2.7 – analyzing graphs of functions and piecewise defined functions tests for symmetry copyright...

2.7 – Analyzing Graphs of Functions and Piecewise Defined Functions

Tests for Symmetry

Note: Origin symmetry is about the line .

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

x-axis symmetry:

Not equivalent

y-axis symmetry: Origin symmetry:


Tests for Symmetry

𝑦= 𝑥2 −9𝑥2+2

Example:

− 𝑦= 𝑥2− 9𝑥2+2

𝑦=(−𝑥 )2− 9

(−𝑥 )2+2

𝑦= 𝑥2 −9𝑥2+2

− 𝑦=(−𝑥 )2− 9

(−𝑥 )2+2

− 𝑦= 𝑥2− 9𝑥2+2

No x-axis symmetry. Equivalent

y-axis symmetry.

Not equivalent

No origin symmetry.


Tests for Symmetry

𝑦= 𝑥2 −9𝑥2+2

Example:

No x-axis symmetry. y-axis symmetry. No origin symmetry.


Even and Odd Functions

y-axis symmetry.

origin symmetry.



Even and Odd Functions

Identify the type of symmetry for each graph

y-axis symmetry.

no symmetry.

Even function

origin symmetry.

Odd function


function


Tests for Even and Odd Functions

odd function

𝑓 (𝑥 )=𝑥3+5 𝑥

Identify the type of symmetry for a function

𝑓 (−𝑥 )=(−𝑥 )3+5 (−𝑥 )

𝑓 (−𝑥 )=−𝑥3− 5 𝑥

𝑓 (−𝑥 )=− (𝑥3+5 𝑥 )

𝑓 (−𝑥 )=− 𝑓 (𝑥 )

function

𝑔 (𝑥 )=− 4 𝑥3+1

𝑔 (−𝑥 )=− 4 (−𝑥 )3+1

𝑔 (−𝑥 )=4 𝑥3+1

𝑔 (−𝑥 ) ≠𝑔(𝑥)

𝑔 (−𝑥 ) ≠−𝑔 (𝑥 )even function

h (𝑥 )=2 𝑥2 −3

h (−𝑥 )=2 (−𝑥 )2 −3

h (−𝑥 )=2𝑥2− 3

h (−𝑥 )=h(𝑥)

Defn: Piecewise Function

A function with different equations for different domains.

Examples

𝑓 (𝑥 )={𝑥+2 𝑖𝑓 𝑥 ≥ 22𝑥 𝑖𝑓 𝑥<2

h (𝑥 )={ 3 𝑥+2 𝑖𝑓 𝑥<−22 𝑥 𝑖𝑓 − 2≤𝑥≤ 3

−2 𝑥+6 𝑖𝑓 𝑥>3


Piecewise Functions

a)

𝑓 (𝑥 )={𝑥+2 𝑖𝑓 𝑥 ≥ 22𝑥 𝑖𝑓 𝑥<2

𝑓 (1 )=2 (1) 𝑓 (1 )=2

b) 𝑓 (3 )=3+2 𝑓 (3 )=5

(1,2 )

(3,5 )

c) 𝑓 (2 )=2+2 𝑓 (2 )=4 (2,4 )


Piecewise Functions

2𝑓 (1 )=2 (2 )=4

𝑓 (𝑥 )=2𝑥 𝑓 (𝑥 )=𝑥+2𝑓 (2 )=2+2=4

a)

𝑓 (𝑥 )={𝑥+2 𝑖𝑓 𝑥 ≥ 22𝑥 𝑖𝑓 𝑥<2

(2,4 ) (2,4 )


Piecewise Functions

a)

𝑓 (𝑥 )={𝑥+2 𝑖𝑓 𝑥 ≥ 22𝑥 𝑖𝑓 𝑥<2

(− ,)b)

(1,2 )(3,5 )

c)

𝑦 𝑒𝑠

(2,4 )(0,0 )

d)

(− ,)

(2,4 )


Piecewise Functions

a) h (2 )=2(2) h (2 )=4

b) h (− 3 )=3 (−3 )+2 h (− 3 )=− 7

(2,4 )

(−3 ,− 7 )

c) h (− 2 )=2(−2)h (− 2 )=− 4 (−2 , − 4 )

h (𝑥 )={ 3 𝑥+2 𝑖𝑓 𝑥<−22 𝑥 𝑖𝑓 − 2≤𝑥≤ 3

−2 𝑥+6 𝑖𝑓 𝑥>3

d) h (3 )=2(3)h (3 )=6 (3,6 )

e) h ( 4 )=−2 ( 4 )+6 h ( 4 )=−2 ( 4 , −2 )


Piecewise Functions

a)

h (𝑥 )={ 3 𝑥+2 𝑖𝑓 𝑥<−22 𝑥 𝑖𝑓 − 2≤𝑥≤ 3

−2 𝑥+6 𝑖𝑓 𝑥>3

(−2 , − 4 ) (−2 , − 4 )(3,6 ) (3,0 )

1

1

2

2

3

3


Piecewise Functions

2

a)

h (𝑥 )={ 3 𝑥+2 𝑖𝑓 𝑥<−22 𝑥 𝑖𝑓 − 2≤𝑥≤ 3

−2 𝑥+6 𝑖𝑓 𝑥>3

(− ,)b)

c)

𝑛𝑜d) ¿

𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 𝑥=3


Piecewise Functions


Greatest Integer Function:

𝑦= 𝑓 (𝑥 )=⟦𝑥 ⟧

For all real numbers x, the greatest integer function returns the largest integer less than or equal to x.

The greatest integer function rounds down a real number to the nearest integer.


Intervals of Increasing, Decreasing and Constant Behavior



Where is the function constant?


Intervals of Increasing, Decreasing and Constant Behavior

Where is the function increasing?

Where is the function decreasing?

(0 ,3 )

(− 4 ,0 )

(3 , 6 )∪(−6 ,− 4 )


Relative (Local) Maximum and Minimum of a Function


Relative (Local) Maxima and Minima of a Function

𝒅𝒆𝒄 .

𝒅𝒆𝒄 .𝒊𝒏𝒄 .𝒊𝒏𝒄 .

Relative Maxima Relative Minima

(1 , 2 ) (−1 , 1 )(3 , 0 )Relative Maxima

Value

2Relative Minima

Value

1𝑎𝑛𝑑0

2.8 – Algebra of Functions and Function Composition

Operations on Functions

2 3For the functions 2 3 4 1

find the following:

f x x g x x

424 23 xx

224 23 xx

31228 325 xxx

2 32 3 4 1x x

2 32 3 4 1x x 2 3(2 3)(4 1)x x

2

3

2 3

4 1

x

x


Function Domains


2

4(a)

2 3

xf x

x x

2(b) 9g x x

(c) 3 2h x x 2

The denominator 0 so find values

where 3 2 0.x x

3 1 0x x

3, 1x x x

The set of all real numbers

Only nonnegative numbers have real

square roots so 3 2 0.x

Find the domain of each of the following functions

(− ∞ ,−1 )∪ (−1 , 3 )∪ (3 , ∞ )

(− ∞ , ∞ )

𝑥≤32

(− ∞ ,32 ]

The Difference Quotient is used to find the average rate of change between two points.

The Difference Quotient also represents to slope of the secant line between two points on a curve.

Secant line

𝑥 𝑥+h

h

(𝑥 , 𝑓 (𝑥 ) )

(𝑥 , 𝑓 (𝑥+h ) )𝑓 (𝑥+h )

𝑓 (𝑥 )

𝑦= 𝑓 (𝑥 )

𝑨𝒗𝒆𝒓𝒂𝒈𝒆𝑹𝒂𝒕𝒆𝒐𝒇 𝑪𝒉𝒂𝒏𝒈𝒆=𝒇 (𝒙+𝒉 )− 𝒇 (𝒙 )

𝒉𝑺𝒍𝒐𝒑𝒆𝒐𝒇 𝒕𝒉𝒆𝑺𝒆𝒄𝒂𝒏𝒕𝒍𝒊𝒏𝒆=

𝒇 (𝒙+𝒉 ) − 𝒇 (𝒙 )𝒉


Difference Quotient

Find the difference quotient

𝑓 (𝑥 )=−2 𝑥2+4

𝑓 (𝑥+h ) − 𝑓 (𝑥 )h

−2 (𝑥+h )2+4 − (− 2𝑥2+4 )h

−2 (𝑥2+2 h𝑥 +h2 )+4 − (−2 (1 )2+4 )h

−2𝑥2 − 4 h𝑥 − 2 h2+4+2𝑥2− 4h

− 4 h𝑥 − 2h2

hh (− 4 𝑥−2h )

h

h𝑇 𝑒𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡 :− 4 𝑥− 2h

→


Difference Quotient

𝑓 (𝑥 )= 9𝑥

𝐿𝐶𝐷 :𝑥 (𝑥+h )

𝑓 (𝑥+h ) − 𝑓 (𝑥 )h

9𝑥+h

−9𝑥

h

𝑥𝑥

∙9

𝑥+h−

9𝑥

∙𝑥+h𝑥+h

h

9 𝑥𝑥 (𝑥+h )

−9𝑥+9 h𝑥 (𝑥+h )

h

9 𝑥− 9𝑥−9 h𝑥 (𝑥+h )

h

− 9hh𝑥 (𝑥+h )

h𝑇 𝑒𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡 :− 9

𝑥 (𝑥+h )

Find the difference quotient


Difference Quotient

The process of combining two or more functions in order to create another function.One function is evaluated at a value of the independent variable and the result is substituted into the other function as the independent variable.The composition of functions f and g is written as:

( 𝑓 ∘𝑔 ) (𝑥 )¿ 𝑓 (𝑔 (𝑥 ) )

The composition of functions is a function inside another function.


Composition of Functions

( 𝑓 ∘𝑔 ) (𝑥 )¿ 𝑓 (𝑔 (𝑥 ) )Given:, find .

( 𝑓 ∘𝑔 ) (𝑥 )= 𝑓 (𝑔 (𝑥 ) )=¿2 (𝑥2+5 )+3

¿2 𝑥2+10+3

2 𝑥2+1 3( 𝑓 ∘𝑔 ) (𝑥 )= 𝑓 (𝑔 (𝑥 ) )=¿

Find .

(𝑔∘ 𝑓 ) (𝑥 )=𝑔 ( 𝑓 (𝑥 ) )=¿(2 𝑥+3 )2+5

4 𝑥2+6𝑥+6 𝑥+9+5

4 𝑥2+12 𝑥+14(𝑔∘ 𝑓 ) (𝑥 )=𝑔 ( 𝑓 (𝑥 ) )=¿



( 𝑓 ∘𝑔 ) (𝑥 )¿ 𝑓 (𝑔 (𝑥 ) )Given:, find .

( 𝑓 ∘𝑔 ) (𝑥 )= 𝑓 (𝑔 (𝑥 ) )=¿(𝑥2+2 )3+ (𝑥2+2 ) −6

Find .

(𝑔∘ 𝑓 ) (𝑥 )=𝑔 ( 𝑓 (𝑥 ) )=¿(𝑥3+𝑥− 6 )2+2

( 𝑓 ∘𝑔 ) (𝑥 )= 𝑓 (𝑔 (𝑥 ) )=¿(𝑥2+2 )3+𝑥2 − 4



2.7 – analyzing graphs of functions and piecewise defined functions tests for symmetry copyright...

Documents

graphs of functions

algebra of functions

piecewise functiona

xaxis symmetry

symmetry copyright

type of symmetry

functionorigin symmetry

equivalentno origin