Unit 2: Investigating Functions
Day 1: Functions & Inverse Functions (2.1 & 6.4)
Special Points of a Graph (5.8)
Day 2: Domain and Range (2.1)
Day 3: Increasing and Decreasing
Day 4: End Behavior (5.2)
Algebra2 Notes: FUNCTIONS AND INVERSE FUNCTIONS
Today we will:
I will:
Algebra2 takes concepts from Algebra 1 and COMBINES and EXTENDS them.
Algebra1
Algebra2
Fractions
3
1 3

7
2 5
Polynomials
Rational Expressions
Fractions
2 1
3 4
Exponents
3x 2 y 4
22 (xy)4
6 xy
Square Roots
25
Solving Equations
3(x – 6) + 7 = 15
Rational Exponents
27
2
3
1
2
1
4 2
( x y)  ( xy )
4
Nth roots
2
3 32
Factoring
x2 – 9x + 20
x 2  4 x  3 x3  8

(2 x  7)2
3x
(2 x  7) 2
x2  4 x  3
x 18 x y
x2
- 36
x2
– 36 = 0
5
6
10 4
96x y z
Factoring
3x3 + 24y6
20 x3 y 2
9 xz 3
Function:
Domain:
Range:
7
x5
3x5 – 24x3 + 48x = 0
x2 + 36 = 0
Relation:
x6
3x5 – 24x3 + 48x
Solving Equations
| 3(x – 6) + 7 | = 15
But our fundamental object is STILL the FUNCTION.
4
Function Notation:
A function can be represented in many ways…..
1. words
2. equation
3.
table of values
x
4.
GRAPH
y
0
1
2
3
4
5
6
Points can be represented as ORDERED PAIRS and with FUNCTION NOTATION:
Examples: Determine whether the relations shown are functions. Explain your answer.
Identify the domain and range.
x
2
4
6
8
y
3
5
6
7
Domain ________
Range _________
Function? ______
x
1
2
3
4
y
5
5
10
10
Domain ________
Range _________
Function? ______
x
2
4
4
8
y
4
8
12
16
Domain ________
Range _________
Function? ______
-1
2
5
4
8
Domain ________
Range _________
Function? _______
-3
-1
1
2
2
4
5
Domain ________
Range _________
Function? _______
A relation is a FUNCTION if it passes _____________________________________________
Examples: Determine whether the relations shown are functions. Explain your answer.
TURN&TALK:
Always/Sometimes/Never
1. A function is a relation.
2. A relation is a function.
3. A graph is a relation.
4. A graph is a function.
5. A line is a relation.
6. A line is a function.
7. A circle is a relation.
8. A circle is a function.
DEFINITION OF AN INVERSE: An inverse function maps the output values back
to their original input values.
In terms of x and y, this means
TURN&TALK: In your own words, what is the relationship between a function and its
inverse?
Inverse of Addition:
Inverse of Division:
INVERSE FUNCTION REPRESENTATIONS:
Table/Mapping Diagram
Original Relation
x
-1
0
y
2
0
Inverse Relation
1
-2
2
-4
x
y
2
-1
0
0
-2
1
-4
2
Inverses on a Graph
1.
 y
Plot the following points and their inverses.
(0, 0)
(1, 2)
(2, 4)
(3, 6)


(4, 8)
Function Notation:


    


Then find the line of reflection.


Is the inverse a function? __________

x





2.
Plot the following points and their inverses.
 y

(-3, 9)
(-2, 4)
(-1,1)
(0, 0)


(1, 1)
(2, 4)
(3, 9)

    
Function Notation:
x






Then find the line of reflection.


Is the inverse a function? __________
The inverse is a function if the original function passes the: ________________
Is this graph a function? __________
Is its inverse a function? __________
 y

Draw a function whose inverse IS a function.



    

x









 y

Draw a function whose inverse IS NOT a function.



    





x







EXAMPLE: Use the calculator to graph both functions. Then determine if they are
inverses!
Steps To graph in a calculator:
f ( x)  2 x  3
1
3
g ( x)  x 
2
2
f ( x)  4 x  2
1
g ( x)  x  2
4
 y
 y







    


x





    









x




Steps To Check the table of values
in a calculator:
Consider this graph…
Is the relation a function?
What is the domain?
What is the range?
Does it have an inverse?
If so, draw the inverse function.
Name 3 ordered pairs that will be in the inverse.

Special Points on a Graph
Just like WE have unique characteristics that identify us, SO DO
FUNCTIONS and RELATIONS.
VOCABULARY:
X-Intercept: point where a graph crosses the x-axis ( y = 0 )
** Also called ZEROS, ROOTS, SOLUTIONS **
Y-Intercept: point where a graph crosses the y-axis ( x = 0 )
Turning Points: Point where the graph changes direction.
Relative Maximum: ANY turning point where the graph changes from increasing to
decreasing.
Relative Minimum: ANY turning point where the graph changes from decreasing to
increasing.
Absolute Maximum: Highest point possible on the whole graph.
Absolute Minimum: Lowest point possible on the whole graph.
Which of these do you see in this function? (btw, is the inverse of this a function?)
With a partner, discuss and identify special points of these graphs.
1.
2.
2.
To find zeros:
To find maximum or minimum
points:
1.
Graph y  3x 4  x 3  10 x 2  2 x  7 using your calculator.
a. Identify any zeros of the function : ___________________________
b. Identify any relative maximums: ___________________________
c. Identify any relative minimums: ____________________________
d. Identify any absolute maximums : __________________________
e. Identify any absolute minimums: __________________________
2.
Using a calculator, graph each of the following.
Estimate the y-intercept and the zeros of the function (if they exist).



3
a) y   x  3  2
b) y  x  2  3
y
y
x

2
c) y  2 x  1  4
y
x
x
Y-intercept:_________
Y-intercept:_________
Y-intercept:_________
Zeros:______________
Zeros:______________
Zeros:______________
Draw a function with zeros at -3, 2, and 5. What special points can you identify on your graph?
y
x
Algebra2 Notes: Domain and Range
Today we will:
I will:
Domain of a Function: Recall that domain is the x-values or input of the function. Therefore you
will read the graph horizontally (left to right)
Note: Read across (left to right) the graph to find the domain
of a function. An arrow that heads right approaches positive
infinity (+ ∞); whereas an arrow that heads left has a domain
that starts at negative infinity (– ∞).
If a function starts and/or stops at a given point then you will
restrict the domain using the x-value of that point.
Use the graph to state the domain of each function.
1.
2.
3.
Domain:
__________________
Domain:
__________________
Domain:
__________________
__________________
__________________
__________________
4.
5.
6.
y
x
Domain:
__________________
Domain:
__________________
Domain:
__________________
__________________
__________________
__________________
Range of a Function: Recall that range is the y-values or output of the function. Therefore you will
read the graph vertically (up and down).
Note: Read the graph vertically (down to up) to find the range
of a function. An arrow that heads up approaches positive
infinity (+ ∞); whereas an arrow that heads down has a range
that starts at negative infinity (– ∞).
If a function starts and/or stops at a given point then you will
restrict the range using the y-value of that point.
Use the graph to state the range of each function.
1.
2.
3.
Range:
__________________
Range:
__________________
Range:
__________________
__________________
__________________
__________________
4.
5.
6.
y
x
Range:
__________________
Range:
__________________
Range:
__________________
__________________
__________________
__________________
Graph the following functions using your calculator. Then identify the
domain and range of each function.
1. y  ( x  3) 2  2
2. y  ( x  2) 3  4
 y
 y






    
x






Domain:


    



Domain:




Range:

3. y  x  3  2

Range:
4. y  x  3  4
 y
 y










    

x

x





Domain:


    







Range:

x





Domain:

Range:
5. Select all of the following equations that have domain of all real numbers and
a range of [-2,  )
1
x3 2
2
b) g( x)  3x  2
2
c) y  2( x  3)  2
d) h(x)  x32
e) f (x)  x3  2
f) g(x) 3 x42
f ( x) 
a)
Let’s talk about graphs with weird breaks in them. Like these…
What is this function doing at the point (1,3)
How about thisy one at (-2,1)?
x
How about this one at x = 2?
Is x=2 in the domain?
Is y=1 in the range?
Continuity: A function is continuous if there are no breaks in the graph. You can
draw it without lifting your pencil.
TURN&TALK: How can a discontinuity affect the DOMAIN and RANGE of a function?
Algebra2 Notes: Increasing and Decreasing Intervals
Today we will:
I will:
Below is the numeric representation of the temperature T (in ºC) at time t in Columbus, OH
during an “Alberta Clipper” cold front.
t
0
1
2
3
4
5
6
7
8
9
10
T
2
0
-2
-4
-6
-8
-6
-4
-2
0
2
Time (t) zero is midnight.
TURN&TALK Questions:
1. From 0 (i.e. midnight) to
5am, the temperature is…
2. From 5am to 10am, the
temperature is…
A. increasing
B. decreasing
C. not changing
3. Around 4am, the
temperature is…
A. increasing
B. decreasing
C. not changing
A.
B.
C.
D.
still increasing
still decreasing
at a maximum
at a minimum
Use the graphs below to answer each question.
4.
5.

y



























y
x
x
















































Is there a maximum point or a minimum
point?______________Why?
_______________
Is there a maximum point or a minimum
point?______________Why?
_______________
When is the graph increasing?_______________
When is the graph increasing?_______________
When is the graph decreasing?_______________ When is the graph decreasing?_______________
In your own words: A function is INCREASING on an interval if
Using function notation:
In your own words: A function is DECREASING on an interval if
Using function notation:
** When identifying the intervals for increasing and decreasing, you decide using y-values,
but answer with the x-values (domain values) of the min and max points from the graph.
6. Find the following using the following graph:
Domain:________________
Range:___________________
x-intercept(s):_________________
y-intercept:__________
Relative minimum:_____________
absolute min:___________
Relative maximum:_____________
absolute max:___________
Increasing interval(s):_______________________________
Decreasing interval(s):________________________________
ALWAYS, SOMETIMES, NEVER
1. A function is increasing.
2. A function is decreasing.
3. A function is neither increasing nor decreasing.
TURN&TALK: How are max and mins (both relative and absolute) related to increasing and
decreasing intervals?
 y

7. Graph the function f x   x  1  4 using the calculator.

a) Does the graph have a maximum point? _________

If so, name it: ________________

    

x






b) Does the graph have a minimum point? _________


If so, name it: ________________

c) Determine the interval where f is increasing. _________________
d) Determine the interval where it is decreasing. ________________
 y
1
2
8. Graph the function f  x    ( x  3)  5 using the calculator.
2


a) Does the graph have a maximum point? _________


If so, name it: ________________
    
b) Does the graph have a minimum point? _________

x






If so, name it: ________________


c) Determine the interval where f is increasing. _________________
d) Determine the interval where it is decreasing. ________________
 y
9. Draw a function that is increasing on (-5, 3)


And decreasing on (3, ∞).


    





x






Algebra2 Notes: End Behavior
Today we will:
I will:
What is different about these functions?
1.
2.
3.
End Behavior:
The behavior of the graph, f(x), as x approaches positive infinity (   ) or
negative infinity (   )
In function notation:
As x    then f (x)  _____
As x    then f (x)  _____
TURN&TALK In your own words, what are we asking when we discuss end behavior?
As x    then f ( x)  _____
As x    then f ( x)  _____
Describe the end behavior of the graph of the function.
1.
2.
As x    then f (x)  _____
As x    then f (x)  _____
As x    then f (x)  _____
As x    then f (x)  _____
Use your calculator to graph the following and determine the end behavior.
3. y = -3x5 – 6x2 + 3x – 2
4. h(x) = 6x8 – 7x5 + 4x
As x    then f (x)  _____
As x    then f (x)  _____
As x    then f (x)  _____
As x    then f (x)  _____
5. Describe in words the end behavior of the following functions:
a.
b.
Use your calculator to find the end behavior of each function.
6.
f ( x)  x 3  3x 2  4 x  1
7.
1
f ( x)  ( x  1)3 ( x  5)
8
As x    then f (x)  _____
As x    then f (x)  _____
As x    then f (x)  _____
As x    then f (x)  _____
8.
Choose the correct letter and answer the additional questions.
Describe the graph’s end behavior…..
As x    then f (x)  _____
As x    then f (x)  _____
On what intervals is the graph …
Increasing:
________________________________
Decreasing:
_______________________________
There are really only 4 options for end behavior:
1.
2.
3.
4.
Draw an example of each: