chapter 1 functions and graphs. quick talk what can you tell me about functions based on your...

CHAPTER 1 FUNCTIONS AND GRAPHS

Quick Talk• What can you tell me about functions based on your

peers’ lessons?

• What can you tell me about graphs based on your peers’ lessons?

Review about factoring• Factor these following:

Challenge: Could you factor these?

You should feel a sudden rush of urge that these can be factored…but how...• 1)

• 2)

• Quick Talk: What does factoring allow you to find?

Potential Answers• Rational Zeros when y=0

A mathematical model is a mathematical

structure that approximates phenomena for the

purpose of studying or predicting their behavior.• There are 3 types of model:

• Numerical Models• Algebraic model• Graphical models

The most basic type: Numerical Model

• Numerical Model: use of numbers or data are analyzed to gain insights into phenomena

Group work: Numerical Model Example #1

Year Total Male Female

1980 456 324 132

1985 480 328 152

1990 532 360 172

1995 690 510 180

2000 719 520 199

• The chart shows growth in the number of engineers from 1980-2000. Is the proportion of female engineers over the years increasing? Explain

Answer• You have to be careful with the last example, the word “proportion”

means what?

• Once we converted all the data into ratios, we see that there is an increase from 1980-1995, but there is a drop from 1990-1995, then from 1995-2000 it increased again. From the data, the peak is at 1990, 32.3%

Year Total Female Actual ratio

1980 456 132 .289

1985 480 152 .317

1990 532 172 .323

1995 690 180 .261

2000 719 199 .277

Algebraic Model• Algebraic model uses formulas or equations to relate

variable quantities associated with the phenomena being studied.

Group Work: Algebraic Model Example #1

Note:• Think about what are you comparing.

• What “formula” can you use? Or do you have to create one?

A pizzeria sells a rectangular 20" by 22" pizza for the same price as its

large round pizza (24" diameter). If both pizzas are the same thickness,

which option gives the most pizza for the money?

Answer

2

2

Compare the areas of the pizzas.

Rectangular pizza: Area 20 22 440 square inches

24Circular pizza: Area 144 452.4 square inches

2

The round pizza is larger and therefore gives more for t

l w

r

he money.

Group Work: Algebraic Model Example #2

• You went to Gamestop, it just happens every game you buy is discounted 15%( ) off the marked price. The discount is taken at the sales counter, and then a state sales tax of 7.5% and local tax of 1% is added on.

• Questions:• 1) If you have $20, could you buy a game marked at

$24.99?• 2) If you are determined to spend no more than $175,

what’s the maximum total value of your marked purchases be?

Answer• 1) Identify your variables (ex: let m=market price, d=discounted price, k=constant,

t=taxes, s=total sale price)

• Your general equation should be:• d=km• s=d+td• When you substitute: s=km+t(km)

• Now determine the values for each variable:• k=.85 m=? d=? t=.085 s=?

Your actual equation (with numbers) should be:• d=.85m• s=.85m+.085(m)

• 1) m=24.99, s=.85(24.99)+.085(24.99), s=21.24+2.12 , s= 23.36

Since you only have $20, 20 < 23.36, therefore, you can not buy the game

• 2) s=175, so 175=.85(m)+.085(m), m=187.17, so the maximum total value of the market price should be at most $187.17

Graphing Model• Graphing model is a visible representation of a numerical

model or an algebraic model that gives insight into the relationships between variable quantities.

Graphical model Example #1• Graph this. What does it look like? Can you find an

algebraic model that fits?

Answer• It is parabolic

• Because (1,0.75) must satisfy the equation, after substitution

Graphical Model example #2Time (t) 0 5 10 15 20

Females (f) 3.8 4.4 5.5 5.9 6.7

Create a graphic model, that fits the best with these data sets.

Answer• When you plot them, it looks like it’s linear, so let’s use a

linear model y=mx+b• m=slope, b=y-intercept

• , b=3.8

• y=.0145x+3.8

In statistics, this is known as “best fit line”

Group work Situation:

• What methods are there to solve for the variable? Use one and find the solution

Answer• Potential ways:• Algebraic• Graphing• Factoring• Quadratic Formula

• I used quadratic formula to solve it. You should have a solution that

Group Work Solve for this equation

• What method did you choose and why?

Answer• X=0 or x=5/2 or x=-2/3

Group Work Solve this

• Describe this graph and find the solutions

Group Work Solve this

• Describe this graph and find the solutions

Homework Practice• Pgs 81-84 #1-9odd, 11-18, 29, 31, 35,43, 45, 48, 50

BASIC PARENT FUNCTIONS

Quick Talk:• What makes a function? What does it consists of?

Answer• One to one / passes vertical line test• Y and x• Equation• Domain• Range

• Note: y is aka f(x)

Quick Talk: From what you just learned, which of these is not a function?

Quick Talk: What are parent functions?

• Parent Function is the simplest function with characteristics. It is without any “transformations” or “shifting”

12 parent functions:• As you are graphing it, please draw an arrow at the end.

The Identity Function

Slide 1- 35

The Squaring Function

Slide 1- 36

The Cubing Function

Slide 1- 37

The Reciprocal Function

Slide 1- 38

The Square Root Function

Slide 1- 39

The Exponential Function

Slide 1- 40

The Natural Logarithm Function

Slide 1- 41

The Sine Function

Slide 1- 42

The Cosine Function

Slide 1- 43

The Absolute Value Function

Slide 1- 44

The Greatest Integer Function

Slide 1- 45

The Logistic Function

Slide 1- 46

Important!!! Read this!!!• I neeeeeeedddddddd you guys to recognize, understand

and know the parent functions!!!!!

• Why? Because it will make finding domain and range easier.

What “shifts” can you possibly have on a parent function?• You can shift a function left or right (horizontal shifts)

• You can shift a function up or down (vertical shifts)

• Function can be steeper or flatter (linear)

• Function can be wider or narrower (horizontal shrink/vertical stretch or horizontal stretch/vertical shrink)

• Function can flip across x or y axis

Group Activity: graphing! Each group will select a member to put it on the whiteboard. With t chart. X:[-5,5]

• Group 1 Group 6•

• Group 2 Group 7•

• Group 3 Group 8•

• Group 4 Group 10•

• Group 5

Shifts review• Parent function

• • Shift right 2 vertical stretch of 3, horizontal shrink of

• • Shift left 8 vertical shrink of , horizontal stretch of 2

• • Shift up 10 Flip across the x-axis

• • Shift down 7 Flip across y-axis

Name the parent function then the shifts

𝑦=32

(𝑥− 5 )2 −9

Answer• Shift right 5• Shift down 9• Vertical stretch of 3/2 or horizontal shrink of 2/3

Homework Practice• P147 #1,4,5,7,9,17,22

Domain and Range• Domain are all the input values of the function.

• Range are all the output of the function.

Looking at Domain and Range Graphically

• What is the parent function?• What is the domain?• What is the range?

Answer• Parent function is a cubic function.

• You always ask yourself “what values of x does the graph cover?”, “what values of y does the graph cover?”

• In this case.

• Domain: • Range: (

Group Work: What is the domain and range?

Answer• Domain: [-5,5]

• Range: [-3,3]

Group Activity:• Find the Domain and Range of the 12 parent functions.

• Note: The answers are not on my slides, I will be showing my answers right now in class. If you miss this, get it from me or another person from your class.

• As you can see from the step function, not every domain or range are continuous.

Slide 1- 61

Example Identifying Points of Discontinuity

Which of the following figures shows functions that are

discontinuous at x = 2?

Slide 1- 62

Continuity

Finding Domain and range Algebraically

How do you find the domain algebraically?

• You have to first determine the possible restrictions.

• In this case, we can never have a negative inside the radical. So we have to set f(x)0 because radical can be 0.

• So , when we solve for x, x2

• Therefore domain is or you can write it as

• We know the parent function is • All the y values or f(x) values are , since there is no

vertical shifts, we can then say the range is or

Group Work: Determining the domain and range

• 1)

• 2)

• 3)

• 1) D: • R:

• 2) D: • R:

• 3) Restriction: in denominator, in numerator• D: • R:

Look at our example #1 and #3• We have something unique. It is called an asymptote.

Slide 1- 68

Horizontal and Vertical Asymptotes

The line is a horizontal asymptote of the graph of a function ( )

if ( ) approaches a limit of as approaches + or - .

In limit notation: lim ( ) or lim ( ) .

The line is a ver

x x

y b y f x

f x b x

f x b f x b

x a

tical asymptote of the graph of a function ( )

if ( ) approaches a limit of + or - as approaches from either

direction.

In limit notation: lim ( ) or lim ( ) .x a x a

y f x

f x x a

f x f x

Asymptotes• Remember the graph will get infinitely close to the

asymptotes but will NEVER intersect with it.

Slide 1- 70

Increasing, Decreasing, and Constant Function on an IntervalA function f is increasing on an interval if, for any twopoints in the interval, a positive change in x results in a positive change in f(x).

A function f is decreasing on an interval if, for any twopoints in the interval, a positive change in x results in a negative change in f(x).

A function f is constant on an interval if, for any two points in

the interval, a positive change in x results in a zero change in f(x).

Slide 1- 71

Increasing and Decreasing Functions

Group work: Find the Increase and Decrease

2

2Given ( ) . Tell the intervals on which ( ) is increasing and the

1intervals on which it is decreasing.

xg x g x

x

Shown in class: How do you graph it?

Slide 1- 74

Answer:

2

2Given ( ) . Tell the intervals on which ( ) is increasing and the

1intervals on which it is decreasing.

xg x g x

x

From the graph, we see that ( ) is increasing on , 1 , increasing on

( 1,0], decreasing on [0,1), and decreasing on (1, ).

g x

Slide 1- 75

Local and Absolute Extrema

A local maximum of a function f is a value f(c) that is greater than or equal to all range values of f on some open interval containing c. If f(c) is greater than or equal to all range values of f, then f(c) is the maximum (or absolute maximum) value of f.

A local minimum of a function f is a value f(c) that is less than or equal to all range values of f on some open interval containing c. If f(c) is less than or equal to all range values of f, then f(c) is the minimum (or absolute minimum) value of f.

Local extrema are also called relative extrema.

In another word• You may have multiple relative/local maximum and

relative/local minimum.

• They are located where the slopes are = 0 (important concept in calculus)

• Absolute “maximum” or “minimum” is where it is the most “top” or “bottom point” of the function.

Slide 1- 77

Lower Bound, Upper Bound and Bounded

A function f is bounded below of there is somenumber b that is less than or equal to everynumber in the range of f. Any such number b iscalled a lower bound of f.

A function f is bounded above of there is somenumber B that is greater than or equal to everynumber in the range of f. Any such number B iscalled a upper bound of f.

A function f is bounded if it is bounded both above and below.

In another word,• Function is bounded below, if there is an “absolute

minimum” or if there is a “floor”

• Function is bounded above, if there is an “absolute maximum” or if there is a “ceiling”

End Behavior Asymptote• End Behavior asymptote describes the characteristic of a function at tail-

ends (both side)

• Find the highest term from both numerator and denominator. Use that term, then divide.

• Example:

• Numerator: Highest power is 3, so I use the term • Denominator: Highest power is 1, so I use the term

• Divide:• So is the end behavior asymptote• End behavior •

Example: Find end behavior asymptote

𝑓 (𝑥)=𝑥

𝑥2− 𝑥−2

Answer:• End behavior asymptote =

• End behavior •

Group work: Find end behavior asymptote

𝑦=3 𝑥

𝑋 3+2𝑥−1

Answer:• End behavior asymptote =

• End behavior •

Symmetry• Symmetry: a line where you can fold one side onto the

other.

Slide 1- 85

Symmetry with respect to the y-axis

Slide 1- 86

Symmetry with respect to the x-axis

Slide 1- 87

Symmetry with respect to the origin

Slide 1- 88

Example Checking Functions for Symmetry

2

Tell whether the following function is odd, even, or neither.

( ) 3f x x

Slide 1- 89

Example Checking Functions for Symmetry

2

Tell whether the following function is odd, even, or neither.

( ) 3f x x

2

2

Solve Algebraically:

Find (- ).

(- ) (- ) 3

3

( ) The function is even.

f x

f x x

x

f x

Group work: Look at the 12 parent functions.

• Find the boundedness, location of local/absolute min/max, where it is increasing, decreasing or constant.

Group work:• For the following examples, find the following:

•

• Domain:• Range:• Continuous:• Increase/decrease:• Symmetric:• Boundedness:• Max/min:• Asymptotes:• End behavior:

Answer shown in class

Homework Practice•

• • For these questions, do the following:

• Domain:• Range:• Continuous:• Increase/decrease:• Symmetric:• Boundedness:• Max/min:• Asymptotes:• End behavior:

BUILDING FUNCTIONS FROM FUNCTIONS

Overview

It is important how you can put functions together.

Slide 1- 96

Sum, Difference, Product, and Quotient

Let and be two functions with intersecting domains. Then for all values

of in the intersection, the algebraic combinations of and are defined

by the following rules:

Sum: ( ) ( )

Differ

f g

x f g

f g x f x g x

ence: ( ) ( ) ( )

Product: ( )( ) ( ) ( )

( )Quotient: , provided ( ) 0

( )

In each case, the domain of the new function consists of all numbers that

belong to both the domain of and

f g x f x g x

fg x f x g x

f f xx g x

g g x

f

the domain of . g

Slide 1- 97

Example Defining New Functions Algebraically

3Let ( ) and ( ) 1. Find formulas of the functions

(a)

(b)

(c)

(d) /

f x x g x x

f g

f g

fg

f g

Group work: • From the last examples, find the following:

• Domain:• Range:• Continuous:• Increase/decrease:• Symmetric:• Boundedness:• Max/min:• Asymptotes:• End behavior:

Slide 1- 99

Composition of Functions

Let and be two functions such that the domain of intersects the range

of . The composition of , denoted , is defined by the rule

( )( ) ( ( )).

The domain of consists of all -values

f g f

g f g f g

f g x f g x

f g x

in the domain of that

map to ( )-values in the domain of .

g

g x f

Note:• Composition is like “function within a function”

Composition Examples

Let ( ) 2 and ( ) 1. Find

(a)

(b)

xf x g x x

f g x

g f x

Answer:

1(a) ( ( )) 2

(b) ( ( )) 2 1

x

x

f g x f g x

g f x g f x

Group Work:• Find the domain of

• Find the domain of

• A)

• B)

Answer shown in class

Group Work:

• Find:

• A)

• B)

Answer

• A) =

• B) =

Working backward

2

Find and such that ( ) ( ( )).

( ) 5

f g h x f g x

h x x

Possible Answers

2

2

One possible decomposition:

( ) and ( ) 5

Another possibility:

( ) 5 and ( )

f x x g x x

f x x g x x

Group Work:• For each function h, find functions f and g such that

h(x)=f(g(x))

• 2)

Answer• 1)

• 2) • or

Word problem• A high-altitude spherical weather balloon expands as it

rises due to the drop in atmospheric pressure. Suppose that the radius r increases at the rate of 0.03 inches per second and that r=48 inches at time t=0. Determine an equation that models the volume V of the balloon at time t and find the volume when t=300 seconds.

Answer

• Substitute r

• Since s=300

• or

Word Problem #2

A satellite camera takes a rectangular shaped picture. The smallest region that can be photographed is a 5km by 7km rectangle. As the camera zooms out, the length l and width w of the rectangle increase at a rate of 2 km/s. How long does it take for the area A to be at least 5 times its original size?

Answer

Since the smallest area is 7 x 5 = 35

5A = 5(35)=175

S=3.63

Implicitly: Something not directly expressed

While this itself is not a function, we can split it into two equations that do define function.

How to solve: you solve for y you should get two equations

Find the domain and range of then graph it.

Group Work: (try factor it!)2 2Describe the graph of the relation 2 4.x xy y

Answer

2 2

2

2 4

( ) 4 factor the left side

2 take the square root of both sides

2 solve for

The graph consists of two lines 2 and 2.

x xy y

x y

x y

y x y

y x y x

Homework Practice• P124 #2,4,6,7,12,16,19,23,24,34,37,44

PARAMETRIC RELATIONS AND INVERSES

Parametric Equation• A function that define both elements of the ordered pair

(x,y) in terms of another variable t.

Example of Parametric

• A)Find the points determined by t=-3,-2,-1,0,1,2,3• B)Graph it• C)Is y a function of x?• D)Find an algebraic relationship between x and y

Answer

• C) Yes, it is because one input has one output• D)

Group Work: Graph Parametric Equation

• A)Find the points determined by t=-3,-2,-1,0,1,2,3• B)Graph it• C)Is y a function of x?• D)Find an algebraic relationship between x and y

Answer

• C) no, because it failed the vertical line test• D)

t (x,y)

-3 (3,-2)

-2 (0,-1)

-1 (-1,0)

0 (0,1)

1 (3,2)

2 (8,3)

3 (15,4)

How to do parametric equation with the calculator.

Slide 1- 127

Inverse Relation

The ordered pair (a,b) is in a relation if and only

if the pair (b,a) is in the inverse relation.

Slide 1- 128

Horizontal Line Test

The inverse of a relation is a function if and only

if each horizontal line intersects the graph of the

original relation in at most one point.

Slide 1- 129

Inverse Function

-1

-1

If is a one-to-one function with domain and range , then the

, denoted , is the function with domain and range

defined by ( ) if and only if ( ) .

f D R

f R D

f b a f a b

inverse

function of f

How do you find the inverse?• You first have to switch the x and y, then solve for y

• Ex:

• Switch the x and y

• Now solve for y

Group work: Find the inverse

𝑓 (𝑥 )= 𝑥𝑥+3

Slide 1- 132

Group Work:

-1 2Find an equation for ( ) if ( ) .

1

xf x f x

x

Slide 1- 133

Answer

-1

2 Switch the and

1

Solve for :

( 1) 2 Multiply by 1

2 Distribute

2 Isolate the terms

( 2) Factor out

Divide by 22

Therefore ( ) .2

yx x y

y

y

x y y y

xy x y x

xy y x y

y x x y

xy x

xx

f xx

Slide 1- 134

The Inverse Reflection Principle

The points (a,b) and (b,a) in the coordinate plane

are symmetric with respect to the line y=x. The

points (a,b) and (b,a) are reflections of each

other across the line y=x.

Slide 1- 135

The Inverse Composition Rule

A function is one-to-one with inverse function if and only if

( ( )) for every in the domain of , and

( ( )) for every in the domain of .

f g

f g x x x g

g f x x x f

Group Work

3 3Show algebraically the ( ) 2 and ( ) 2 are inverse functions.f x x g x x

Homework Practice• P135 #2,5,9-12, 14,16,22,27,29,33

MODELING WITH FUNCTIONSPlease bring your book

Quick Talk: Name the following formulas

• 1) Volume of Sphere

• 2) Volume of Cone

• 3) Volume of Cylinder

• 4) Surface Area of Sphere

• 5) Surface Area of Cone

• 6) Surface Area of Cylinder

• 7) Area of circle

• 8) Circumference

Answer• 1) • 2) • 3)• 4) • 5) • 6) • 7) • 8)

Slide 1- 141

Example A Maximum Value Problem

A square of side inches is cut out of each corner of an 8 in. by 15 in. piece

of cardboard and the sides are folded up to form an open-topped box.

(a) Write the volume as a function of .

(b) Find th

x

V x

e domain of as a function of .

(c) Graph as a function of over the domain found in part (b) and use

the maximum finder on your grapher to determine the maximum volume

such a box can hold.

(d) How b

V x

V x

ig should the cut-out squares be in order to produce the box of

maximum volume?

Answer

(a) The width 8 2 and the length 15 2 . The depth is when the

sides are folded up.

8 2 15 2

x x x

V x x x

(b) The depth of must be nonnegative, as must the side length and width.

The domain is [0,4] where the endpoints give a box with no volume.

x

3

The maximum occurs at the point (5/3, 90.74).

The maximum volume is about 90.74 in. .

(d) Each square should have sides of one-and-two thirds inches.

Group Work• Grain is leaking through a hole in a storage bin at a constant

rate of 5 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of the cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour?

Answer

2

3 3

The volume of a cone is 1/ 3 . Since the height always equals the radius,

1/ 3 . When 12 inches, the volume will be (1/ 3) 12 576 in. .

One hour later, the volume will have grown by (60 mi

V r h

V h h V

3 3

3

3

3

3

n)(5 in. / min) 300 in .

The total volume will be 576 300 in .

1/ 3 576 300

3 576 300

3 576 300

12.63 inches

h

h

h

h

I hope you bring your book• Classwork

• P160 #1-35 every other odd