Question

Transformations of f(x)=|3x+9|-6

Answer 1

The function f(x) = |3x + 9| - 6 is the result of horizontal compression, a leftward shift by 3 units, and a downward shift by 6 units applied to the base absolute value function.

Step-by-step explanation:

Transformations of the function f(x) = |3x + 9| - 6 involve shifts and reflections based on the modifications made to the base function, which is the absolute value function. When analyzing transformations, it is important to consider horizontal shifts, vertical shifts, reflections, and dilations. For f(x), the term +9 within the absolute value indicates a horizontal shift to the left by 3 units, since it is in the form f(x) = |3(x + 3)|. The -6 outside of the absolute value indicates a downward shift of 6 units.

To fully understand the transformations, let us look at the function step by step. The multiplication by 3 inside the absolute value, before the shift, corresponds to a horizontal compression by a factor of 1/3. Then, we apply the horizontal shift mentioned earlier, followed by the vertical shift. No reflection across the x-axis is present since there is no negative sign in front of the absolute value.

Here is a sequence to visualize the transformations:

1. Start with the base function, f(x) = |x|.
2. Apply a horizontal compression by multiplying the input by 3: f(x) = |3x|.
3. Shift the graph horizontally to the left by 3 units: f(x) = |3x + 9|.
4. Shift the graph downwards by 6 units to get the final function: f(x) = |3x + 9| - 6.