17. $$ f(x)=|2 x|+4 $$; horizontal shrink by a factor of $$ \frac{1}{2} $$

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UpStudy AI · Accepted Answer

To perform a **horizontal shrink** by a factor of $$ \frac{1}{2} $$ on the function $$ f(x) = |2x| + 4 $$, follow these steps:

### Understanding Horizontal Shrink

A **horizontal shrink** by a factor of $$ k $$ involves compressing the graph of the function towards the y-axis by multiplying the input $$ x $$ by $$ \frac{1}{k} $$. Specifically:

- **Shrink Factor ($$ k $$)**: $$ \frac{1}{2} $$
- **Transformation Rule**: Replace $$ x $$ with $$ \frac{x}{k} $$ or equivalently $$ x \times \frac{1}{k} $$.

### Applying the Transformation

Given:
$$ f(x) = |2x| + 4 $$

1. **Determine the Transformation:**
   - Shrink factor $$ k = \frac{1}{2} $$ implies $$ \frac{1}{k} = 2 $$.

2. **Replace $$ x $$ in the Function:**
   $$
   f_{\text{new}}(x) = |2 \times (2x)| + 4
   $$
   
3. **Simplify the Expression:**
   $$
   f_{\text{new}}(x) = |4x| + 4
   $$

### Resulting Function

After performing a horizontal shrink by a factor of $$ \frac{1}{2} $$, the transformed function is:
$$
f_{\text{new}}(x) = |4x| + 4
$$

### Graphical Interpretation

- **Original Function $$ f(x) = |2x| + 4 $$:** 
  - V-shaped graph with a vertex at $$ (0, 4) $$.
  - Steeper slopes due to the coefficient 2 inside the absolute value.

- **Transformed Function $$ f_{\text{new}}(x) = |4x| + 4 $$:**
  - Vertex remains at $$ (0, 4) $$.
  - Even steeper slopes due to the increased coefficient 4, indicating a greater horizontal compression.

### Visualization

Here's how the functions compare:

- **Original Function:**
  $$
  f(x) = |2x| + 4
  $$

- **After Horizontal Shrink:**
  $$
  f_{\text{new}}(x) = |4x| + 4
  $$

![Graph Comparison](https://i.imgur.com/graph_comparison_example.png)

*Note: The image above is a placeholder. When plotting, you'll see that the transformed function $$ f_{\text{new}}(x) $$ is narrower than the original $$ f(x) $$ due to the horizontal shrink.*

### Summary

- **Original Function:** $$ f(x) = |2x| + 4 $$
- **After Horizontal Shrink by $$ \frac{1}{2} $$:** $$ f_{\text{new}}(x) = |4x| + 4 $$

This transformation compresses the graph horizontally by a factor of $$ \frac{1}{2} $$, making it steeper and narrower.
After a horizontal shrink by a factor of $$ \frac{1}{2} $$, the function becomes $$ f(x) = |4x| + 4 $$.

; horizontal shrink by a factor of

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Understanding Horizontal Shrink

Applying the Transformation

Resulting Function

Graphical Interpretation

Visualization

Summary

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