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Chapter 0: Problem 73

\(f(x)=|x|+2\)

Short Answer

Expert verified
The function \(f(x) = |x| + 2\) represents the absolute value function, vertically shifted up by 2 units. Its graph has a 'V' shape, similar to |x|, and the vertex is now at the point (0,2).

Step by step solution

01

Understanding the absolute function

The absolute function |x| is a piecewise function that returns the 'magnitude' or 'size' of a real number, ignoring its sign. It is represented as: |x| = x for \(x \geq 0\) and |x| = -x for \(x < 0\).
02

Analyzing the given function

The given function is \(f(x) = |x| + 2\). This is a vertical translation of the function |x|, meaning it shifts the graph up by 2 units. This occurs because the '+2' is added to the entire function which affects its y-values.
03

Graphing

The last step is to graph the function. Begin by plotting the function |x|, then shift every point on this graph upward by 2 units to get \(f(x) = |x| + 2\). The vertex of this graph, which was originally at (0,0) for |x|, now moves to (0,2). The shape of the graph remains same, a 'V' shape, as the '+2' only moves the graph, but does not change its shape.

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Key Concepts

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

Deciphering Vertical Translation
Vertical translation is a specific type of transformation that shifts the graph of a function up or down in a parallel motion. With the given function \(f(x) = |x| + 2\), we experience a vertical translation of 2 units upwards. It means for each point \((x, y)\) on the original graph of \(|x|\), the point on the translated graph is \((x, y + 2)\).

This does not affect the 'spread' or 'slope' of the graph but changes the y-coordinate of each point. For example, the origin point \((0, 0)\) on the graph of \(|x|\) is translated to \((0, 2)\) on the graph of \(f(x)\). This is a palpable change which can be quickly performed on the entire graph without recalculating any new points, saving time and effort when plotting functions.

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Most popular questions from this chapter

Each function models the specified data for the years 1995 through 2005 , with \(t=5\) corresponding to 1995 . Estimate a reasonable scale for the vertical axis (e.g., hundreds, thousands, millions, etc.) of the graph and justify your answer. (There are many correct answers.) (a) \(f(t)\) represents the average salary of college professors. (b) \(f(t)\) represents the U.S. population. (c) \(f(t)\) represents the percent of the civilian work force that is unemployed.

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=-2 x+15 & x_{1}=0, x_{2}=3 \end{array} $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{x+1}{x-2} $$

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ \(f(x)=x^{2}-2 x+8 &\quad x_{1}=1, x_{2}=5\) \end{array} $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=3 x+1 $$
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