parabola. The vertex is the point where the parabola changes direction, and having it at the origin means it is located at the coordinate point (0,0).
This placement ensures that any transformations or translations from the standard form equation are unnecessary.
Thus, the equation of the parabola becomes simplified without needing to account for any shifts along the x-axis or y-axis. This centrality helps us focus solely on the other parameters like the focus and direction of opening.

Focus of a Parabola

The focus of a parabola is a point that is used to define its particular shape. It lies on the axis of symmetry of the parabola.
For a parabola that opens sideways, like in the problem given, the focus will have coordinates (h + p, k) if the vertex is at (h, k).

• For the given exercise, with the vertex at the origin (0,0), the focus is at (2,0).
• The value of 'p', in this case, is the horizontal distance from the vertex, which equals 2.

The focus ensures that every point on the curve is equidistant from the focus and the directrix, another critical line in parabola geometry.

Standard Form of Parabola

The standard form of a parabola varies depending on its orientation. For a sideways opening parabola with its vertex at the origin, the equation is written as:
\[ y^2 = 4px \]
Here, the y term is squared, indicating the parabola opens to the right if 'p' is positive or to the left if 'p' is negative.
In the given exercise, since the focus is at (2,0), implying a positive 'p', the parabola opens to the right.

• This specific alignment of the standard form equation reflects how the graph will dynamically represent itself.
• The equation becomes straightforward to interpret against various algebraic problems.

Calculation of p-value

Calculating the 'p' value is crucial because it helps determine the definitive direction and positioning of the parabola.
The p-value represents the distance from the vertex to the focus or from the vertex to the directrix. For the given problem:

• The vertex is at the origin (0,0), and the focus is at (2,0). Thus, the p-value is simply the x-coordinate difference from the point of origin to the focus, resulting in \( p = 2 \).
• Inserting this value into the standard form \( y^2 = 4px \), we get \( y^2 = 8x \).

Knowing the p-value helps complete the equation, thereby allowing for easy plotting and construction of the parabola's graph.