Graph $$ f(x) $$ sketch the following 1. $$ 2 f(x) $$ 2. $$ f(2 x) $$ $$ 3 f(x-1)+2 $$ 4. $$ -2 f(2 x+4) $$

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To graph the function $$ f(x) $$ and its transformations, we will follow these steps:

1. **Graph $$ 2 f(x) $$**: This transformation vertically stretches the graph of $$ f(x) $$ by a factor of 2.

2. **Graph $$ f(2 x) $$**: This transformation compresses the graph of $$ f(x) $$ horizontally by a factor of 2.

3. **Graph $$ 3 f(x-1) + 2 $$**: This transformation shifts the graph of $$ f(x) $$ to the right by 1 unit, vertically stretches it by a factor of 3, and then shifts it up by 2 units.

4. **Graph $$ -2 f(2 x + 4) $$**: This transformation compresses the graph of $$ f(x) $$ horizontally by a factor of 2, shifts it to the left by 2 units, vertically stretches it by a factor of 2, and reflects it across the x-axis.

To create these graphs, we need to know the original function $$ f(x) $$. Please provide the function $$ f(x) $$ so that we can proceed with the graphing.
To graph the function $$ f(x) $$ and its transformations: 1. **Double the height**: $$ 2 f(x) $$ makes the graph twice as tall. 2. **Halve the width**: $$ f(2 x) $$ makes the graph twice as narrow. 3. **Shift right and stretch**: $$ 3 f(x-1) + 2 $$ moves the graph right by 1 and up by 2, then stretches it vertically by 3. 4. **Reflect and stretch**: $$ -2 f(2 x + 4) $$ moves the graph left by 2, compresses it horizontally by 2, stretches it vertically by 2, and flips it upside down. Please provide the original function $$ f(x) $$ to complete the graphs.

Graph
sketch the following

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