Figure 28 shows the graph of f(x)=|x|+1 . Match the functions (a)-(e) with their graphs (i)-(v).

step 1 Okay, we have to match the graphs with the functions. Now, here is the original functions, and w

Figure 28 shows the graph of f(x)=|x|+1 . Match the...

Figure 28 shows the graph of $f(x)=|x|+1 .$ Match the functions $(\mathrm{a})-(\mathrm{e})$ with their graphs $(\mathrm{i})-(\mathrm{v})$. $$ \begin{array}{ll}{\text { (a) } f(x-1)} & {\text { (b) }-f(x)} \\ {\text { (d) } f(x-1)-2} & {\text { (e) } f(x+1)}\end{array} \quad(\mathrm{c})-f(x)+2 $$

Figure 28 shows the graph of $f(x)=|x|+1 .$ Match the functions $(\mathrm{a})-(\mathrm{e})$ with their graphs $(\mathrm{i})-(\mathrm{v})$.

$$
\begin{array}{ll}{\text { (a) } f(x-1)} & {\text { (b) }-f(x)} \\ {\text { (d) } f(x-1)-2} & {\text { (e) } f(x+1)}\end{array} \quad(\mathrm{c})-f(x)+2
$$

Key Concepts

Absolute Value Function

The absolute value function, typically written as |x|, creates a V-shaped graph with a vertex at the origin. When a constant is added, as in |x| + 1, the entire graph shifts vertically. This function provides a fundamental example for studying and visualizing basic transformations of functions.

Horizontal Translation

Horizontal translations involve shifting the graph of a function left or right by adding or subtracting a constant within the function's argument. For example, f(x-1) shifts the graph to the right, while f(x+1) shifts it to the left. This concept is key to understanding how modifications to the input variable affect the location of key features of the graph.

Vertical Translation

Vertical translations occur when a constant is added to or subtracted from the entire function, shifting the graph upward or downward respectively. An example is f(x) - 2, where the entire graph moves downward by 2 units. This transformation affects the y-values directly, altering the vertical placement of the graph without changing its shape.

Reflection about the x-axis

A reflection across the x-axis involves multiplying the function by -1, which inverts the graph vertically. In the function -f(x), each y-value becomes its negative, flipping the original graph about the x-axis. This transformation is important when analyzing symmetry and changes in the direction of the graph.

Compound Transformations

Compound transformations combine two or more individual transformations, such as translations and reflections, into one operation. For instance, f(x-1)-2 includes a horizontal translation to the right by 1 unit and a vertical translation downward by 2 units. Understanding compound transformations is essential for accurately predicting the overall effects on the graph when multiple changes are applied simultaneously.

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