Unit 2: Investigating

Functions

Day 1 Functions & Inverse Functions (2.1 & 6.4 in textbook)

Day 2 Special Points of a Graph (5.8 in textbook)

Day 3 Domain and Range (2.1 in textbook)

Day 4 Increasing and Decreasing

Day 5 End Behavior (5.2 in textbook)

ALGEBRA IS COOL!

Schedule of Upcoming Classes

Day 1 A Mon 8/31 Functions & Inverse Functions (2.1 & 6.4 in textbook) B Tues 9/1

Day 2 A Wed 9/2 Special Points of a Graph

(5.8 in textbook) B Thurs 9/3

Day 3 A Fri 9/4 Domain and Range (2.1 in textbook) B Tues 9/8

Day 4 A Wed 9/9 Increasing and Decreasing B Thurs 9/10

Day 5 A Fri 9/11 End Behavior (5.2 in textbook) B Mon 9/14

Day 6 A Tues 9/15 Review: Unit 2 B Wed 9/16

Day 7 A Thurs 9/17

Quiz: Unit 2 B Fri 9/18

Absent?

See Ms. Huelsman AS SOON AS POSSIBLE to get work and any help you need.

Notes are always posted online.

You may also email Ms. Huelsman at Kelsey.huelsman@lcps.org!

____

Need Help?

Ms. Huelsman and Mu Alpha Theta are available to help Monday, Tuesday, Thursday, and Friday mornings in L506 starting at 8:15.

Ms. Huelsman is also available after school until 4:30.

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Need to make up a test/quiz?

Math Make Up Room is open Mon/Tues/Thurs/Fri mornings and Mon/Wed/Thurs afternoons.

Schedule is posted around the math hallway & in Ms. Huelsman’s classroom

Algebra2 Notes: FUNCTIONS AND INVERSE FUNCTIONS Algebra 2 takes concepts from Algebra 1 and COMBINES and EXTENDS them.

Algebra1 Algebra2 Fractions Polynomials 37

1 32 5+ 2 4 3x x− + 2(2 7)x +

Rational Expressions 2 3

2

4 3 8(2 7) 3

x x xx x− + −

++

Fractions Exponents 23

14

22 (xy)4 2 43

6x yxy

Rational Exponents

2327

1 14 22 4( ) ( )x y xy+

Square Roots

25 3 32 218x x y

Nth roots

6 10 45 96x y z 3 2

3

209

x yxz

64

7 5

xx

Factoring x2 – 9x + 20 x2 - 36

Factoring 3x3 + 24y6 3x5 – 24x3 + 48x

Solving Equations 3(x – 6) + 7 = 15 x2 – 36 = 0

Solving Equations | 3(x – 6) + 7 | = 15 3x5 – 24x3 + 48x = 0

But our fundamental object is STILL the FUNCTION. Relation: A mapping or a pairing of ___________________________________________________ It can be in the form of ________________________________________________________ Function: A special kind of ________________________________ What’s special about it? Each _________________ can have only ______________________ Domain: Range:

Objective: Today you will understand functions and their inverses So that you can identify, define and graph inverses.

Function Notation: A function can be represented with an equation, A table of values, or a graph

x y

0

1

2

3

4

5

6

Points can be represented as ORDERED PAIRS and with FUNCTION NOTATION:

Examples: Identify the domain and range of each relation shown. Then determine whether it is a function. Explain your answer.

Domain ________ Domain ________ Domain ________ Range _________ Range _________ Range _________ Function? ______ Function? ______ Function? ______

A relation is a FUNCTION if it passes _____________________________________________

(This means you can also VISUALIZE a graph of the ordered pairs given!)

x y 1 5 2 5 3 10 4 10

x y 2 4 4 8 4 12 8 16

Examples: Determine whether the relations shown are functions. Explain your answer.

Decide with a partner: 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 _________________________ (x, y) ( , ) In terms of functionality, it UNDOES what the function DOES. Inverse of Addition: Inverse of Division: INVERSE FUNCTION REPRESENTATIONS: Table/Mapping Diagram

Original Relation Inverse Relation x -1 0 1 2 y 2 0 -2 -4

x 2 0 -2 -4 y -1 0 1 2

−10 −8 −6 −4 −2 2 4 6 8 10

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−10 −8 −6 −4 −2 2 4 6 8 10

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Inverses on a Graph

1. Plot the following points and their inverses. (0, 0) (1, 2) (2, 4) (3, 6) (4, 8)

Find the line of reflection based on the graph.

Is the inverse a function? __________

Why? 2. Plot the following points and their inverses. (-3, 9) (-2, 4) (-1,1) (0, 0)

(1, 1) (2, 4) (3, 9) Function Notation:

Then find the line of reflection.

Is the inverse a function? __________ Why? Discuss with a partner: 1. List some ways that examples 1 and 2 were similar and different (Hint: Were both the original graphs functions? Were both inverses functions? Did the inverses have anything in common?)

−10 −8 −6 −4 −2 2 4 6 8 10

−10

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−10 −8 −6 −4 −2 2 4 6 8 10

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The inverse is a function if the original function passes the: ___________________________ Is this graph a function? __________ Is its inverse a function? __________

Without even drawing the inverse, I know whether or not it will be a function!

Draw a function whose inverse IS a function. Draw a function whose inverse IS NOT a function.

EXAMPLE: Use the calculator to graph both functions. Then determine if they are inverses!

23

21)(

32)(

−=

+=

xxg

xxf

Consider this graph… Is the relation a function? Is the inverse a function? If so, draw the inverse function. Name 3 ordered pairs that will be in the inverse.

Steps To graph in a calculator:

Steps To Check the table of values in a calculator:

−10 −8 −6 −4 −2 2 4 6 8 10

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Algebra2 Notes: Special Points on a Graph Objective: Today you will learn to identify special points on a graph So that you can compare and contrast functions.

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: Turning point where the graph changes from increasing to decreasing. Relative Minimum: Turning point where the graph changes from decreasing to increasing. Absolute Maximum: Highest point possible on the graph. Absolute Minimum: Lowest point possible on the graph. Which of these do you see in this function? (btw, is the inverse of this a function?)

With a partner, discuss and identify the following special points of these graphs:

x-intercept y-intercept turning points relative / absolute max / min 1. 2. 1. Graph 72103 234 ++−+= xxxxy using your calculator. Then sketch the graph below:

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: __________________________

How to find maximum or minimum points:

How to find zeros:

2. Using a calculator, graph each of the following. Find the y-intercept (use the table… x = _________)

and the zeros of the function (if they exist).

a) y = − x + 3 + 2 b) y = x − 2( )3 − 3 c) y = 2 x +1( )2 − 4

Draw a function with zeros at -3, 2, and 5. What other special points can you identify on your graph?

y

x

y

x

y

x

Y-intercept:_________ Zeros:______________

Y-intercept:_________ Zeros:______________

Y-intercept:_________ Zeros:______________

y

x

Algebra2 Notes: Domain and Range Objective: Today you will learn to identify the domain and range of a graph So that you can compare function families. 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. Domain: __________________ __________________

2. Domain: __________________ __________________

3. Domain: __________________ __________________

4. Domain: __________________ __________________

5. Domain: __________________ __________________

6. Domain: __________________ __________________

x

y

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 (bottom to top) 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. Range: __________________ __________________

2. Range: __________________ __________________

3. Range: __________________ __________________

4. Range: __________________ __________________

5. Range: __________________ __________________

6. Range: __________________ __________________

x

y

Graph the following functions using your calculator. Then identify the domain and range of each function. 1. 2)3( 2 −+= xy 2. 4)2( 3 +−= xy Domain: Domain:

Range: Range:

3. 23 −+= xy 4. 43 −+= xy Domain: Domain:

Range: Range:

5. Select all of the following equations that have domain of all real numbers and a range of [-2, ∞ )

a)

1( ) 3 22

f x x= + − b) ( ) 3 2g x x= − c) 22( 3) 2y x= − −

d) ( ) 3 2h x x= − − e) 3( ) 2f x x= − f) 3( ) 4 2g x x= + −

−10 −8 −6 −4 −2 2 4 6 8 10

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−10 −8 −6 −4 −2 2 4 6 8 10

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−10 −8 −6 −4 −2 2 4 6 8 10

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Let’s talk about those weird breaks in the graph. Like these…

What is this function doing at the point (1,3) How about this one at (-2,1)?

How about this one at x = 2?

Continuity: A function is continuous if there are no breaks in the graph. You can draw it

without lifting your pencil.

Types of Discontinuities:

• Point (Removeable) Discontinuity: If it is discontinuous at a single point.

• Asymptotic Discontinuity: The function approaches a line, called an asymptote,

but never reaches it.

Discuss: Does a discontinuity affect the DOMAIN and RANGE of a function?

x-coordinate of Discontinuity

Type of Discontinuity

(point, asymptotic)

x

y

Quick Questions: DOMAIN AND RANGE

Question Answer A Answer B

1

Determine the DOMAIN of the polynomial function whose graph is shown.

(-∞ , ∞ )

(-2, ∞ )

2

Determine the RANGE of the polynomial function whose graph is shown.

All real numbers -4 < y < 4

3

[ -4, ∞ ) means…

x > - 4 x > - 4

4

Determine the RANGE of the polynomial function whose graph is shown.

( - ∞, ∞ ) ( - ∞, 1 ]

5

Determine the function whose domain is [ 4, ∞ )

34 +−= xy 3)4( 2 +−= xy

Algebra2 Notes: Increasing and Decreasing Objective: Today you will learn to identify the ranges of x for which y is either getting larger (increasing) or smaller (decreasing) So that you can analyze real-world models. 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. Questions: 1. From 0 (i.e. midnight) to 5am, the temperature is…

A. increasing B. decreasing C. not changing

2. From 5am to 10am, the temperature is…

A. increasing B. decreasing C. not changing

3. Around 4am, the temperature is…

A. still increasing B. still decreasing C. at a maximum D. at a minimum

Use the graphs below to answer each question. 4.

Is there a maximum point or a minimum point?______________Why? _______________ When is the graph increasing?_______________

When is the graph decreasing?_______________

5.

Is there a maximum point or a minimum point?______________Why? _______________ When is the graph increasing?_______________

When is the graph decreasing?_______________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

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−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

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A function is INCREASING on an interval if

Using function notation:

A function is DECREASING on an interval if

Using function notation:

** When identifying the intervals for increasing and decreasing, you reference the domain

values of the minimum and maximum 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):________________________________ Draw a function to the right and answer the following statements with ALWAYS, SOMETIMES, NEVER

1. My function is increasing.

2. My function is decreasing.

3. My function is neither increasing nor decreasing.

−10 −8 −6 −4 −2 2 4 6 8 10

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Discuss:

How are max and mins (both relative and absolute) related to increasing and decreasing intervals?

7. Graph the function ( ) 41 −−= xxf using the calculator.

a) Does the graph have a maximum point? _________

If so, name it: ________________

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. ________________

8. Graph the function ( ) 5)3(21 2 ++−= xxf using the calculator.

a) Does the graph have a maximum point? _________

If so, name it: ________________

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. ________________

9. Draw a function that is increasing on (-5, 3)

And decreasing on (3, ∞).

Algebra2 Notes: End Behavior Objective: Today you will learn to describe the behavior of a function as values in its domain approach positive and negative infinity So that you can compare function families. Discuss: 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 ( ∞− ) When looking at the graph think …”

As x gets bigger, where are the arrows of the graph going? As x gets smaller, where are the arrows of the graph going?

In function notation:

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

To finish the first statement: As you move to the right, The end of the graph is going up or down?

To finish the second statement: As you move to the left, The end of the graph is going up or down?

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 graph the following and determine the end behavior.

6. 3 2( ) 3 4 1f x x x x= + + + 7. 31( ) ( 1) ( 5)8

f x x x= + −

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. There are really only 4 options for end behavior: 1. 2. 3. 4. Draw an example of each:

Describe the graph’s end behavior…..

As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____

On what intervals is the graph … Increasing: ________________________________ Decreasing: _______________________________

Quick Questions: Polynomial END BEHAVIOR Question Answer A Answer B

1

Describe the end behavior of the polynomial function whose graph is shown.

As x -∞ , f(x) -∞ As x +∞ , f(x) -∞

As x -∞ , f(x) +∞ As x +∞ , f(x) +∞

2

Describe the end behavior of the polynomial function whose graph is shown.

As x -∞ , f(x) -∞ As x +∞ , f(x) +∞

As x -∞ , f(x) +∞ As x +∞ , f(x) -∞

3 As x -∞ , f(x) +∞ . This means…

The graph goes down as you look far right.

The graph goes up as you look far left.

4 As x +∞ , f(x) -∞ . This means…

The graph goes down as you look far right.

The graph goes up as you look far left.

5

Which polynomial graph is being described? As x -∞ , f(x) -∞ . As x +∞ , f(x) -∞ .

6

Which polynomial graph is being described? As x -∞ , f(x) +∞ . As x +∞ , f(x) -∞ .

7 odd degree and positive leading coefficient As x +∞ , f(x) +∞

As x +∞ , f(x) -∞

8 even degree and negative leading coefficient As x +∞ , f(x) +∞

As x +∞ , f(x) -∞

9 f(x) = -2x5 + 3x4 – 4x3 + 5x2 – 6x + 7 As x -∞ , f(x) -∞

As x -∞ , f(x) +∞

10 g(x) = 13x8 – 11x6 + 9x4 – 7x2 – 5 As x -∞ , f(x) -∞

As x -∞ , f(x) +∞

Unit 2 Functions: Quick Questions Review

QUESTION ANSWER A ANSWER B

1 A graph represents a function if it passes the: Horizontal line test Vertical line test

2 A table represents a function if the x-values: Repeat Do not repeat

3 A relation is a function if every input Has more than one output Has exactly one output

4 A function has an inverse if: The x-values do not repeat It passes the horizontal line test

5 A function and its inverse will be reflected over: The line y=x The x-axis

6 A relative minimum can be smaller than the absolute minimum. True False

7 A zero is: An x-intercept A y-intercept

8 What does y = 10 look like? A point A horizontal line

9 A continuous graph: Has arrows at the ends of it Has no breaks in it

10 “All real numbers except 5” in interval notation:

( , ), ( , 5) (5, )

11 Definition: An asymptote is… A vertical line A line that the graph approaches

12 When you read a graph, you read it from Left to right Right to left

13 When looking at intervals of increasing and decreasing, you must look at intervals of

x y

14

refers to the

_____ side of the graph:

Left Right

15

Which way is the right side going?

Up Down