Piecewise-defined Functions
Algebra 1
Name:
Date:
Teacher Guide
Piecewise-defined functions are ideal for situations that show a pattern for a while, and then
suddenly behave differently. We use domain values (the independent variable) to indicate when
the change takes place. Today we will describe the behavior of a piecewise graph, determine the
shape of a piecewise-defined function given a description of the situation or the equation, and
write the equation for a piecewise-defined function from a graph.
Exercise: High-intensity Interval Training
One method to get the most out of your workout is High-intensity Interval Training (HIIT).
People who exercise using the HIIT model alternate between spurts of high-energy sprints and
medium-energy recovery periods. The graph below shows an example of the first ten minutes of
a HIIT workout session.
Describe what you notice about the graph. Express changes in the graph in relation to the
intervals (domain values) over which they occur.
When the person exercising is transition between speeds, there is a diagonal line to represent
the change in speed. Once the person reaches the speed he wants, it is a constant line over
the interval. Then it transitions to the next speed.
Disease Prevention: Malaria
A Sub-Saharan African village has struggled with malaria for years. This life-threatening
disease is transmitted through the bites of infected mosquitoes. The number of people in the
village infected by malaria grew steadily from about 6,350 cases in 1998 to about 13,560 cases in
2005. In 2005, an aid organization helped the village equip its residents with indoor pesticide
sprays and pesticide-treated mosquito nets to drape over their beds. The number of cases of
malaria dropped by 65% over each of the next five years so that there were only 71 new cases of
malaria reported in 2010.
How would you define your variables? t = year, M(t) = new reported cases of malaria
What would the shape of a graph drawn to represent this situation look like? The graph
would start as a positive line between the years 1998 and 2005. Beginning in 2005, the
graph would show an exponential decay curve.
Would the graph follow the same pattern or shape for the entire domain? If not, where would
the change take place? Why would this happen? The change would happen at t = 2005. This
change would happen because of the prevention methods brought by the aid organization.
Function Behavior: Determining the Shape of the Graph
Describe how the sections of each piecewise-defined function will behave. Indicate where in the
domain a change will happen. Determine whether the graph pieces will connect or be
discontinuous.
Show all work to help justify your response. You can graph the functions to help justify.
A.
10
2 x 2  12x  20 , x  4

f x   4
,4  x  7
x  1
,x7

The first piece of this function is a quadratic function. It
will be a parabola that opens upwards. It will go on
forever as x , hit its vertex at (3, 2) and end with a
closed circle at (4, 4).
The second piece is a constant linear function. It will
be a horizontal line beginning with an open circle at
(4, 4) and ending with an open circle at (7, 4). It will
be connected to the first piece.
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
2
4
6
8 10
-2
-4
-6
-8
-10
The third piece is a positive-correlation linear function.
It will start with a closed circle at (7, 6) and continue
forever as x   . It will not connect to the second
piece.
10
8
6
4
B.
 0.5x  1 ,8  x  0
f x    x
,x0
2
2
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
The first piece of this function is a negative-correlation
linear function. It begins with an open circle at (-8, 5)
and ends with an open circle at (0, 1).
The second piece of this function is an exponential
growth curve. It begins with a closed circle at (0, 1) and
continues forever as x   . The second and first pieces
connect.
Outdoors: Hiking a Trail
Luis and Matt are on a camping trip with some friends. They decide to go on a hike to some
nearby waterfalls while the rest of the group stays near the camp site. It takes the boys about 30
minutes to walk the relatively flat path to the start of the trail, two miles from camp. They stop
at the trail shop for 10 minutes to use the restrooms and buy a snack. They start up the steep,
rocky path. It is slow-going and the boys make it only a mile in 20 minutes before they need a
rest. Luis and Matt take the opportunity to eat their snacks. Feeling revived after 15 minutes,
they take 35 minutes to hike the remaining two miles to the waterfalls. The boys take photos of
the waterfalls and relax for 30 minutes before deciding it is time to head back to camp. Coming
down the trail is easier. They make it back to the trail shop in 45 minutes. Luis takes 5 minutes
to pick out a postcard to send to his sister, and they take 30 minutes to rejoin their friends at the
campsite.
How far away from camp did Luis and Matt travel? 2 + 1 + 2 = 5 miles
How long were Luis and Matt away from the campsite?
30 + 10 + 20 + 15 + 35 + 30 + 45 + 5 + 30 = 220 minutes = 3 hours, 40 minutes
Graph this situation. Define your variables and label the axes. Choose an appropriate scale.
t = time away from camp (minutes), D(t) = distance from camp (miles)
Write the piecewise-defined function to represent your graph and this situation.










Dt   









1
t
15
2
,
0 t  30
,
30 t  40
1
t
20
3
,
40t  60
,
60t 75
2
9
t
35 7
5
,
75  t  110

3
43
t
45
3
,
1
44
t
15
3
,
2

, 110  t  140
140 t  185
, 185 t  190
190  t  220
**Recommendation: Assign groups of students one non-constant part of the function to find.
Then present to class.