Use transformations of the absolute value function, $$ f(x)=|x| $$, to graph the function $$ h(x)=|x+4|+5 $$. What transformations are needed in order to obtain the graph of $$ h(x) $$ from the graph of $$ f(x) $$ ? Select all that apply. $$ \square $$ A. Horizontal stretch/shrink $$ \square $$ B. Vertical shift $$ \square $$ C. Vertical stretch/shrink $$ \square $$ D. Horizontal shift $$ \square $$ E. Reflection about the $$ x $$-axis $$ \square $$ F. Reflection about the $$ y $$-axis

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We start with the base function:

$$
f(x)=|x|
$$

and want to graph the transformed function:

$$
h(x)=|x+4|+5
$$

**Step 1: Horizontal Shift**

The expression $$|x+4|$$ inside the absolute value shows a horizontal translation. In general, replacing $$ x $$ with $$ x+c $$ shifts the graph to the left by $$ c $$ units if $$ c > 0 $$.

Since we have $$ x+4 $$, this corresponds to a shift $$4$$ units to the left.

**Step 2: Vertical Shift**

The additional $$ +5 $$ outside the absolute value corresponds to a vertical translation. In general, adding a constant outside the function shifts the graph upward by that constant if it is positive.

Thus, the $$+5$$ shifts the graph upward by $$5$$ units.

**Conclusion:**

The transformations needed to obtain the graph of $$ h(x) $$ from $$ f(x) $$ are:

- Horizontal shift (to the left by 4 units)
- Vertical shift (upward by 5 units)

Thus, the correct selections are:

$$
\square \textbf{B. Vertical shift} \quad \text{and} \quad \square \textbf{D. Horizontal shift}
$$
To graph $$ h(x) = |x+4| + 5 $$ from $$ f(x) = |x| $$, you need to perform a horizontal shift 4 units to the left and a vertical shift 5 units up. So, select options **B. Vertical shift** and **D. Horizontal shift**.

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