Absolute Value Function

I've been condensing my notes into a couple pages per topi. I plan to do this kind of post for all of the functions, so I hope you like it and hope it helps with anyone trying to review the basics :)


Absolute Value Function 
Y= a |bx-h| + k


Basic absolute values:
|4| =  4         |1-5| = 4            |n • m| = |n| • |m|
|-3| = 3         |-6 x 2| = 12     |n/m| = |n| / |m|


If |?| = a and a > 0, there are 2 possible answers : -a and a
Example: |a| = 66, a can equal 66 or - 66


|?| = -1  IS IMPOSSIBLE
Simplifying parameter b :

To simplify parameter b, simply make it into parameter a by multiplying the absolute value of b with the value of a. If  parameter h is involved, factor out parameter b. 

Examples:  

f'(x) = 2|-3x|        g(x)= 4 |2x-2| +3            h(x)= |2x|
f(x)= 6 |x|            g(x)= 4 |2 (x-1)| + 3       h(x)= 2 |x|
                              g(x) = 8 |x-1| + 3
  
TO FIND THE VERTEX WE MUST FIRST SIMPLIFY PARAMETER B


Properties of f(x)= a |x-h|+ k


If a > 0
Domain = R
Range = [k, ∞
Vertex (h,k)
Min: k
Slopes of sides: +a, -a 

If a< o

Domain = R

Range = -∞, k]

Vertex (h,k)

Max: k

Slopes of sides: +a, -a 

Finding the rule:

1. Write template : y= a |x-h|+ k

2. Fill in given parameters

3. Substitute given point

4. Solve for missing parameters

Example 1: Given point (9, 17) and vertex (3,5)

1. y= a |x-h|+ k

2. y= a |x-3|+ 5

3. 17= a |9-3|+ 5

4. 17= 6a +5

      a=2

f(x)= 2 |x-3|+ 5

Example 2: Given zeroes -5 and 3, and minimum value of -2

* Because the vertex h is between the 2 zeroes, add and divide by 2 : (-5+3)/ 2 = -1

    vertex (-1, -2)

1. y= a |x-h|+ k

2. y= a |x+1|- 2

3. 17= a |3+1|- 2 

4. 2= 4a

      a= 0.5

f(x)= 0.5 |x+1|- 2

Finding the zero(es)

In an absolute value function, there can be 0, 1 or 2 zeroes. 

* change result of absolute value so you have both a negative and positive answer before you solve for x 

Example:  f(x)= |x+2| - 6

0= |x+2| -6

6=x+2

AND

-6= x+2

x= 4 or x= -8

Solving inequalities

In an absolute value function, there are 4 different possibilities for the answer of an inequality, they are:

a) 1 interval   X E [ 1st zero, 2nd zero]

b) 2 intervals   X E -∞, 1st zero] U [ 2nd zero, ∞

c) The entire X axis

d) The empty set X E ∅ 


Example: Rule < 0

a)

b)

c)

d)