FIND OF VERTEX OF ABSOLUTE VALUE FUNCTION

Any absolute value function will be in the form 

y = a|x - h| + k

Here (h, k) is vertex.

Vertex can be minimum or maximum point.

• If the curve opens up, then it will have minimum point at vertex.
• If the curve opens down, then it will have maximum point at vertex.

Here a represents slope,

• If a is positive, then the curve will open up.
• If a is negative, then the curve will open down.

Find the vertex of absolute value function given below and find the direction of opening.

Problem 1 :

y = 1/4│x + 4│- 9

Solution :

y = 1/4│x + 4│- 9

Comparing the given function with

y = a │x - h│+ k

y = 1/4 │x - (-4)│- 9

Vertex (h, k) = (-4, -9)

a = 1/4

It is positive, so it will open up.

Problem 2 :

y = -2│x + 1│+ 6

Solution :

y = -2│x + 1│+ 6

y = a │x - h│+ k

y = -2 │x - (-1)│+ 6

Vertex (h, k) = (-1, 6)

a = -2

It is negative, so it opens down.

Problem 3 :

y = 4│x - 3│

Solution :

y = 4│x - 3│

Compare with

y = a │x - h│+ k

y = 4 │x - 3│+ 0

Vertex (h, k) = (3, 0)

a = 4

It is positive, so it will open up.

Problem 4 :

y = -1/2│x│+ 3

Solution :

y = -1/2│x│+ 3

Compare with 

y = a │x - h│+ k

y = -1/2 │x - 0│+ 3

Vertex (h, k) = (0, 3)

a = -1/2

It is negative, so it will open down.

Problem 5 :

y = -5│x - 8│- 5

Solution :

y = -5│x - 8│- 5

Vertex (h, k) = (8, -5)

a = -5

It is negative, so it will open down.