Problem 48 - Precalculus: Functions and Graphs

Question

Find an equation for the indicated half of the parabola. Right half of \((x-4)^{2}=y-5\)

Step-by-Step Solution

Answer

The right half is given by \\( y = (x-4)^2 + 5 \\) for \\( x \geq 4 \\).

Step 1: Understand the Parabola Equation

The given equation \( (x-4)^2 = y-5 \) represents a parabola that opens upward. In standard form, this is \( (x-h)^2 = 4p(y-k) \). Therefore, \( h = 4 \), \( k = 5 \), and the equation indicates that the parabola is vertical.

Step 2: Convert to Solved Equation for y

To find an equation where \( y \) is the subject, we solve the equation \( (x-4)^2 = y-5 \) for \( y \). Adding 5 to both sides gives: \( (x-4)^2 + 5 = y \). This expresses \( y \) in terms of \( x \).

Step 3: Determine the Right Half

The right half of the parabola corresponds to \( x \geq 4 \), because this ensures the \( x \) values are greater than or equal to the vertex's x-coordinate (which is 4), thus only the right side of the parabola is considered.

Step 4: Write the Final Equation

Combine the solved equation for \( y \) with the domain restriction for the right half: \( y = (x-4)^2 + 5 \) for \( x \geq 4 \). This is the equation for the right half of the parabola.

Key Concepts

Right Half of Parabola

To find the equation for the right half of a parabola, you must first understand the concept of a vertex. The vertex of the given parabola \((x-4)^2 = y-5\) is located at the point \((h, k)\), which in this case is \((4,5)\). Parabolas are symmetrical down the vertical axis passing through this vertex.

For the right half of the parabola, we consider only the section where \(x\) is greater than or equal to \(4\). In other words, we only take into account \(x\)-values that are to the right side of the vertex or equal to the vertex on the x-axis.

• The x-values start at 4, since the vertex is at \((4,5)\).
• The parabola consists of all points with \(x\geq 4\).

Understand that the notion of 'half' refers to restricting the domain to focus only on the part of the graph that lies on one side of the vertex, hence the term 'half'. This approach helps in simplifying symmetry calculations or determining paths in physical scenarios.

Vertical Parabola

When you see a parabola equation such as \((x-4)^2 = y-5\), it is important to know its orientation. Parabolas are generally either vertical or horizontal, depending on how they open within the 2D plane.

In this case, the equation is centered around \((x-h)^2\). This structure indicates a vertical parabola.

• This parabola opens upwards, which is typical when \((x-h)^2\) is set equal to a \(y\)-expression.
• A vertical parabola has its axis of symmetry aligned with the y-axis.
• The vertex of this parabola, at \((4,5)\), is also its minimum point since it opens upwards.

Vertical parabolas are commonly found in mathematical problems involving quadratic equations, physics trajectories, and real-world phenomena such as the path of projectiles or parabolic antennas.

Converting to y-form

Converting an equation of a parabola to what is called "y-form" simply means expressing the parabola such that \(y\) is isolated on one side of the equation. The given standard form equation is \((x-4)^2 = y-5\). Solving this to make it "y=" involves basic algebraic manipulation.

To isolate \(y\), add \(5\) to both sides of the equation:

• Start with \((x-4)^2 = y - 5\).
• Add \(5\) to each side, resulting in \((x-4)^2 + 5 = y\).

This conversion is simple, yet powerful, as it allows you to easily calculate \(y\) for any \(x\), making the function ready for graphing or further analysis.

By converting \(y\)-form, you also make it clearer the dependency of \(y\) on changes in \(x\), which is essential when considering transformations, translations, and various applications in calculus and applied mathematics.