Linear and Absolute Value Functions - Graphing Transformations and Characteristics
Using the linear function to the right, answer the following:
y
10
 ,  
or x 

Domain:

Range:

Increasing Interval:

Decreasing Interval: none

End Behavior:
 ,  
or
8
6
4
y
 ,  
2
–10 –8
–6
–4
–2
–2
as x  , f  x   
as x  , f  x   
2
4
6
8
10
–4
–6
–8
–10
Use the linear function in ActivInspire to answer the corresponding questions.
1. Shift the graph up and down. You can move the graph by selecting the pointer
clicking on the function and dragging it up and down.
then
a. Does shifting the graph (changing the y-intercept) change any of the 5 characteristics
initially found? List any characteristics changed and how they were affected.
A vertical shift (meaning a change to the y-intercept) does not affect the domain,
range, increasing/decreasing intervals, or end behavior
2. Rotate the graph to change the slope. You can rotate the graph by clicking on the function
and selecting the rotation icon
(it appears upon selecting the object) and then
dragging the object.
a. Does rotating the graph (changing the slope) change any of the 5 characteristics
initially found? List any characteristics changed and how they were affected.
Negative slope – decreasing interval only  ,   and end behavior is now
as x  , f  x   ; as x  , f  x   
Zero slope (horizontal line) – no increasing or decreasing interval (it’s constant) and
end behavior is as x  , f  x   0; as x  , f  x   0 (don’t worry about
constant graphs)
3.
What conclusions can be made about how linear graphing transformations (i.e. changes to
the graph) affect the characteristics of a linear graph?
All linear graphs have the same domain and range:  ,  
All increasing linear graphs have an increasing interval of  ,   and the same end
behavior of as x  , f  x   ; as x  , f  x    .
All decreasing linear graphs have a decreasing interval of  ,   and the same end
behavior of as x  , f  x   ; as x  , f  x   
x
y
10
8
Using the absolute value function to the right, answer the
following:
 Domain:  ,   or x 

Range: 0,  or

Increasing Interval:


y0
 0,  
Decreasing Interval:  ,0 
as x  , f  x   
End Behavior:
as x  , f  x   
6
4
2
–10 –8
–6
–4
–2
–2
2
4
6
8
10
x
–4
–6
–8
–10
Use the absolute value function in ActivInspire to answer the corresponding questions.
4. Shift the graph up and down. Does shifting the graph (changing the y-coordinate) change
any of the 5 characteristics initially found? List any characteristics changed and how they
were affected.
Shifting the graph vertically (changing the y-coordinate) affects the range.
5. Shift the graph left and right. Does shifting the graph (changing the x-coordinate) change
any of the 5 characteristics initially found? List any characteristics changed and how they
were affected.
Shifting the graph horizontally (changing the x-coordinate) affects the intervals
.
6. Stretch the graph to change the slope. You can stretch the graph by clicking on the function
and selecting the surrounding circle icons
(it appears upon selecting the object) and
then dragging the circles.
a. Does stretching the graph (changing the slope) change any of the 5 characteristics
initially found? List any characteristics changed and how they were affected.
Stretching the graph to have a wide or narrow opening function does not affect
domain, range, increasing/decreasing intervals, or end behavior.
7. Rotate the graph to point the graph down. Does rotating the graph (changing the slope)
change any of the 5 characteristics initially found? List any characteristics changed and how
they were affected.
Negative slope – range is now  ,0 and end behavior is x  , f  x   ; x  , f  x   
and intervals are changed
8.
What conclusions can be made about how linear graphing transformations (i.e. changes to
the graph) affect the characteristics of a linear graph?
All absolute value functions have the same domain:  ,   .
All positive absolute value functions have the same end behavior and all negative absolute
value functions have the same end behavior.
Increasing/decreasing intervals are affected by changes to the x-coordinate (shifts left and
right).
Range is affected by changes to the y-coordinate (shifts up and down).