Question

Find the equation of the parabola defined by the given information. Sketch the parabola. Vertex: (2,3)\(;\) directrix: \(x=4\)

Short answer

The equation is \((y - 3)^2 = 8(x - 2)\). The parabola opens to the left.

Step-by-step solution

Identify Parabola Characteristics

The vertex of the parabola is given as the point \((2, 3)\). The directrix is a vertical line given by \(x = 4\). Since the directrix is vertical, the parabola opens horizontally.

Determine the Direction and Equation Form

The parabola opens to the left because the directrix \(x = 4\) is to the right of the vertex \((2, 3)\). The standard form of a horizontally opening parabola is:\[(y - k)^2 = -4p(x - h)\]Here, \((h, k)\) is the vertex of the parabola, and \(p\) is the distance from the vertex to the directrix.

Calculate the Distance 'p'

The distance from the vertex to the directrix \(x = 4\) is 2 units because the vertex is at \(x = 2\). If the parabola opens to the left, \(p\) is negative. Thus, \(p = -2\).

Substitute Values into the Standard Form

Substitute \((h, k) = (2, 3)\) and \(p = -2\) into the equation for a horizontal parabola:\[(y - 3)^2 = -4(-2)(x - 2)\] Simplify this to get the equation of the parabola:\[(y - 3)^2 = 8(x - 2)\]

Verify and Sketch the Parabola

The parabola has its vertex at (2,3), opens to the left, and its directrix is the line \(x = 4\). Use these characteristics to sketch the parabola. The axis of symmetry is horizontal and passes through \(y = 3\).

Key concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial as it is essentially the starting point. The vertex is the peak point or the lowest point of a parabola depending on the direction it opens. In this context, it acts as a reference point to determine the shape and position of the parabola on the graph.

• For our exercise, the vertex is at the point \(2, 3\), which is pivotal in forming the parabola equation.
• It serves as the \((h, k)\) in the parabola's standard form equation, specifically, \((h, k) = (2, 3)\).
• Knowing the vertex helps in sketching the parabola accurately on a graph.

The vertex also allows you to understand other properties of the parabola, like its axis of symmetry, which is a line that splits the parabola into two mirror images. For a horizontally opening parabola, this axis is a horizontal line passing through the y-value of the vertex.

Directrix of a Parabola

The directrix is an important component in defining a parabola. It is a line that, together with the focus, helps form the parabola's shape. In simple terms, a parabola is a set of points equidistant from a fixed point, the focus, and a line, the directrix.

• In this exercise, the directrix is given as the equation \(x = 4\).
• It lies vertically, indicating that our parabola opens in the horizontal direction.
• The directrix is always perpendicular to the axis of symmetry of the parabola.

For our specific example, the vertex is to the left of the directrix, leading the parabola to open leftwards. The distance from the vertex to the directrix is crucial for determining the value of \(p\), which influences the equation's form and graph orientation.

Horizontal Parabola Standard Form

The standard form of the parabola varies depending on its opening direction. For a horizontally oriented parabola, the standard form is \[(y - k)^2 = 4p(x - h)\]. Here, \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus or directrix.

• For our specific parabola exercise, \((h, k) = (2, 3)\), and \(p = -2\).
• The negative \(p\) value indicates the parabola opens to the left.
• Plugging the values into the equation, we formed \((y - 3)^2 = 8(x - 2)\).
• This equation helps in accurately sketching and verifying the parabola's position and direction.

Understanding how the standard form changes with the parabola orientation is essential for correctly plotting and analyzing parabolas on a graph, allowing deeper insight into their mathematical and practical applications.