HOW TO FIND MAXIMUM OR MINIMUM OF A ABSOLUTE VALUE FUNCTION

Identify the vertex.

• Determine if the graph opens up or down.
• Determine if the graph has a maximum or minimum and its value.
• Decide if the graph is narrower, wider, or the same width as the parent graph.

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.