https://file.aiquickdraw.com/mathbot/file/1728880781987z497s787.jpg

Let's go through the problems from the image and solve them one by one.

### 54. \( m(x) = |x| + 6 \)
This represents a vertical shift of the absolute value function \( f(x) = |x| \). Specifically, it's shifted **6 units up**.
- **Domain**: All real numbers, \( (-\infty, \infty) \)
- **Range**: \( [6, \infty) \)

### 55. \( p(x) = |x - 4| \)
This represents a **horizontal shift** to the right by 4 units.
- **Domain**: All real numbers, \( (-\infty, \infty) \)
- **Range**: \( [0, \infty) \)

### 56. \( q(x) = 4|x| \)
This represents a **vertical stretch** by a factor of 4.
- **Domain**: All real numbers, \( (-\infty, \infty) \)
- **Range**: \( [0, \infty) \)

### 57. \( r(x) = \frac{1}{4}|x| \)
This represents a **vertical shrink** by a factor of \( \frac{1}{4} \).
- **Domain**: All real numbers, \( (-\infty, \infty) \)
- **Range**: \( [0, \infty) \)

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### 58. Horizontal translation 9 units right
To translate \( g(x) = |x| \) 9 units to the right, the equation would be:
- \( g(x) = |x - 9| \)

### 59. Vertical shrink by a factor of \( \frac{1}{6} \)
To vertically shrink the graph of \( g(x) = |x| \) by a factor of \( \frac{1}{6} \), the equation would be:
- \( g(x) = \frac{1}{6} |x| \)

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### 60. Compare the graphs and find \( h \)

The graph shows a translation of \( f(x) = |x| \) to the left by 2 units, so \( h = -2 \). The equation would be:
- \( p(x) = |x + 2| \)

### 61. Compare the graphs and find \( a \)

The second graph shows a vertical stretch. Since the graph is stretched vertically by a factor of 2, \( a = 2 \). The equation would be:
- \( d(x) = 2|x| \)

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Would you like more details on any of these steps?

### Relative Questions:
1. How does shifting an absolute value function horizontally affect its graph?
2. What is the difference between vertical stretching and shrinking of a function?
3. How would the range change if you applied a vertical shift down to \( f(x) = |x| \)?
4. What is the transformation rule for reflecting \( f(x) = |x| \) over the x-axis?
5. Can you explain how to find the vertex of the function \( g(x) = |x - h| + k \)?

### Tip:
When dealing with transformations of absolute value functions, always pay attention to whether you're adding, subtracting, multiplying, or dividing inside or outside the absolute value to determine the type of shift or stretch/shrink.

Graph the function. Compare the graph to the graph of f(x) = |x|. Find the domain and range. Solve various transformation problems and compare graphs.

Learn how to graph and transform absolute value functions such as f(x) = |x|. Explore vertical shifts, horizontal shifts, and vertical stretches with detailed examples and explanations for Grades 9-12 students.

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