Use transformations of the absolute value​ function, f left parenthesis x right parenthesis equals StartAbsoluteValue x EndAbsoluteValue​, to graph the function h left parenthesis x right parenthesis equals StartAbsoluteValue x plus 3 EndAbsoluteValue plus 4.
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Part 1
What transformations are needed in order to obtain the graph of​ h(x) from the graph of​ f(x)? Select all that apply.
A.
Reflection about the​ y-axis
B.
Reflection about the​ x-axis
C.
Vertical shift
D.
Horizontal shift
E.
Horizontal​ stretch/shrink
F.
Vertical​ stretch/shrink
Part 2
Use the graphing tool to graph the equation.



To determine the transformations needed to graph \( h(x) = |x+3| + 4 \) from \( f(x) = |x| \), we analyze the equation step by step.

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### Part 1: Identifying Transformations  
We start with the basic absolute value function \( f(x) = |x| \) and compare it to \( h(x) = |x+3| + 4 \).  

1. **Inside the absolute value \( |x+3| \):**  
   - The \( +3 \) indicates a **horizontal shift** to the left by 3 units.  
   - Remember: Adding a constant inside \( |x + c| \) moves the graph left by \( c \).

2. **Outside the absolute value \( +4 \):**  
   - The \( +4 \) indicates a **vertical shift** upward by 4 units.  
   - Adding a constant outside the function \( f(x) + c \) moves the graph up by \( c \).

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### Final Answer:  
The transformations are:  
- **C. Vertical shift** (up 4 units)  
- **D. Horizontal shift** (left 3 units)  

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### Part 2: Graphing the Function

To graph \( h(x) = |x+3| + 4 \):

1. Start with the parent function \( f(x) = |x| \), which has a "V" shape centered at \( (0,0) \).
2. **Shift it left 3 units** so the vertex moves to \( (-3, 0) \).
3. **Shift it up 4 units** so the vertex moves to \( (-3, 4) \).
4. The graph retains its "V" shape, opening upwards.

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### Instructions for Graphing Tool:  
1. Locate the new vertex at \( (-3, 4) \).  
2. Use the standard slope of the absolute value function:  
   - To the right, the slope is \( +1 \) (rise over run: 1 unit up for every 1 unit right).  
   - To the left, the slope is \( -1 \) (rise over run: 1 unit up for every 1 unit left).  
3. Draw the "V" shape starting from \( (-3, 4) \).

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### Recap:  
- Horizontal shift: left 3 units  
- Vertical shift: up 4 units  
- No reflection or stretch/shrink.  

Do you need further clarification, or help using the graphing tool? Let me know!  

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### Related Questions:  
1. What is the effect of a negative sign inside \( |x| \), such as \( |x-3| \)?  
2. How does the graph of \( y = |x| - 5 \) differ from \( y = |x| \)?  
3. If \( h(x) = -|x| \), what transformations occur?  
4. How would a vertical stretch affect the graph of \( y = |x| \)?  
5. How can you write a transformation for a graph shifted right by 2 units and down by 3 units?

### Tip:  
Remember that transformations inside the absolute value \( |x \pm c| \) affect the **horizontal direction**, while transformations outside affect the **vertical direction**.

Use transformations of the absolute value function, f(x) = |x|, to graph the function h(x) = |x+3| + 4.

Learn how to graph the absolute value function h(x) = |x+3| + 4, including identifying the necessary transformations from the parent function f(x) = |x|. This problem focuses on horizontal and vertical shifts, key concepts in graphing absolute value functions. Suitable for students in Grades 8-10.

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