MM2A1. Students will investigate step and MM2A1. Students will investigate step and piecewise functions, including greatest piecewise functions, including greatest integer and absolute value functions.integer and absolute value functions.

b. Investigate and explain characteristics of a variety of piecewise functions b. Investigate and explain characteristics of a variety of piecewise functions including domain, range, vertex, axis of symmetry, zeros, intercepts, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, points of discontinuity, intervals over which the function is extrema, points of discontinuity, intervals over which the function is constant, intervals of increase and decrease, and rates of change.constant, intervals of increase and decrease, and rates of change.

c. Solve absolute value equations and inequalities analytically, graphically, and c. Solve absolute value equations and inequalities analytically, graphically, and by using appropriate technology.by using appropriate technology.

Absolute Value FunctionsGeneral Form: y = a | x – h | + k

1. The graph is V-shaped

2. Vertex of the graph: (h, k)

note: opposite of h in general form

3. “a” acts as the slope for the right hand side (the left side is the opposite)

Absolute Value FunctionsParent Graph: y = | x |

x y ordered pair

What effect does each one have on the parent graph? y = a | x – h | + k

Determines if graph opens up (+)or down (-)

Determines if graph is

fatter 0 < a < 1

or skinnier a > 1

Moves the graph left (+)

or right (-)

Moves the graph up (+)

or down (-)

1. y = 2 |x - 2| + 3 4. y = 1/3 |x| + 5

2. y = -|x + 5| - 6 5. y = |x|

3. y = -2|x + 2|

Steps for Graphing:Steps for Graphing:1) Find and plot the vertex (opposite of h,

k)

2) Find and sketch the axis of symmetry

3) 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!

run

risea

Graphing Absolute Value Functionsexample 1

32 xy

Vertex: ( , )

Slope: ________

Graphing Absolute Value Functionsexample 2

11 xy

Vertex: ( , )

Slope: ________

Graphing Absolute Value Functionsexample 3

322

1 xy

Vertex: ( , )

Slope: ________

Steps for writing an equation when Steps for writing an equation when given an absolute value graph.given an absolute value graph.

1) Identify the vertex (opposite of h, k)2) Determine if “a” will be positive or

negative (opens up or down)3) 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!)

4) For the final answer: substitute “a” and the vertex (opposite of h, k) back into

khxay

Example 1

Vertex: ( , )

A is: positive / negative

Slope: ________

Equation:

y =

Example 2

Vertex: ( , )

A is positive / negative

Slope: ________

Equation:

y =

Writing Absolute Value Equations as Piecewise Functions from a Graph

Step 1:Locate the vertex and draw a vertical dashed line to represent the breaking point of the graph

This is the AXIS OF SYMMETRYThis x-value will be your DOMAIN RESTRICTION for the inequality

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 + brise

run

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.)

Step 5: Just like a piecewise function is

organized, write the equation of the RIGHT piece first including the DOMAIN RESTRICTION

_________________( )

_________________f x

Step 6:

Worksheet Side 1 Try these on your own!

EOCT Challenge!

Match the absolute value equation to its piecewise function

2y x

x 2 if x 0. y

-x 2 if x 0b

x-2 if x 2. y

-x-2 if x 2a

x - 2 if x 2. y

-x 2 if x 2c

x 2 if x 2. y

-x-2 if x 2d

-x-2 if x 0. y

x-2 if x 0e

Convert Absolute Value Equations to Piecewise Functions

Step 1: Split absolute value into two separate

equations One with the same slope as original One with the opposite slope as original

( ) 2 1 3f x x

( ) 2 1 3f x x ( ) 2 1 3f x x

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

( ) 2 1 3f x x

( ) 2 1 3f x x

( ) 2 1 3f x x

Step 3: Change into y=mx+b form Identify slope

a = m (slope)

Identify y-intercept f(0)= b

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

_________________( )

_________________f x

Homework

Complete Writing Absolute Value as Piecewise worksheet.