1. Graph this piecewise function.

f(x) = 3x – 1 if x < -2 ½x + 1 if x > -2

2. Write an equation for thispiecewise function.

{

Algebra II 1

Absolute Value

Algebra II

1.6 E

The graph of an absolute value function always forms a “V”!

Standard Form for Abs. Value

y = a|x – h| + kAlgebra II 3

y = a|x – h| + kVertex: (opposite of h, same as k)Line of Symmetry (LOS): x = opposite hReflections:

If “a” is negative the “V” will open down

If “a” is positive: the “V” will open upTransformations: (stretch & shrink)

If the |a| is bigger than one, then the graph will be skinnier.

If the |a| is less than one, the graph will be wider.

Algebra II 4

1. y = 3 |x – 5| + 1 2. y = ½ |x + 2| + 3

Vertex: (5, 1)

Vertex: (-2, 3)

Opens up Opens up

skinnier wider

Algebra II 5

3. y = -2 |x + 5| - 3 4. y = - |x – 3| + 3

Vertex: (-5, -3)

Vertex: (3, 3)

Opens down Opens down

Skinnier Same

Algebra II 6

1. Determine and graph the vertex

2. Graph the line of symmetry (LOS)

3. Make a table and graph the points

choose the next two x-values to the right of your vertex

4. Reflect your points over the LOS

5. Draw your “V”Algebra II 7

Graph:y = |x – 2| + 1

Opens up

Same width as |x|

Vertex: (2,1)

LOS: x = 2

Table

Reflect

x

3

4

y

2

3Algebra II 8

Graph:y = -½|x + 3| – 1

Opens down

Wider than |x|

Vertex: (-3,-1)

LOS: x = -3

Table

Reflect

x

-2

-1

y

-1.5

-2Algebra II 9

Graph:y = 3|x + 1| – 3

Opens up

Skinnier than |x|

Vertex: (-1,-3)

LOS: x = -1

Table

Reflect

x

0

1

y

0

3Algebra II 10

Graph:y = - |x – 3| + 3

Opens down

Same width as |x|

Vertex: (3,3)

LOS: x = 3

Table

Reflect

x

4

5

y

2

1Algebra II 11

1. Identify the vertex

2. Identify another point on the graph

3. Plug both the vertex and the point into y = a|x – h| + k (standard form)

4. Find “a” it is the only variable left!

5. Plug “a” and the vertex into standard form

y = a|x – h| + k (keep y and x here)

Algebra II 12

Write an absolute value function for: What is the vertex?

(1,0) this is h and k

(2, -2) is x and y

Plug these points in and solve for a.y = a|x – h| + k

-2 = a|2 – 1| + 0

-2 = a|1| + 0

-2 = a + 0

-2 = a

y = -2|x – 1| + 0Algebra II 13

Write an absolute value function for: What is the vertex?

(-2, -1) this is h and k

(-1, 2) is x and y

Plug these points in and solve for a.y = a|x – h| + k

2 = a |-1 + 2| – 1

2 = a |1| – 1

2 = a – 1

3 = a

y = 3|x + 2| – 1Algebra II 14

Write an absolute value function for: What is the vertex?

(-1, 2) this is h and k

(1,1) is x and y

Plug these points in and solve for a.y = a|x – h |+ k

1 = a|1 + 1| + 2

1 = a|2| + 2

1 = 2a + 2

-1 = 2a

-½ = a

y = -1/2 |x + 1| + 2Algebra II 15

Write an absolute value function for: What is the vertex?

(2, -2) this is h and k

(5, 0) is x and y

Plug these points in and solve for a.y = a|x – h| + k

0 = a|5 – 2| – 2

0 = a|3| – 2

2 = 3a

⅔ = a

y = ⅔|x – 2| – 2Algebra II 16

Algebra II

Real World Application

Algebra II

Real World Application

1. Graph: y = 2|x - 3| -1

2. Write an absolute value function of this graph:

y = -2| x – 1| + 4

Algebra II 19