Question

Find the equation of the parabola defined by the given information. Sketch the parabola. Focus: (1,1)\(;\) vertex: (1,2)

Short answer

The equation is (y - 2)^2 = -4(x - 1).

Step-by-step solution

Understand the Problem

We are given the focus and vertex of a parabola. The focus is (1, 1), and the vertex is (1, 2). This parabola is vertical because the x-coordinates of the focus and vertex are the same.

Identify the Parabola Type

Since the x-coordinates of the focus and vertex are the same, the parabola opens either upward or downward. Here, the vertex is at (1, 2), while the focus is at (1, 1), which is below the vertex. Therefore, this parabola opens downward.

Determine the Directrix

For a vertically opening parabola, the directrix is a line parallel to the x-axis. Since the distance from the vertex to the focus is 1 unit downward (from y = 2 to y = 1), the directrix is 1 unit above the vertex, at y = 3.

Use the Standard Form Equation

The equation of a vertical parabola with vertex (h, k) is given by (y - k)^2 = 4p(x - h) . In this case, h = 1, k = 2, where p = -1 (because focus is below the vertex). Substituting, (y - 2)^2 = -4(x - 1) .

Sketch the Parabola

Draw the vertex at (1, 2), focus at (1, 1), and directrix y = 3. Sketch the parabola opening downward from the vertex, with the axis of symmetry being x = 1.

Key concepts

Vertex and Focus

The vertex and focus of a parabola are crucial in defining its shape and position. The vertex is the point where the parabola changes direction, typically considered the "peak" or "bottom" if the parabola is vertical.
The focus is a specific point that, together with the directrix, helps in defining the parabola. For any point on the parabola, the distance to the focus is equal to the distance to the directrix.
In a vertical parabola, if the x-coordinates of the vertex and the focus are the same, the parabola opens either upwards or downwards. This becomes important when determining the equation of the parabola from its vertex and focus. Remember that:

• If the focus is "below" the vertex, the parabola opens downwards.
• If the focus is "above" the vertex, it opens upwards.

Understanding these positions assists in constructing the parabola's equation and ensuring it correctly represents the situation.

Parabola Properties

Parabolas are U-shaped curves that have several important characteristics and properties. Knowing these properties helps in deducing the equation of a parabola and understanding its graphical representation. Here are a few key properties:

• Axis of Symmetry: This is a vertical or horizontal line that divides the parabola into two mirror-image halves. For vertical parabolas, the axis of symmetry is a line parallel to the y-axis, passing through the vertex.
• Directrix: A line perpendicular to the axis of symmetry used in the geometric definition of the parabola. It is equidistant from the vertex as the focus, but located in the opposite direction of the opening.
• Direction of Opening: Determined by the position of the focus relative to the vertex. This can be up, down, left, or right.

When defining these properties within equations, especially in the standard form \((y - k)^2 = 4p(x - h)\) for vertical parabolas, knowing \(h\), \(k\), and \(p\) helps determine how wide or narrow the parabola is, and in what direction it opens.

Sketching Parabolas

Drawing a parabola involves carefully placing the vertex, focus, and directrix on a coordinate plane. Using these key points allows for an accurate depiction of the parabola. Here's a step-by-step guide:

• Start by plotting the vertex, which is given as a fundamental point.
• Next, plot the focus. It determines the direction in which the parabola opens. The vertex and focus determine whether the curve goes upwards or downwards for a vertical parabola.
• Draw the directrix. This line is parallel to the axis of symmetry and an equal distance away from the vertex as the focus, but on the opposite side.

After positioning these elements, sketch the curve, ensuring it passes through the focus and curves around it, maintaining equidistance with the directrix throughout. The symmetry of the parabola should be evident, with equal spacing on either side of the vertex along the axis of symmetry, ensuring a proportional and accurate sketch.