Maximum or minimum of a absolute value function will be at vertex.
The absolute value function will be in the form
y = a|x - h| + k
Here (h, k) is the vertex, that would be maximum or minimum based on the direction of opening it may be decided.
How to find direction of opening ?
The sign of a will decide the direction of opening.
Identify the vertex.
Problem 1 :
y = -|x + 1|
Solution :
y = -|x + 1|
y = a|x - h| + k
y = -1|x - (-1)| + 0
The vertex (h, k):
(-1, 0)
Open: UP/DOWN
a = -1, the curve will open down.
Maximum/Minimum:
The maximum is at x = -1.
Narrower/Wider/Same :
Comparing with parent function y = |x|
y = -|x + 1|
Coefficient of absolute value function is same as the given function. So, width of this curve will be same.
Problem 2 :
y = 7|x - 3| - 4
Solution :
y = 7|x - 3| - 4
y = a|x - h| + k
The vertex (h, k) :
(3, -4)
Open: UP/DOWN :
a = 7, the curve will open up.
Maximum/Minimum :
The minimum is at x = 3.
Narrower/Wider/Same :
a = 7 which is greater than 1. So, width of this curve will be narrower.
Problem 3 :
y = -2/3|x - 1|
Solution :
y = -2/3|x - 1| + 0
y = a|x - h| + k
The vertex (h, k) :
(1, 0)
Open: UP/DOWN
a = -2/3, the curve will open down.
Maximum/Minimum:
The maximum is at x = 1
Narrower/Wider/Same :
a = 2/3 < 1, so it is wider.
Problem 4 :
y = 5/2|x + 9| - 1
Solution :
y = 5/2|x + 9| - 1
y = a|x - h| + k
y = 5/2|x - (-9)| - 1
The vertex (h, k) :
(-9, -1)
Open: UP/DOWN :
a = 5/2, the curve will open up.
Maximum/Minimum :
The minimum is at x = -9
Narrower/Wider/Same :
a = 5/2 > 1, so the curve is narrower.
Problem 5 :
y = 3/4|x + 3| - 6
Solution :
y = 3/4|x + 3| - 6
y = a|x - h| + k
y = 3/4|x - (-3)| - 6
The vertex (h, k) :
(-3, -6)
Open: UP/DOWN
a = 3/4, the curve will open up.
Maximum/Minimum:
The minimum is at x = -3
Narrower/Wider/Same:
a = 3/4 < 1, so it is wider.
Problem 6 :
y = -|x| + 5
Solution :
y = -|x| + 5
y = a|x - h| + k
y = -1|x - 0| + 5
The vertex (h, k) :
(0, 5)
Open: UP/DOWN :
a = -1, the curve will open down.
Maximum/Minimum :
Maximum is at x = 0
Narrower/Wider/Same :
a = 1. so it is same.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM