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)