Hello Ken,
For absolute value symbols, we want whatever is inside of them to turn out positive.
For example |5| = 5 and |-5| = 5.
So, suppose 2x-1 is already positive.
Then we wouldn't need the absolute value symbol, and we can just write the equation as
y = (4/3)(2x-1),
→ y = (8/3)x - (4/3), which you graph using point-slope.
Now suppose 2x-1 is negative. So, to get rid of the absolute value symbol, we can put a negative sign in front of the 2x-1 to turn it positive, i.e.
y = (4/3)(-(2x - 1))
→ y = (4/3)(-2x+1), by distributing the negative sign,
→ y = (-8/3)x + (4/3).
Now, absolute value equations are a 'v - shape', and the 'x' value of the vertex will be found by setting what is inside the absolute value equal to zero, i.e.
2x - 1 = 0
→ 2x = 1
→ x = 1/2.
When you plug this in for x, you get y = 0
→ the coordinate of the vertex is (1/2 , 0).
So you will graph the second equation we came up with from left to right, but only till you get to x = 1/2.
Then you will graph the first equation, starting at x = 1/2, and continuing to the right.
Why an I using the second equation for the left side of the 'v' and the fist equation for the right side of the 'v'?
From the original equation given,
4/3 is positive
→ the 'v' shape is 'face up'
→ you will be coming down on the left side, and
going up on the right side.
The second equation has a negative slope value, which will bring us down, and the first equation has a positive slope, which will take us back up.
If you message me directly, I can send you a hand written explanation with the graph drawn, in a PDF file.
Thank you for the question, and I hope the above helps.
~Michael E.
