AI Tutor
If you add 33 to a number, it gets bigger. So why does the graph of y=∣x+3∣y = |x + 3| shift backwards to the left instead of to the right?
Inside the absolute value bars, every horizontal change does the exact opposite of what your instincts tell you.

The Parent Function

The parent function is the simplest version of a function family, written as f(x)=∣x∣f(x) = |x|.
Its graph forms a sharp V-shape with its sharpest point, called the vertex, sitting right at the origin (0,0)(0, 0).
What happens when we modify the input variable xx directly before the absolute value evaluates it?

Horizontal Shifts: f(x - c)

Replacing xx with x−cx - c creates a horizontal shift, which slides the entire graph sideways along the x-axis without changing its size or shape.
To find where the new vertex lands, set the inside equal to zero: x−c=0x - c = 0, which means x=cx = c.
Because you must solve x−c=0x - c = 0, subtracting a positive number cc moves the vertex to the right, while adding a number moves it to the left.
A common mistake is seeing y=∣x−4∣y = |x - 4| and plotting the vertex at (−4,0)(-4, 0) because minus feels negative.