Question

What is the function g(x) that results when translating the function f(x) = |x| to the left 12 units and reflecting it over the x-axis?

Answer 1

To create the function g(x) from f(x) = |x|, translate f(x) left by 12 units and reflect it over the x-axis to get g(x) = -|x+12|.

Step-by-step explanation:

The original function f(x) = |x| can be transformed by translating it to the left by 12 units and reflecting it across the x-axis. The left translation is achieved by replacing x with (x + 12), which translates the graph to the left by 12 units since it is the opposite direction of the addition inside the function. To reflect it across the x-axis, we multiply the function by -1 to get the negative of the original value for any x. Thus, the resulting function, denoted as g(x), after applying the described transformations would be g(x) = -|x+12|.

Answer 2

g(x) = -|x + 12|

Step-by-step explanation:

Translation to left ≡ f(x + 12)

f(x + 12) = |x + 12|

Reflection of a function f(x) about the x - axis is keeping the x value same but negating the y values

Reflected f'(x) = -y = -f(x)

Taken together

g(x) = -(|x + 12|)

Check out the graphs attached to make these transformations clearer