Problem 35 - 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 \(f\). \(g (x) = \sqrt{3x} \)

Step-by-Step Solution

Answer

The parent function is \(f(x) = \sqrt{x}\), where \(x ≥ 0\). The function \(g(x) = \sqrt{3x}\) is the result of a horizontal compression of the parent function by a factor of 1/3. The graph starts from the origin and increases more steeply than the graph of the parent function. The function \(g(x)\) can be written in terms of the parent function as \(g(x) = f(3x)\).

Step 1: Identify the Parent Function

The parent function of \(g(x) = \sqrt{3x}\) is \(f(x) = \sqrt{x}\), where \(x ≥ 0\). This is because the square root function is the simplest form of the given function.

Step 2: Describe the Transformations

The transformation from the parent function \(f(x) = \sqrt{x}\) to \(g(x) = \sqrt{3x}\) involves a horizontal compression by a factor of 1/3. This is because the x-value in the given function is multiplied by 3 inside the square root, which compresses the graph horizontally by a factor of 1/3.

Step 3: Sketch the Graph

The graph of the function \(g(x) = \sqrt{3x}\) starts from the origin (0,0) and increases slowly. Because of the horizontal compression, the graph increases more steeply than the parent function \(f(x) = \sqrt{x}\).

Step 4: Write in Terms of Parent Function

The function \(g(x)\) can be written in terms of its parent function \(f(x)\) as \(g(x) = f(3x)\). This shows that \(g(x)\) is the result of applying the function \(f\) to \(3x\) instead of \(x\).

Key Concepts

Parent Functions

Parent functions represent the simplest form of a set of functions sharing common characteristics. They act as a baseline or starting point for graph transformations. By modifying parent functions, we can generate a wide variety of other functions.Some common parent functions include:

• Linear function: \( f(x) = x \)
• Quadratic function: \( f(x) = x^2 \)
• Square root function: \( f(x) = \sqrt{x} \)
• Cubic function: \( f(x) = x^3 \)

Each parent function has its unique shape and properties. By knowing the parent function, it's easier to understand how transformations such as shifting, reflecting, or compressing affect the graph. For example, in the function \( g(x) = \sqrt{3x} \), the parent function is \( f(x) = \sqrt{x} \). This is the most basic representation of a square root function.

Square Root Functions

A square root function is a function of the form \( f(x) = \sqrt{x} \), where the input \( x \) should be non-negative because square roots of negative numbers are not real numbers.Characteristics of square root functions:

• The starting point, also called the vertex, is at the origin \((0, 0)\) when non-transformed.
• The graph is a curve that increases slowly without bound as \( x \) increases.
• It always lies above the x-axis, as the square roots of real numbers are non-negative.

In the exercise, the function \( g(x) = \sqrt{3x} \) is a transformation of the parent square root function. Shifting, reflecting, or compressing can significantly alter its graph from the original \( f(x) = \sqrt{x} \).

Horizontal Compression

Horizontal compression occurs when a function's graph is squeezed closer to the y-axis. This transformation affects the x-values by making them closer together. In mathematical terms, if a function \( f(x) \) is horizontally compressed, it can be expressed as \( f(cx) \), where \( c > 1 \).When dealing with horizontal compression, it's crucial to understand how it alters a graph:

• The factor \( c \) inside the function indicates the degree of compression. A value of \( c = 3 \) means each x-coordinate is compressed by a factor of \( \frac{1}{3} \).
• For square root functions, as in our example \( g(x) = \sqrt{3x} \), compression means the curve of the function becomes steeper and closer to the origin compared to its parent function \( f(x) = \sqrt{x} \).
• Horizontal compressions do not change the y-values of the points on the graph but merely reposition the existing points closer horizontally.

Understanding horizontal compression is crucial for graphing transformations correctly, as it affects how we interpret data or solve problems by visually analyzing the function's graph.