Special Functions and Graphs

Algebra II …………… Sections 2.7 and 2.8

SWBAT identify, graph and write absolute value

functions

graph a function based on translations and transformations

graph and find the value of a piecewise defined function

graph the greatest integer function

Vocabulary absolute value function vertex transformation translation parent function reflection piecewise function step function

Absolute Value Function symmetric about y-

axis for every (x, y) there

is a point (-x, y) vertex - point where

sides come together parent function -

basic function, vertex is (0, 0)

General Equation

vertex: (h, k)

horizontal shift: x - h = 0

vertical shift: k

shape of graph remains the same

y ax h k

Examples

For each equation, state the vertex, describe the translation and sketch the graph

2 3f x x

4y x

1g x x

Transformations

shape of the graph changes stretch = more narrow shrink = wider vertical stretch:

vertical shrink:

reflection (about x-axis):

where 1y a f x a

where 1 1y a f x a

y f x

vertical ___________ by factor of ______

vertex is ___________

find point on each side of vertex

12f x x

reflected about _____________

vertical _________ by factor of _______

vertex is __________

two more points

2y x

vertex is ________

reflected about _________

vertical __________ by factor of _______

two more points

3 2 1f x x

Write the equation of the graph

Sketch additional graphs Use the graph to

sketch (a)

(b)

12y f x

1 2y f x

Piecewise Function defined by different expressions, each with a

separate domain

ex]

Find f(-3) and f(1)

2 3 , 2

2 5 , 2

x xf x

x x

Graph the function:

2 3 , 1

2 1 , 1

x xf x

x x

Write a piecewise function from the graph

Write the piecewise function from the graph

Greatest Integer Function (step function) greatest integer not

larger than the given value

y x