Problem 55 - Precalculus with Limits

Question

In Exercises 51-60, find the standard form of the equation of the parabola with the given characteristics. Vertex: \((4, 3) \quad\) focus: \((6, 3)\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \( (y - 3)^2 = 8*(x - 4)\).

Step 1: Compute the distance between the vertex and the focus

The distance between the vertex (h, k) = (4, 3) and the focus (h, k+p) = (6, 3) equals 'p', so in this case, p = 6 - 4 = 2.

Step 2: Determine the direction of opening

The parabola opens to the right if p > 0 and to the left if p < 0. Here, p > 0, so the parabola opens to the right.

Step 3: Find the standard form of the equation

The standard form for a parabola that opens to the right or left is \((y - k)^2 = 4p(x - h)\). Substituting h = 4, k =3, p = 2 into the equation, we get \( (y - 3)^2 = 4*2*(x - 4) \), which simplifies into \( (y - 3)^2 = 8*(x - 4)\).

Key Concepts

Vertex

In a parabola, the vertex is a key point where the curve changes direction. Picture it as the pointy tip of the U-shaped graph. In general, it's the midpoint between the focus and the directrix, acting as a balance for the curve. For a parabola described mathematically, the vertex is often given in the form

• \((h, k)\).

For example, in our problem, the vertex is at \((4, 3)\). This tells us where the parabola is centered in relation to the coordinate plane. Every parabola has exactly one vertex, and understanding its position helps us determine how the parabola behaves.

Focus

The focus of a parabola is a special point. It lies "inside" the curve of the parabola, and every point on the parabola is equidistant from the focus and a line called the directrix. The focus plays a crucial role in the parabolic shape, influencing how narrow or wide the parabola is.

• In our example, the focus is \((6, 3)\).
• This particular orientation suggests that the parabola opens horizontally.

Knowing the focus alongside the vertex allows us to visualize the depth and stretch of the parabola, aiding in drafting its precise equation.

Standard Form

The standard form of a parabola's equation makes it easier to identify key components, such as the vertex and the focus, quickly. For parabolas that open horizontally (right or left), the standard form is expressed as:

• \((y - k)^2 = 4p(x - h)\).

Here,

• \((h, k)\)
is the vertex, and
• '\(p\)'
indicates the distance between the vertex and the focus. In our exercise, substituting
• \(h = 4\), \(k = 3\), and \(p = 2\)
gives us
• \((y - 3)^2 = 8(x - 4)\).
Easily plotting this form of the equation, we can see at a glance, the structure and orientation of the parabola.

Parabola Orientation

Understanding the orientation of the parabola helps in predicting its overall shape on a graph. Parabolas can open in four possible directions: upward, downward, to the left, or to the right. Orientation depends on the sign and position of \(p\):

• If \(p > 0\), the parabola opens right (for horizontal) or up (for vertical).
• If \(p < 0\), it opens left (for horizontal) or down (for vertical).

In this exercise, with \(p = 2\) (a positive value), our parabola opens to the right. This key feature allows us to visualize the parabola's trajectory in a coordinate plane, ensuring accuracy in graphing or further calculations.