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.