Problem 54 - Algebra 2

Question

Write an equation of a parabola with a vertex at \((1,1)\) focus at \((1,0)\)

Step-by-Step Solution

Answer

The equation of the parabola with a vertex at \((1,1)\) and focus at \((1,0)\) is \((x-1)^2 = -4(y-1)\)

Step 1: Determine the Orientation of Parabola

The vertex of the parabola is at \((1, 1)\) and the focus is at \((1, 0)\). Since they have the same x-coordinate, it indicates that the parabola opens downwards.

Step 2: Find the Directrix of the Parabola

The directrix is a line that is symmetrically opposite to the focus with respect to the vertex. For a parabola that opens downwards, the directrix is a horizontal line above the vertex. We know that the focus is 1 unit below the vertex, so the directrix will be 1 unit above the vertex. Therefore, the equation of the directrix will be \(y=2\).

Step 3: Writing the Equation

For a parabola that opens downwards, its general equation is \((x-h)^2 = -4p(y-k)\), where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus or the directrix. Here, \((h, k) = (1, 1)\) and \(p = 1\). Substituting these values in the equation, we get \((x-1)^2 = -4(y-1)\)

Key Concepts

Vertex of Parabola

The vertex of a parabola is a crucial element. It serves as the central point from which the curve extends. The vertex can be thought of as the "tip" or the "corner" of the parabola. For equations in the vertex form

• the parabola appears as \( y = a(x-h)^2 + k \), where
• \((h, k)\) represents the vertex coordinates.
• Given the vertex \((1, 1)\), the parabola's path is shaped around this point.

It's important because it also determines the parabola's axis of symmetry—a vertical line passing through the vertex point. In this case, \(x = 1\) is the axis of symmetry. Knowing the vertex helps us not only to find the orientation of the parabola but also to sketch it accurately on a graph.

Focus of Parabola

The focus of a parabola is a fixed point that defines the curve in conjunction with the directrix, a straight line.

• The parabola's shape is determined by how it bends around this point.
• Each point on the parabola is equidistant from the focus and the directrix.
• In the given problem, the focus is \((1, 0)\).

This means the parabola opens downwards as the focus is below the vertex. The distance from the vertex to the focus is known as \(p\). For the parabola in question, \(p = 1\), calculated as the difference in the y-coordinates \((1 - 0 = 1)\).
This distance is integral to formulating the parabola's equation, impacting the sign and coefficient of the terms in the equation.

Orientation of Parabola

The orientation of a parabola tells you in which direction it "opens" or extends.

• In standard form, parabolas can open upwards or downwards (if they are vertical)
• or to the left or right (if they are horizontal).

In our case, the problem specifies that the vertex and focus have the same x-coordinate \((1)\), which points to a vertical orientation. Since the focus is at \((1, 0)\) below the vertex at \((1, 1)\), this specific parabola opens downwards.
The orientation affects the formulation of the parabolic equation. For vertical parabolas that open downwards, the standard equation form is \((x-h)^2 = -4p(y-k)\). The negative sign is due to its direction downwards, indicating a negative curvature. Knowing the orientation helps in graphing and understanding the parabola's behavior visually.