Math 3
5-1 and 5-2 Function Families
Name ______________________________
In this investigation, you will be working towards the following learning objectives:
 I can use limit notation to describe the end behavior of functions
 I understand the basic function families and their important characteristics
 I can find average rate of change for specified intervals of a function
1. Your family is driving 250 miles to the amusement park Kings Island in Cincinnati, Ohio. Fill in the
below table for the hours it will take to make the trip based on the average speed given.
speed (mph)
10
20
25
40
50
60
70
100
time (hours)
2. In the past, you have learned the equation D  r  t . Rearrange this equation to write an equation for
time in terms of rate.
3. Graph the data from the table on the axes at right.
4. In the future, cars will be allowed to go much faster. How long would it take to get to Kings Island if
we could travel
200 mph?
500 mph?
1000 mph?
50,000 mph?
5. As the value of r increases, what value does t appear to approach?
6. Even given your answer to problem (5), could we ever get to Kings Island instantaneously? Explain.
7. The time it takes to get a job done is determined by the number of workers. For instance, if one worker
could get the job done in 6 hours, 2 workers could get the work done in 3 hours, 3 workers in 2 hours,
6
and so on. This situation can be modeled by the function t ( w)  . Graph t ( w) on your calculator.
w
Sketch it below (both branches).
8. Description of the end behavior of d:
lim t( w)
w
=
_____
lim t( w)
w 
= _____
9. Restaurants cook pizzas in very hot ovens, but as soon as the pizza comes out of the oven, it begins
cooling. Assume room temperature is 70 degrees. Below is the graph of the temperature of one pizza
measure in two minute intervals after coming out of the oven.
Find
lim f ( x)
x
9. Find the end behavior of the below graphs. Write your answers in limit notation.
lim f  x  
x 
lim f  x  
x 
Linear Functions
I.
General Rule:
Parent Function:
f ( x)  a  bx
𝑓(𝑥) = 𝑥
Domain:
Range:
Symmetries (if any):
How do the parameters affect the function?
For the parent function, the average rate of change from
x  1 to x  4 is . . .
End Behavior (parent):
Quadratic Functions
II. General Rule:
f ( x)  ax 2  bx  c
Parent Function:
𝑓(𝑥) = 𝑥 2
𝑓(𝑥)
Domain:
Range:
x
Symmetries (if any):
How do the parameters (a and c) affect the function?
End Behavior (parent – limit notation!):
For the parent function, the average rate of change from
x  1 to x  4 is . . .
Exponential Functions
III. General Rule:
f ( x)  a  b x
Parent Function:
𝑓(𝑥) = 2𝑥
𝑓(𝑥)
Domain:
Range:
x
Symmetries (if any):
How do the parameters affect the function?
For the parent function, the average rate of change from
x  1 to x  4 is . . .
End Behavior (parent – limit notation!):
Inverse Variation (odd powers)
IV. General Rule:
f ( x) 
k
xr
Parent Function:
1
𝑓(𝑥) = 𝑥
𝑓(𝑥)
Domain:
Range:
x
Symmetries (if any):
For the parent function, the average rate of change from
x  1 to x  4 is . . .
Inverse Variation (even powers)
V. General Rule:
k
f ( x)  r
x
End Behavior (parent):
Parent Function:
1
𝑓(𝑥) = 𝑥 2
𝑓(𝑥)
Domain:
Range:
x
Symmetries (if any):
For the parent function, the average rate of change from
x  1 to x  4 is . . .
End Behavior (parent):
Square Root Function
VI. General Rule:
f ( x)  x
Domain:
Range:
Symmetries (if any):
For the parent function, the average rate of change from
x  1 to x  4 is . . .
End Behavior (use limit notation)
Absolute Value Function
Notes:
VII. General Rule:
f ( x)  x
Domain:
Range:
Symmetries (if any):
For the parent function, the average rate of change from
x  1 to x  4 is . . .
End Behavior (use limit notation)