Parallel-Axis Theorem

Shift the rotation axis away from the center of mass and watch the moment of inertia increase.

Center-axis inertia, ICM1.50 kg·m²
Axis shift, d0.50 m
Added inertia, Md²1.00 kg·m²
Shifted-axis inertia, I2.50 kg·m²
Increase over ICM66.7%

Uniform rod

4.00 kg
Increasing M scales both ICM and Md².
2.00 m
For this rod, ICM = ML²/12.
+0.50 m
The sign chooses left/right. Moment of inertia depends on d².
I = ICM + Md²,    ICM = ML²/12
Axis shifted 0.50 m to the right: Every mass element is, on average, farther from the new axis, so the moment of inertia is larger.

Shift the axis

CM CM axis shifted axis d = +0.50 m L = 2.00 m
center-of-mass axis shifted parallel axis
ICM
1.50
+ Md²
1.00
Total I
2.50

I versus axis displacement

d I current axis d = 0 gives the minimum I
Why a parabola? The extra term is Md². Therefore shifting the axis by +d or −d gives the same increase in moment of inertia.
Predict before moving the axis

Compare d = +1 m and d = −1 m. Will the two values of I be different?