Problem 35 - Calculus Early Transcendentals

Question

Find an equation for the conic that satisfies the given conditions. Parabola, vertex \((2,3), \quad\) vertical axis, passing through \((1,5)\)

Step-by-Step Solution

Answer

The equation of the parabola is \((x - 2)^2 = \frac{1}{2}(y - 3)\).

Step 1: Identify the Parabola Equation Format

Since the parabola has a vertical axis, it follows the format \[ (x - h)^2 = 4p(y - k) \]where \((h, k)\) is the vertex. Here, \(h = 2\) and \(k = 3\). So the equation becomes \[ (x - 2)^2 = 4p(y - 3) \].

Step 2: Use the Given Point to Find p

Substitute the point \((1, 5)\) into the equation to find the value of \(p\). We have \[ (1 - 2)^2 = 4p(5 - 3) \].Simplifying gives \[ 1 = 8p \].Solving for \(p\), we get \[ p = \frac{1}{8} \].

Step 3: Write the Final Equation

With \(p = \frac{1}{8}\), substitute back into the parabola form to get the equation: \[ (x - 2)^2 = \left(\frac{1}{2}\right)(y - 3) \].Rewriting gives the final equation: \[ (x - 2)^2 = \frac{1}{2}(y - 3) \].

Key Concepts

Vertex Form of Parabola

Understanding the vertex form of a parabola is pivotal for graphing and solving problems involving parabolic equations. For a parabola with a vertical axis, the vertex form is given by:\[(x - h)^2 = 4p(y - k)\]where

• \( (h, k) \) is the vertex of the parabola.
• \( p \) is the distance from the vertex to the focus and thence to the directrix.

The vertex form is particularly useful because it directly reveals key features of the parabola—namely, its vertex and the parameter \( p \). The vertex \( (h, k) \) provides a fixed point around which the parabola is centered, while the parameter \( p \) indicates the steepness and direction of the parabola. Direction here refers to whether the parabola opens upwards, downwards, or sideways, depending on whether the axis of symmetry is vertical or horizontal. In summary, understanding and using the vertex form of a parabola allows for quick insights into the shape and position of the curve.

Vertical Axis of Parabola

When we talk about a parabola having a vertical axis, it means the parabola opens either upwards or downwards. This is a key aspect of the orientation, determined by the term \( 4p \) in the equation:\[(x - h)^2 = 4p(y - k)\]Here are some important points about a vertical axis:

• The right side of the equation contains the \( y \)-variable, indicating the axis is parallel to the y-axis.
• The parabola is symmetric about a vertical line that passes through its vertex.
• If \( p > 0 \), the parabola opens upwards; if \( p < 0 \), it opens downwards.

A vertical axis of symmetry is a significant property because it determines the direction in which the parabola "grows." Additionally, a vertical axis helps in deriving other properties such as the focus and directrix of the parabola that are aligned with this axis. Understanding this concept makes it easier to visualize and draw the parabola correctly.

Solving for Parameter p

The parameter \( p \) in the parabola equation provides essential information regarding the parabola's width and direction. Here’s how to solve for \( p \):To find \( p \), we use a point that the parabola passes through, not just the vertex. Substituting this point into the equation helps solve for \( p \). For example, if the vertex is \((h, k)\) and the parabola passes through a point \((x_1, y_1)\), then substitute these into the vertex form:\[(x_1 - h)^2 = 4p(y_1 - k)\]Simplify this equation to solve for \( p \):

• Subtract \( h \) from \( x_1 \) and square the result.
• Subtract \( k \) from \( y_1 \).
• Set the two sides equal and solve for \( 4p \), and then divide to isolate \( p \).

Understanding how to determine \( p \) ensures accurate graphing and solving of parabolic expressions, as it defines the parabola's openness and focal properties. Knowing \( p \) thereby aids in predicting and validating the trajectory and symmetry the parabola will exhibit.