Question 14: Graph each of the following absolute functions:
a. y = |x – 4|
b. y = |-4-x|
c. y = |2x + 5|
d. y = -|x|

Solution

a. y = |x – 4| By definition, the given function is, Now (x-4) ≥ 0 ⇒ x ≥ 4.
and x-4 < 0 ⇒ x < 4
If x = 4, y = 0. Therefore, the graph will consist of two lines that meet at (4, 0).
Giving values to x and finding the corresponding values of y, the following data is obtained.

 x 4 5 6 7 y 0 1 2 3

And

 x 3 2 1 0 y 1 2 3 4

(b) y = |-4-x|

When we take the absolute value bars off, we get the equation as; Now if -4-x ≥ 0 ⇒ x ≥ -4
If -4-x < 0 ⇒ x < -4.
Now giving appropriate values to x in both segments and finding the corresponding value of y and then plotting it, (c) y = |2x + 5|

Take the absolut bars off, If 2x+5 ≥ 0 ⇒ x ≥-5/2
If 2x+5 < 0, ⇒ x < -5/2
So giving appropriate value to x and finding the corresponding value of y,
If x = -2 (>-5/2), y = 2(-2) + 5 = 1
If x = -3 (<-5/2), y = -[2(-3) + 5] = -(-6+5) = +1 d. y = – |x|

By definition the above function is Hence substituting values for x and finding y
If x = 0, y = 0
If x = 1, y = -1
If x = 2, y = -2
If x = 3, y = -3
AND
If x = -1, y = -1
If x = -2, y = -2
If x = -3, y = -3
Plotting on the graph 