Question

Given a parabola opening upward with vertex located at \((h, k)\) and focus located at \((h, k+p)\), where \(\rho\) is a constant, the equation for the parabola is given by $$ y=\frac{1}{4 p}(x-h)^{2}+k $$ This is the standard form of a parabola.

Short answer

It's a parabola opening upward with vertex at \((h, k)\) and focus at \((h, k+p)\).

Step-by-step solution

Identifying Parameters

Given the parabola equation \( y = \frac{1}{4p}(x-h)^{2} + k \), we must identify the parameters: \(h\), \(k\), and \(p\). \(h\) and \(k\) are the coordinates of the vertex, while \(p\) is the distance from the vertex to the focus along the axis of symmetry (vertical line passing through \(h\)). The focus is located at \((h, k + p)\).

Equation Components

In the equation \( y = \frac{1}{4p}(x-h)^{2} + k \), the \((x-h)^{2}\) term implies that any x-value is shifted relative to \(h\), the vertex x-coordinate. The term \(\frac{1}{4p}\) is the coefficient impacting the parabola's width, indicating how quickly it opens upwards. The \(+ k\) term moves the entire graph up or down, depending on the vertex y-coordinate \(k\).

Understanding Focus and Directrix

With the focus at \((h, k + p)\), this parabola has a corresponding directrix line at \(y = k - p\). The distance from any point on the parabola to the focus equals the perpendicular distance from that point to the directrix, which is a key characteristic of parabolas.

Graphing the Parabola

To graph the parabola, plot the vertex at \((h, k)\). Identify the focus \((h, k+p)\) and draw the directrix line \(y = k-p\). The parabola opens upward, and additional points can be found by selecting x-values near \(h\) and calculating y-values using the parabola's equation.

Summary of Parabola Characteristics

The standard parabola equation is \( y = \frac{1}{4p}(x-h)^{2} + k \) with vertex \((h, k)\), focus \((h, k+p)\), and opening upwards, characterized by its symmetry around the vertical line \(x = h\). The directrix complements the focus at \(y = k-p\), ensuring the parabola's reflective property.

Key concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point that determines its position on the coordinate plane. In the standard form equation of a parabola, given by \( y = \frac{1}{4p}(x-h)^2 + k \), the vertex is located at the coordinates \((h, k)\). This point is significant because it represents the parabola's minimum or maximum point, depending on its orientation.

When a parabola opens upwards, as is the case in the equation above, the vertex is the lowest point on the curve. If the parabola opens downwards, the vertex becomes the highest point. Understanding the position of the vertex helps in sketching the graph of the parabola and predicting the path it follows.

Remember these key points about the vertex:

• The line of symmetry for the parabola, where the curve is mirror-imaged on either side, passes through the vertex at \(x = h\).
• The y-coordinate \(k\) shifts the parabola vertically, determining how high or low it appears on the graph.

By manipulating \(h\) and \(k\), you can position the parabola anywhere on the coordinate grid.

Focus of a Parabola

The focus is an essential element associated with a parabola, fundamentally defining its shape and direction. For the standard form \(y = \frac{1}{4p}(x-h)^2 + k \), the focus is found at the point \((h, k+p)\).

This focus point works in harmony with the parabola's directrix, contributing to the parabola's unique property: every point on the parabola is equidistant from the focus and the directrix. This property is key to understanding the reflective characteristics of parabolas.

Important details about the focus to consider:

• The focus lies along the axis of symmetry, which is the vertical line \(x = h\) for this equation.
• The distance \(p\) from the vertex to the focus determines not only the position of the focus but also affects how "wide" or "narrow" the parabola appears; a larger \(p\) value indicates a wider parabola, whereas a smaller \(p\) makes it more narrow.

Developing a solid comprehension of the focus assists in graphing parabolas with accuracy and understanding their reflective nature.

Directrix of a Parabola

The directrix is a fixed, straight line that interacts closely with the focus to define the parabola's structure. In the context of the standard parabola equation \( y = \frac{1}{4p}(x-h)^2 + k \), the directrix is situated at \( y = k - p \).

The directrix plays a vital role in ensuring every point on the parabola maintains an equal distance to both the focus and itself. This is a defining feature of parabolic curves and is crucial for their characteristic reflective properties.

Key aspects of the directrix include:

• It lies parallel to the x-axis if the parabola opens upwards or downwards, which complements where the focus is positioned.
• The directrix line, while not part of the parabola itself, is fundamental for accurately determining the parabolic path, serving as a reference line opposite the focus.

In understanding the directrix, not only do you reinforce the concept of the parabola's shape, but you also deepen your appreciation for its mathematical properties.