Deriving Kinematic Equations using Velocity vs Time Graph

Derivation of the Kinematics Equations for Uniformly Accelerated Motion
This derivation is based on the properties of a velocity-time graph for uniformly accelerated motion where the
slope of the graph represents the acceleration
graph's area represents the displacement
Equation #1: slope = acceleration
Starting with the slope
gives us our first equation:
In this equation
a represents the object's uniform acceleration
t represents the time interval over which the object's velocity changed
vf represents the object's final velocity at the end of the time interval
vo represents the object's initial velocity at the beginning of the time interval
Equation #2: rearrange equation #1 for vf
Equation #3: area = displacement
Equation #4: multiply equation #1 by equation #3
Equation #1:
Equation #3:
Equation #5: substitute equation #2 into equation #3
Equation #2:
Equation #3: