1. Standard
MM2A1. Students will investigate step and piecewise functions, including greatest integer and absolute value functions.
b. Investigate and explain characteristics of a variety of piecewise functions including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, points of discontinuity, intervals over which the function is constant, intervals of increase and decrease, and rates of change.
c. Solve absolute value equations and inequalities analytically, graphically, and by using appropriate technology.

2. Absolute Value Functions
General Form: y = a | x – h | + k
Characteristics:
1. The graph is V-shaped
Vertex of the graph: (h, k)
note: opposite of h in general form
“a” acts as the slope for the right hand side (the left side is the opposite)

3. Absolute Value Functions
Parent Graph: y = | x |
x y ordered pair
Graph Transformations
What effect does each one have on the parent graph? y = a | x – h | + k
Determines if graph is
fatter 0 < a < 1
or skinnier a > 1
Moves the graph up (+)
or down (-)
Moves the graph left (+)
or right (-)
Determines if graph
opens up (+)
or down (-)

4. Determine the vertex of the following functions.
State whether the graph will open up or down.
y = 2 |x - 2| y = 1/3 |x| + 5
y = -|x + 5| y = |x|
y = -2|x + 2|

5. Steps for Graphing: Find and plot the vertex (opposite of h, k)
Find and sketch the axis of symmetry
Use “a” to find the slope and the next 2 points.
4) Using symmetry, plot 2 additional points and connect them to your vertex to create a “V” shaped graph!

6. Graphing Absolute Value Functions example 1
Vertex: ( , )
Slope: ________

7. Graphing Absolute Value Functions example 2
Vertex: ( , )
Slope: ________

8. Graphing Absolute Value Functions example 3
Vertex: ( , )
Slope: ________

9. Steps for writing an equation when given an absolute value graph.
Identify the vertex (opposite of h, k)
Determine if “a” will be positive or negative (opens up or down)
Find a point to the right of the vertex that the graph passes through exactly and count the slope from the vertex to the point. This is “a” (the slope!)
For the final answer: substitute “a” and the vertex (opposite of h, k) back into

10. Example 1 Vertex: ( , ) A is: positive / negative Slope: ________
Equation:
y =

11. Example 2 Vertex: ( , ) A is positive / negative Slope: ________
Equation:
y =

12. Writing Absolute Value Equations as Piecewise Functions from a Graph

13. Step 1:
Locate the vertex and draw a vertical dashed line to represent the breaking point of the graph This is the AXIS OF SYMMETRY This x-value will be your DOMAIN RESTRICTION for the inequality

14. Steps 2-3:
Write the equation of the RIGHT piece first
Find your slope by counting on the graph
Find your y-intercept by looking at the graph
(Remember: you may have to see where it WOULD cross the y-axis if the graph does not)
y = mx + b

16. Step 4:
To write the equation of the LEFT piece, make the slope from the RIGHT piece NEGATIVE
Find your new y-intercept by looking at the graph (this is where the line does or WOULD cross the y-axis.)

17. Step 5:
Just like a piecewise function is organized, write the equation of the RIGHT piece first including the DOMAIN RESTRICTION

18. Step 6:

19. Worksheet Side 1
Try these on your own!

20. EOCT Challenge!
Match the absolute value equation to its piecewise function

21. Convert Absolute Value Equations to Piecewise Functions

22. Step 1: Split absolute value into two separate equations
One with the same slope as original
One with the opposite slope as original

23. Step 2: Substitute “0” for x and solve for
y-intercept, so that f(0)= b

24. Step 3: Change into y=mx+b form
Identify slope
a = m (slope)
Identify y-intercept
f(0)= b

25. Step 4: Identify the Domain Restriction
Look at the original absolute value equation, the OPPOSITE of H is your axis of symmetry.
Just like graphing, this is your domain restriction
Put into piecewise form

26. Complete Writing Absolute Value as Piecewise worksheet.
Homework
Complete Writing Absolute Value as Piecewise worksheet.