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Chapter 1: Problem 54

\(\mathrm{G}\) 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 \(f\). $$ g(x)=\sqrt{3 x+1} $$

Short Answer

Expert verified
The parent function is \(f(x) = \sqrt{x}\). There are two transformations from \(f\) to \(g\): a vertical stretch by a factor of 3 and a horizontal shift of 1 unit to the left. The function \(g\) can be written in terms of \(f\) as \(g(x) = f(3x + 1)\).

Step by step solution

01

Identify the parent function \(f\)

The parent function \(f\) in this case would be \(f(x) = \sqrt{x}\). This is because \(f(x)\) is the simplest form of the square root function.
02

Describe the sequence of transformations

The function \(g(x)=\sqrt{3x+1}\) has a vertical stretch by a factor of 3 and a horizontal shift 1 unit to the left from the parent function \(f(x)=\sqrt{x}\). This observation helps in understanding the sequence of transformations.
03

Sketch the graph

To sketch the graph, plot values of \(g(x)\) against a set of \(x\) values. The horizontal shift implies that the curve of the graph will begin from -1 on the \(x\)-axis as the square root function is undefined for negative numbers. Further, the vertical stretch by a factor of 3 implies a steeper graph as compared to the original function.
04

Write \(g\) in terms of \(f\)

Use the transformations identified earlier to write \(g\) in terms of \(f\). This can be expressed as \(g(x) = f(3x + 1)\)

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Key Concepts

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

Parent Function Identification
When working with algebraic functions, the term 'parent function' refers to the simplest function that maintains the definition of a particular family of functions. Identifying the parent function is the first step in understanding complex functions such as g(x) = \( \sqrt{3x + 1} \), presented in our exercise.

For the given function g(x), we see that its basic form is rooted in the square root function, f(x) = \( \sqrt{x} \). This square root function is the parent because it is the simplest form without any transformations such as shifts, stretches, or reflections. Recognizing the parent function provides a foundation upon which we build to understand the transformations that create more complex variations.
Function Transformations
Function transformations are various operations that modify the parent function to produce a new function which can be visualized as a 'changed' graph. Each transformation corresponds to a particular adjustment in the equation. For our function g(x) = \( \sqrt{3x + 1} \), there are two transformations:
  • A horizontal shift 1 unit to the left, which is shown by the +1 inside the square root.
  • A vertical stretch by a factor of 3, indicated by the coefficient 3 inside the square root.

Recognizing these transformations is crucial. The horizontal shift suggests a change in the x-values where the function begins, and the vertical stretch alters the steepness or 'rate of change' of the function relative to its parent.
Graph Sketching
Sketching the graph of a function is a visual representation of the function's behavior. Graph sketching for g(x) = \( \sqrt{3x + 1} \) incorporates our understanding of the parent function and its transformations. To sketch g(x), we start by considering the starting point, usually where x equals zero. However, due to the horizontal shift, the curve will actually start at x = -1.

Next, we note the steepness of the curve. The vertical stretch by a factor of 3 indicates that for every x, the curve will rise thrice as fast as the parent function f(x) = \( \sqrt{x} \). By applying these two transformations, we can determine the shape and position of the new function on a graph.
Function Notation
Function notation is a way to express the relationship between variables and functions systematically and concisely. It helps us in connecting and comparing different functions. Using function notation for our problem, g(x) is expressed in terms of the parent function f(x), incorporating the transformations. We can write g(x) = f(3x + 1), where f(x) = \( \sqrt{x} \).

This notation tells us that to find g(x), we apply the function f to the expression (3x + 1) instead of just x. It effectively captures the transformations in a compact, mathematical way, making it easier to analyze the relationship between g and its parent function f.

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Most popular questions from this chapter

Find a mathematical model for the verbal statement. \(h\) varies inversely as the square root of \(s\).

On a yardstick with scales in inches and centimeters, you notice that 13 inches is approximately the same length as 33 centimeters. Use this information to find a mathematical model that relates centimeters \(y\) to inches \(x\). Then use the model to find the numbers of centimeters in 10 inches and 20 inches.

Mathematical models that involve both direct and inverse variation are said to have _____ variation.

The winning times (in minutes) in the women's 400-meter freestyle swimming event in the Olympics from 1948 through 2008 are given by the following ordered pairs. $$\begin{array}{lll} (1948,5.30) & (1972,4.32) & (1996,4.12) \\ (1952,5.20) & (1976,4.16) & (2000,4.10) \\ (1956,4.91) & (1980,4.15) & (2004,4.09) \\ (1960,4.84) & (1984,4.12) & (2008,4.05) \\ (1964,4.72) & (1988,4.06) & \\ (1968,4.53) & (1992,4.12) & \end{array}$$ A linear model that approximates the data is \(y=-0.020 t+5.00,\) where \(y\) represents the winning time (in minutes) and \(t=0\) represents \(1950 .\) Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain. (Source: International Olympic Committee)

Determine whether the variation model is of the form \(y=k x\) or \(y=k / x,\) and find \(k .\) Then write \(a\) model that relates \(y\) and \(x\). $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline y & 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \hline \end{array} $$
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