Let $$ f(x)=|x| $$. Find a formula for a function $$ g $$ whose graph is obtained from $$ f $$ after the following sequence of t;onsformations: (1) Shift left 3 units. (2) Stretch vertically by a facto of 2 . (3) Reflect across the $$ x $$-axis. (4) Shift down 1 unit. $$ g(x)=-2|x+3|-1 $$ $$ g(x)=-2|x-3|-1 $$ $$ g(x)=2|-x+3|-1 $$ $$ g(x)=-\frac{1}{2}|x+3|-1 $$

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To determine the correct formula for the function $$ g(x) $$ after the given sequence of transformations applied to $$ f(x) = |x| $$, let's analyze each transformation step by step:

1. **Shift Left 3 Units:**
   - Shifting a function $$ f(x) $$ to the left by 3 units results in $$ f(x + 3) $$.
   - Therefore, $$ f(x) = |x| $$ becomes $$ |x + 3| $$.

2. **Stretch Vertically by a Factor of 2:**
   - Stretching vertically by a factor of 2 multiplies the function by 2.
   - Thus, $$ |x + 3| $$ becomes $$ 2|x + 3| $$.

3. **Reflect Across the $$ x $$-Axis:**
   - Reflecting across the $$ x $$-axis changes the sign of the function.
   - Therefore, $$ 2|x + 3| $$ becomes $$ -2|x + 3| $$.

4. **Shift Down 1 Unit:**
   - Shifting the function down by 1 unit subtracts 1 from the entire function.
   - So, $$ -2|x + 3| $$ becomes $$ -2|x + 3| - 1 $$.

Combining all these transformations, the final function $$ g(x) $$ is:

$$
g(x) = -2|x + 3| - 1
$$

**Answer:** $$ g(x)=-2|x+3|-1 $$
$$ g(x) = -2|x + 3| - 1 $$

Let . Find a formula for a function whose graph is obtained from after the
following sequence of t;onsformations: (1) Shift left 3 units. (2) Stretch vertically by a facto
of 2 . (3) Reflect across the -axis. (4) Shift down 1 unit.

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Let . Find a formula for a function whose graph is obtained from after the following sequence of t;onsformations: (1) Shift left 3 units. (2) Stretch vertically by a facto of 2 . (3) Reflect across the -axis. (4) Shift down 1 unit.