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Which absolute value function has a graph that is wider than the parent function, [tex]f(x)=|x|[/tex], and is translated to the right 2 units?

A. [tex]f(x)=1.3|x|-2[/tex]
B. [tex]f(x)=3|x-2|[/tex]
C. [tex]f(x)=\frac{3}{4}|x-2|[/tex]
D. [tex]f(x)=\frac{4}{3}|x|+2[/tex]

Sagot :

GinnyAnswer
To determine which absolute value function has a graph that is wider than the parent function [tex]\( f(x) = |x| \)[/tex] and is translated to the right 2 units, we need to examine each given function step by step.

### Step 1: Understanding the Parent Function

The parent function is [tex]\( f(x) = |x| \)[/tex]. This function has a V-shaped graph with the vertex at the origin (0,0) and opens upward.

### Step 2: Identifying a Wider Graph

A graph of an absolute value function is wider than the parent function if the coefficient of the absolute value is between -1 and 1 (excluding 0). Coefficients greater than 1 or less than -1 make the graph narrower.

### Step 3: Translation to the Right

To translate the graph of an absolute value function to the right by 2 units, the function must include [tex]\(|x - 2|\)[/tex] as part of its argument. This shifts the vertex of the V-shaped graph from [tex]\((0,0)\)[/tex] to [tex]\((2,0)\)[/tex].

### Evaluating the Given Functions

Let's examine each provided function:

1. [tex]\( f(x) = 1.3|x| - 2 \)[/tex]
- The coefficient [tex]\(1.3\)[/tex] is greater than 1, meaning the graph is narrower, not wider.
- There is no shifting term [tex]\(x - 2\)[/tex], so it is not translated to the right.
- This function does not meet the criteria.

2. [tex]\( f(x) = 3|x - 2| \)[/tex]
- The coefficient 3 is greater than 1, making the graph narrower.
- The term [tex]\(x - 2\)[/tex] indicates the graph is translated to the right by 2 units.
- This function only partially meets the criteria (translated right but not wider).

3. [tex]\( f(x) = \frac{3}{4}|x - 2| \)[/tex]
- The coefficient [tex]\(\frac{3}{4}\)[/tex] is between 0 and 1, meaning the graph is wider.
- The term [tex]\(x - 2\)[/tex] indicates the graph is translated to the right by 2 units.
- This function meets both criteria: wider graph and translated right.

4. [tex]\( f(x) = \frac{4}{3}|x| + 2 \)[/tex]
- The coefficient [tex]\(\frac{4}{3}\)[/tex] is greater than 1, making the graph narrower.
- There is no shifting term [tex]\(x - 2\)[/tex], so it is not translated to the right.
- This function does not meet the criteria.

### Conclusion

The function that has a graph which is wider than the parent function [tex]\( f(x) = |x| \)[/tex] and translated to the right 2 units is:

[tex]\[ f(x) = \frac{3}{4}|x-2| \][/tex]
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