Use transformations of the absolute value​ function,

f left parenthesis x right parenthesis equals StartAbsoluteValue x EndAbsoluteValuef(x)=x​,

to graph the function

h left parenthesis x right parenthesis equals StartAbsoluteValue x plus 1 EndAbsoluteValue plus 3h(x)=x+1+3.

To graph the function \( h(x) = |x + 1| + 3 \) using transformations of the basic absolute value function \( f(x) = |x| \), we need to understand how each transformation modifies the original graph. Here's how to proceed:

### Step-by-Step Explanation:
1. **Start with the basic absolute value function**:
   - The basic graph of \( f(x) = |x| \) is a V-shaped graph that has a vertex at the origin (0, 0) and opens upwards.

2. **Apply the transformation \( x + 1 \)** inside the absolute value**:
   - The transformation \( |x + 1| \) represents a horizontal shift. Since the "+1" is inside the absolute value function, it shifts the graph 1 unit **to the left**. This moves the vertex from (0, 0) to (-1, 0).

3. **Apply the vertical shift of +3**:
   - The "+3" outside of the absolute value function shifts the graph **upwards** by 3 units. This moves the vertex from (-1, 0) to (-1, 3).

### Final Graph:
- The resulting graph of \( h(x) = |x + 1| + 3 \) is a V-shaped graph with a vertex at the point (-1, 3).
- The slope of the arms of the V remains the same as that of the basic absolute value function, with the left arm having a slope of -1 and the right arm having a slope of +1.

### Summary of Transformations:
1. Horizontal shift: 1 unit to the left.
2. Vertical shift: 3 units upwards.

Would you like a detailed graph or further explanation of how to visualize the graph?

Here are 5 relative questions to expand on this information:
1. How does changing the coefficient inside the absolute value function affect the graph?
2. What happens to the graph if you subtract a value outside the absolute value function?
3. Can you describe the graph of \( h(x) = -|x + 1| + 3 \)?
4. How does combining horizontal and vertical transformations change the graph of any function?
5. What are other functions that have a V-shaped graph, and how do they compare to absolute value functions?

**Tip:** Always handle horizontal transformations (inside the function) first, as they impact the x-values directly, followed by vertical transformations (outside the function) that affect the y-values.

Use transformations of the absolute value function, f(x) = |x|, to graph the function h(x) = |x + 1| + 3.

Learn how to graph the function h(x) = |x + 1| + 3 using transformations of the basic absolute value function f(x) = |x|. This solution provides a step-by-step explanation of horizontal and vertical shifts and is ideal for students in Grades 8-10.

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