Problem 53 - Precalculus with Limits

Question

In Exercises 25-54, \(g\) is related to one of the parent functions described in Section 1.6. (a) Identify the parent function \(f\). (b) Describe the sequence of transformations from \(f\) to \(g\). (c) Sketch the graph of \(g\). (d) Use function notation to write \(g\) in terms of \(f\). \(g (x) = \sqrt{\frac{1}{2}x} - 4\)

Step-by-Step Solution

Answer

The parent function is \(f(x) = \sqrt{x}\). The transformations applied are a horizontal compression by a factor of 1/2 and a vertical shift down by 4 units. The function \(g\) in terms of \(f\) is \(g(x) = f(\frac{1}{2}x) - 4\).

Step 1: Identifying the Parent Function

A parent function is the simplest function with the defining characteristics of the family. In this case, the function \(f(x) = \sqrt{x}\) seems to be the parent function since the rest of the function can be derived from applying transformations to \(f(x)\).

Step 2: Describing the Sequence of Transformations

Given \(g(x) = \sqrt{\frac{1}{2}x} - 4\), a number of transformations have been performed on the parent function \(f(x) = \sqrt{x}\). First, a horizontal compression by a factor of 1/2 is performed, denoted by \(\frac{1}{2}x\) inside the square root. After that, a vertical translation (shift) of 4 units downwards is performed, denoted by the -4 outside the square root.

Step 3: Sketching the Graph of g(x)

Begin with the graph of the parent function, \(f(x) = \sqrt{x}\), which starts at the origin and increases slowly. After that, compress the graph horizontally by a factor of 1/2, i.e. each x-coordinate will be half of its original value. Lastly, shift the graph 4 units downwards, which translates the point (0,0) to (0,-4). Optionally, plot some significant points to further define the shape of the plot.

Step 4: Expressing g in terms of f

To express function \(g\) in the terms of parent function \(f\), we need to replace the variable x in the parent function with the transformations applied to variable x in the function \(g\). The compressed and shifted function can be written as \(g(x) = f(\frac{1}{2}x) - 4\)

Key Concepts

Parent Functions

Parent functions serve as the foundational building blocks in the world of math functions. They are the simplest form of functions within a family and are used to generate other functions through transformations. For example, the parent function for all square root functions is \(f(x) = \sqrt{x}\). This function, starting at the origin \((0,0)\), gradually increases and displays key characteristics, such as how it only accepts non-negative x-values.
This foundational form, \(f(x) = \sqrt{x}\), is crucial because it makes it easier to identify changes such as shifts or stretching in other related functions. Using parent functions helps us quickly see how transformations modify the graph and behavior of the function.

Horizontal Compression

Horizontal compression is a type of transformation that affects the x-values of a graph without altering the y-values directly. Imagine squeezing the graph toward the y-axis. It's like pressing both sides of a rubber band inward. In function notation, horizontal compression occurs when you multiply x by a factor greater than 1 inside the function.
NNIn the case of \(g(x) = \sqrt{\frac{1}{2}x}\), the compression factor is \(\frac{1}{2}\). This means every x-coordinate in the parent function \(f(x) = \sqrt{x}\) is halved. When this compression is applied, the graph becomes more condensed horizontally, indicating that the x-values reach certain points faster than they normally would.

• Compression Factor: \(\frac{1}{2}\)
• Affects x-coordinates, not y-coordinates directly

Vertical Translation

Vertical translation shifts the entire graph of a function up or down. This transformation modifies the y-values of the graph directly, without affecting the x-values. It can be seen as lifting or lowering the graph along the y-axis.
In our example, the graph of \(g(x) = \sqrt{\frac{1}{2}x} - 4\) experiences a vertical translation. The "-4" outside the square root indicates that every y-coordinate of the function \(\sqrt{\frac{1}{2}x}\) has been shifted down by 4 units. This translates the graph down along the y-axis:

• Translation Direction: Downward
• Magnitude: 4 units

As a result, points like the origin \((0,0)\) in the parent function are moved to new positions \((0,-4)\). Vertical translation is like translating the height of every point on the graph.

Function Notation

Function notation is a mathematical shorthand used to depict operations, relationships, and transformations of functions clearly and concisely. It uses symbols such as \(f(x)\) to denote the function and its input/output.
In the transformation of the parent function \(f(x) = \sqrt{x}\) to \(g(x) = \sqrt{\frac{1}{2}x} - 4\), function notation helps us express these changes systematically. The final function can be described using:\[g(x) = f\left(\frac{1}{2}x\right) - 4\]
This notation highlights the sequence of transformations: first, applying horizontal compression by manipulating the input \(\frac{1}{2}x\), then applying a vertical shift of 4 units downwards. Function notation is vital as it provides a clear and unified way to express and visually analyze how a function has evolved from its parent function.