Problem 52 - Algebra 2

Question

Write an equation of a parabola with a vertex at \((1,1)\) directrix \(y=-\frac{1}{2}\)

Step-by-Step Solution

Answer

The equation of the parabola with vertex at (1, 1) and directrix \(y = -\frac{1}{2}\) is \(y = 6x - 5\).

Step 1: Determine the distance between the vertex and the directrix

The distance 'p' between the vertex (1, 1) and the directrix \(y = -\frac{1}{2}\) is \(1 - (-\frac{1}{2}) = \frac{3}{2}\). This is also the distance from the vertex to the focus.

Step 2: Determine the coordinates of the focus

Since the parabola opens upwards, we add the distance 'p' to the y-coordinate of the vertex to get the y-coordinate of the focus. So, the focus is at \((1, 1 + \frac{3}{2}) = (1, \frac{5}{2})\).

Step 3: Write the standard form of the equation

The standard form equation of a parabola with its vertex at the origin opening upwards is \(y = 4px\). But since the vertex is not at the origin and is at (1, 1), the equation becomes \(y - 1 = 4p(x - 1)\).

Step 4: Substitute 'p' and simplify

Substitute \(p = \frac{3}{2}\) into the equation and simplify.\(y - 1 = 4*\frac{3}{2}*(x - 1)\)Simplify and rearrange to get \(y = 6x - 5\)

Key Concepts

Vertex

The vertex of a parabola is a unique point where the curve changes direction. It represents the "turning point" of the parabola. In this exercise, the vertex is at \((1,1)\). The vertex defines the symmetry of the parabola and is critical when determining the parabola's shape and orientation.

• It is the midpoint between the focus and the directrix.
• The coordinates of the vertex help to define the equation of the parabola.

Understanding the vertex is vital because it sets the foundation for forming the equation of the parabola in standard form. The vertex not only helps in graphing the parabola accurately but also in calculating other elements like the focus and directrix.

Directrix

The directrix is a fixed line that is paired with a focus to define a parabola. This line is used as a reference to ensure that the distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. In our problem, the directrix is given as \(y = -\frac{1}{2}\).

• It is always perpendicular to the axis of symmetry of the parabola.
• The position of the directrix, relative to the vertex, affects the shape and equation of the parabola.

The radial symmetry around the vertex forms the core of understanding the geometric properties of parabolas, making the directrix a critical component of parabola formation.

Focus

The focus of a parabola is a point from which distances define the shape of the parabola. It works together with the directrix. In this problem, after calculating, the focus is at \((1, \frac{5}{2})\).

• The focus lies on the axis of symmetry of the parabola.
• It is always "inside" the curve of the parabola when you consider its opening direction.

The focus is crucial because it influences both the steepness of the parabola and its orientation. Recognizing where the focus lies allows you to fully describe the curvature and formation of the parabola.

Standard Form Equation

The standard form of a parabola's equation relates directly to its geometric parts: vertex, directrix, and focus. In this particular exercise, the equation is derived based on the vertex's and focus's positions. Normally, for a parabola vertical in nature, the equation takes the form \((y - k)^2 = 4p(x - h)\).

• \(h\) and \(k\) represent the coordinates of the vertex.
• \(p\) is the distance between the vertex and the focus, which relates to the directrix as well.

For our exercise, after substituting and simplifying, the final equation becomes \(y = 6x - 5\). This equation tells us the full story of the parabola, describing exactly how it will appear on a graph. Mastering this standard form is fundamental as it provides a blueprint to build parabolas of all types.