Hello!
Let's go through some formula to distinguish the changes that happen to a graph's equation when they are transformed.
Vertical shifts have this type of formula:
f(x) + k → up k units and f(x) - k → down k units
Horizontal shifts have this type of formula:
f(x + k) → left k units and f(x - k) → right k units
Reflections have this type of formula:
-f(x) → reflect over x-axis and f(-x) → reflect over y-axis
Vertical stretches have this type of formula:
a · f(x) where a > 1
Vertical compressions have this type of formula:
a · f(x) where a < 1
Horizontal stretches have this type of formula:
f(a · x) where a > 1
Horizontal compressions have this type of formula:
f(a · x) where a < 1
With that in mind, we can write our transformed absolute value function.
Since the equation is vertically stretched by a factor of 2, a = 2.
y = 2|x|
Also, since the function is shifted left 3 units, k = -3.
y = 2|x + 3|
Finally, the function is also shifted down 4 units, so k = -4.
y = 2|x + 3| - 4
Therefore, the equation is y = 2|x + 3| - 4.
