Problem 4 - Algebra 2

Question

Write an equation for a graph that is the set of all points in the plane that are equidistant from the given point and the given line. $$ F(0,-8), y=8 $$

Step-by-Step Solution

Answer

The equation of the graph that represents all points in the plane that are equidistant from the given point F(0,-8) and the given line y=8, is given by \(x^2=32y\).

Step 1: Determine the vertex

The vertex of the parabola is halfway between the focus F(0,-8) and the directrix y=8. Therefore, the vertex coordinates will be given by (0, average between -8 and 8), or (0,0).

Step 2: Calculate the value of p

The value p is the distance from the vertex to the focus or the directrix. The distance from the vertex (0,0) to the focus (0,-8) is 8 units. So, p equals to 8.

Step 3: Write the equation of the parabola

The general equation for a parabola opening upwards (as it's the case here) is: \((x-h)^2=4p(y-k)\). Substitute h=0, k=0 and p=8 in the equation and it becomes \(x^2=32y\).

Key Concepts

Focus and Directrix

When studying parabolas, a crucial concept to grasp is the focus and directrix. These two elements define the parabola's shape and orientation in the plane. The **focus** is a fixed point, while the **directrix** is a fixed line. Each point on the parabola is equidistant to the focus and the directrix.

This balance of distances is vital because it is by this property that parabolas are constructed. For example, if you have a focus **F(0, -8)** and a directrix **y = 8**, the set of all points that maintain this equidistant nature will form a parabola.

In summary:

• Focus: A special point acting as a fixed distance reference
• Directrix: A line serving as another fixed distance reference

This setup helps determine the position and shape of the parabola.

Distance Formula

To determine the points on a parabola, we need to understand how to measure distances in the plane. The **distance formula** is a powerful tool for this. It calculates the distance between two points ( \(x_1, y_1\) and \(x_2, y_2\) ) in the 2-dimensional plane and is given by:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

This formula allows us to find how far any point is from the focus and the directrix. For example, take the vertex at (0, 0), the focus at (0, -8), and the directrix **y = 8**. We can use the distance formula to assert that a point on the parabola is equidistant from both the focus and the directrix.

This understanding is crucial for calculating the vertex and writing the equation of a parabola. It helps confirm equal distances from the specific points and lines that define the parabolic shape.

Vertex of Parabola

The **vertex of a parabola** is a key point that helps in constructing its equation. It is located exactly halfway between the focus and the directrix.

To find the vertex, you calculate the midpoint between the y-coordinate of the focus and the directrix. For instance, with a focus at **(0, -8)** and a directrix at **y = 8**, the vertex will be located at the average of the two: **(0, 0)**. This is derived by computing the midpoint:\[y_{vertex} = \frac{-8 + 8}{2} = 0\]

The vertex is essential as it serves as the point around which the parabola is symmetric. This location is used to determine the general form of the parabola's equation. In an upward-opening parabola, this form is:\[(x-h)^2 = 4p(y-k)\]
where \(h\) and \(k\) are the coordinates of the vertex.

• Vertex: (0, 0) for the given example
• Equation form: Anchored by the positions of the vertex, focus, and directrix

Understanding these connections ensures precise determination of the parabola's equation.