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Chapter 0: Problem 28

\(g\) is related to one of the parent functions described in this chapter. (a) Identify the parent function \(f\). (b) Describe the sequence of tranformations from \(f\) to \(g\). (c) Sketch the graph of \(g\). (d) Use function notation to write \(g\) in terms of \(f\). $$g(x)=\sqrt{\frac{1}{4} x}$$

Short Answer

Expert verified
The parent function of \(g(x) = \sqrt{\frac{1}{4} x}\) is \(f(x) = \sqrt{x}\). The transformation from \(f(x)\) to \(g(x)\) is a horizontal dilation by a factor of 4. The graph of \(g(x)\) begins at the origin and stretches rightwards. Using function notation, \(g\) can be written in terms of \(f\) as \(g(x) = f(4x)\).

Step by step solution

01

Identify the Parent Function

The parent function \(f(x)\) is \(f(x) = \sqrt{x}\) because the function \(g(x)\) includes a square root operation.
02

Describe the Sequence of Transformations

The transformation from the parent function \(f(x)\) to \(g(x)\) can be described as a horizontal dilation by a factor of 4. This happens because the function \(g(x)\) includes \(\sqrt{\frac{1}{4} x}\), and the \(\frac{1}{4}\) in front of the \(x\) inside the square root means that the graph of the function will be less steep, stretching horizontally.
03

Sketch the Graph of g(x)

The graph of this function begins at the origin (0,0) and will stretch to the right (positive x direction). This is understood from the fact that all square roots are undefined for negative inputs, and the horizontal dilation decreases the slope, resulting in a stretched graph along the x-axis.
04

Write g in Terms of f

Using function notation, \(g(x)\) can be written in terms of the parent function \(f(x) = \sqrt{x}\) as \(g(x) = f(4x)\). This is read as 'g of x equals f of 4x', reflecting the horizontal dilation from \(f\) to \(g\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parent Functions
Understanding parent functions is essential as they serve as blueprints for more complex functions. Simply put, parent functions are the simplest form of functions in their family, displaying the most basic characteristics of a function type without any transformations.

For instance, the parent function for square root functions is represented as \( f(x) = \.sqrt{x} \), which graphs as a gentle curve starting from the origin and rising into the first quadrant. Knowing the parent function helps in predicting the general shape of related functions before applying any transformations.
Horizontal Dilation

Stretching and Compressing Along the X-axis

Horizontal dilation refers to the stretching or compressing of a graph along the x-axis. It is one of the most common transformations of functions. A function is stretched horizontally if it is multiplied by a factor between 0 and 1, resulting in a wider graph. Conversely, when the function is multiplied by a number greater than 1, it compresses and appears narrower.

In our example, \( g(x) = \.sqrt{\frac{1}{4} x} \) is a horizontal dilation of \( f(x) = \.sqrt{x} \) by a factor of 4; this means that each x-coordinate on the graph of \( f(x) \) is divided by 1/4, which stretches the graph horizontally.
Function Notation
Function notation, written as \( f(x) \), serves as a way to name a function and indicate the variable used as input. The notation offers a streamlined method to refer to functions, allowing complex operations and transformations to be written succinctly.

By using function notation, we can communicate how a function is manipulated. For example, writing \( g(x) = f(4x) \) tells us that for every input into function \( g \), the function \( f \) is evaluated at four times that input value, which is key to understanding how \( g \) relates to its parent function \( f \).
Sketching Graphs

Visualizing Functions and Their Transformations

Sketching graphs is a vital skill for visual learners and helps all students understand the behavior of functions. The graph of a function depicts all possible input-output pairs and offers insight into the function's characteristics, such as continuity, slope, and intercepts.

When sketching the graph of \( g(x) = \.sqrt{\frac{1}{4} x} \), one starts at the origin and sketches a curve rising into the first quadrant with a horizontal dilation. This visual representation solidifies the concept that all square root functions will have similar curves, amplified by the transformation applied.

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Most popular questions from this chapter

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