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Chapter 1: Problem 6

Identify the function family to which \(f\) belongs. Compare the graph of \(f\) to the graph of its parent function. \(f(x)=3\)

Short Answer

Expert verified
The function \(f(x) = 3\) belongs to the family of constant functions. Its parent function is \(y = 0\). The graph of \(f(x) = 3\) is a horizontal line which is shifted upwards by 3 units from the graph of its parent function.

Step by step solution

01

Identify Function Family

The first step in solving this problem is to identify the function family that the function \(f(x) = 3\) belongs to. This function is a simple linear function of the form \(y = c\), where \(c\) is a constant. So, it belongs to the family of constant functions.
02

Identify Parent Function

The next step is to identify the parent function of this function family. Parent functions are the simplest form of each family of functions. For this function family, the parent function is \(y = 0\). This is a straight horizontal line along the x-axis.
03

Compare the graphs

The final step is to compare the graph of the function \(f(x) = 3\) to the graph of its parent function \(y = 0\). The graph of the function \(f(x) = 3\) is also a horizontal line, but it is shifted upwards by 3 units from the parent function.

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Key Concepts

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

Parent Function
The concept of a parent function is key in understanding the behavior of function transformations. In essence, a parent function is the most basic form of a function family. In the case of constant functions, the parent function is notably simple: it is represented by the equation \(y = 0\). This tells us that for any value of \(x\), the output is always \(0\). The graph of this function is a horizontal line that sits directly on the x-axis. This simplicity allows us to easily predict variations when transformations, such as translations or stretches, are applied to the function.
Graph Comparison
When we talk about graph comparison, what we're doing is analyzing how the graph of a specific function differs from or resembles its parent function. For the constant function \(f(x) = 3\), we begin by comparing it to its parent function, \(y = 0\). Both graphs are horizontal lines. However, there's a distinct difference in their vertical positions.
  • The parent function \(y = 0\) is on the x-axis.
  • The function \(f(x) = 3\) is 3 units above the x-axis.

This upward shift indicates that all outputs of \(f(x) = 3\) are consistently greater than those of the parent function by 3. Thus, while maintaining the same overall shape, the two graphs are positioned differently in the coordinate plane.
Horizontal Translation
Horizontal translation of a function involves shifting the graph left or right on the x-axis. However, for constant functions, horizontal translations do not apply as they manifest, regardless of \(x\), as a single, unchanging value. Consider the function \(f(x) = 3\). Vertical shifts characterize changes to constant functions. In this scenario, what you see is a vertical translation of 3 units up from its parent function, \(y = 0\).
  • Horizontal shifts do not affect constant functions, because there is no dependence on \(x\).
  • Only vertical translations alter constant functions, as demonstrated by the constant addition here.

Thus, while understanding horizontal translations is essential for various function types, its relevance diminishes when examining constant functions like \(f(x) = 3\). Rather, the focus here is on vertical adjustments.

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Most popular questions from this chapter

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