1. Graph this piecewise function. f(x) = 3x – 1if x < -2 ½x + 1if x > Write an equation for this piecewise 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| + k Algebra II 3

y = a|x – h| + k  Vertex: (opposite of h, same as k)  Line of Symmetry (LOS): x = opposite h  Reflections:  If “a” is negative  the “V” will open down  If “a” is positive:  the “V” will open up  Transformations: (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| y = ½ |x + 2| + 3 Vertex: (5, 1)Vertex: (-2, 3) Opens up skinnierwider Algebra II 5

3. y = -2 |x + 5| y = - |x – 3| + 3 Vertex: (-5, -3)Vertex: (3, 3) Opens down SkinnierSame 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 3 Algebra II 8

Graph: y = -½|x + 3| – 1 Opens down Wider than |x| Vertex: (-3,-1) LOS: x = -3 Table  Reflect x -2 y Algebra 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 3 Algebra 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 1 Algebra 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| = a|1| = a = a y = -2|x – 1| + 0 Algebra 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| – 1 Algebra 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| = a|2| = 2a = 2a -½ = a y = -1/2 |x + 1| + 2 Algebra 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| – 2 Algebra II 16

Algebra II Real World Application

Algebra II Real World Application

1. Graph: y = 2|x - 3| Write an absolute value function of this graph: y = -2| x – 1| + 4 Algebra II 19