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