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Lesson 2 – Exponents, Functions and Function Operations - MiniLesson

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Lesson 2 - Mini-Lesson Section 2.1 – Properties of Exponents

What is an exponent? An exponent is a number in the “superscript” location and identifies the number of times the base number is to be multiplied times itself. For example:

!23 =2⋅2⋅2 In this situation, the exponent 3 is attached to the base 2. Raising 2 to the 3rd power indicates that we are to multiply the base 2 times itself a total of 3 times. Here is how we would perform that multiplication:

!23 =2⋅2⋅2= 4 ⋅2=8

Zero Exponent Property

For any nonzero real number !a , !!a0 =1 . Notes about this property:

• !00 is undefined • !a is the base of the exponential expression

Problem 1 MEDIA EXAMPLE – Zero Exponent Property Simplify the following expressions using the Zero Exponent Property.

a) !50 = b) !−50 =

c) !!5x0 != ! d) !! 5x( )0 =


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Problem 2 WORKED EXAMPLE – Zero Exponent Property Simplify the following expressions using the Zero Exponent Property.

a)

!20 =1!

b)

!

−20 != −1( )20= −1( ) 1( )!= −1!

c)

!!

−2x0 = −2! 1( ) != −2

!!Note:!x0 =1;!assume!that!x ≠0

d)

!!−2x( )0 != !1!!

!Note:!The!base!is!−2x;!assume!that!x ≠0

Negative Exponent Property For any real number !!a≠0,!!b≠0 :

!!a−m = 1

am1a−m = am a

b⎛⎝⎜

⎞⎠⎟

−m

= ba

⎛⎝⎜

⎞⎠⎟

m

Problem 3 MEDIA EXAMPLE – Negative Exponent Property Rewrite each of the following with only positive exponents. Variables represent nonzero quantities.

a) !!x −3 = b)

!!

1

x−12

=

c) !!12x −3 = d) !2−3 =

e) !416

⎛⎝⎜

⎞⎠⎟

−12= f) !!3x −4 =


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Problem 4 WORKED EXAMPLE – Negative Exponent Property Rewrite each of the following with only positive exponents. Variables represent nonzero quantities.

a)

!!y−4 = 1

y4 b)

!!1x −2 = x

2

c)

!!

14x −5 =

1x54

Note:!The!coefficient!4!does!not!move!

= x5

4

d)

!

4−12 = 1

412

= 14

= 12

e)

!!

2x −5 = 2x5

Note:!The!coefficient!2!does!not!move.

f)

!

13

⎛⎝⎜

⎞⎠⎟

−3

= 31

⎛⎝⎜

⎞⎠⎟

3

= 3( )3=3⋅3⋅3= 9⋅3=27

Problem 5 You Try – Zero Exponent Property and Negative Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible.

a) !!7a0 = b) !! 7a( )0 =

c) !!5a−2

3 = d)

!!

7

a−12

=

e) !! 7a( )−1 = f) !!7a

⎛⎝⎜

⎞⎠⎟

−1

=


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Multiplication Property:

For any real number !!a, !am ⋅an = am+n

Problem 6 MEDIA EXAMPLE – Multiplication Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible.

a) !!n3n9 = b) !!b5·b–4·b=

c) !!5x2 7x3( ) = d) !!5x2 + !7x3 =

Problem 7 WORKED EXAMPLE – Multiplication Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. a)

!!q7q5 = q7+5

= q12 b)

!!

a4a−52b= a

4+(−52)b

= a82 − 52b

= a32b

c)

!!

4 y5 7xy−3( ) = 4 ⋅7( ) x( ) y5 y−3( )= 4 ⋅7( ) x( ) y5 + (−3)( )= 28( ) x( ) y2( )=28xy2

d) !!4 y5 +7 y−3 = completely!simplified


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Raising an Exponent to an Exponent Property:

For any real number !!a,

!am( )n = amn

Problem 8 MEDIA EXAMPLE – Raising an Exponent to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible.

a) !! x3( )5 = b) !!−2x x

2( )3 =

Problem 9 WORKED EXAMPLE – Raising an Exponent to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible.

a)

!!

2x2 x3( )4 =2x2x12=2x2+12=2x14

b)

!!

−4x2 x3( )−23 = −4x2x

3 i −23⎛⎝⎜

⎞⎠⎟

= −4x2x −2

= −4x2+(−2)= −4x0= −4 1( )= −4


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Product and Quotient to an Exponent Property: For any real number !!a,

PRODUCT:

!! ab( )n = !anbn

QUOTIENT:

!!ab

⎛⎝⎜

⎞⎠⎟

n

= an

bn, b≠0

Problem 10 MEDIA EXAMPLE – Product to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible.

a) !! 5x( )2 = b) !!−8ab

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⎛

⎝⎜⎞

⎠⎟

2

!= c) !!5n4 −3n3( )–2 !=

Problem 11 MEDIA EXAMPLE – Quotient to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible.

a) !57

⎛⎝⎜

⎞⎠⎟

2

= b) !!x5

y−3⎛

⎝⎜⎞

⎠⎟

4

=

Problem 12 WORKED EXAMPLE – Product to an Exponent Property and Quotient to an Exponent Property

Simplify the following expression. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible.

!!

–1a12b−2⎛

⎝⎜⎞

⎠⎟

4

= ! −1( )4 a12

⎛

⎝⎜⎞

⎠⎟

4

b−2( )4

=1 a42

⎛

⎝⎜⎞

⎠⎟b−8( )

= a2

b8


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The Division Property:

For any nonzero real number !!a,

!!am

an= a(m−n)

Problem 13 MEDIA EXAMPLE – The Division Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible.

a) x4

x20= b) !!

4a10b56ab−2 =

Problem 14 WORKED EXAMPLE – The Division Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible.

a)

!!

y12

y6= y12−6

1= y6

b)

!!

y6

y12= y6−12

1

= y−6

1= 1y6

c)

!!

3a−1b5

9ab6 = 1a−1b5

3ab6 = a−1b5

3a1b6 =a−1−1b5−6

3 = a−2b−1

3 = 13a2b1 =

13a2b

Note:!This!problem!shows!horizontal!format!which!is!also!anacceptable!way!to!write!your!solutions.!


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Problem 15 You Try – Properties of Exponents Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. Show all steps as in the media examples.

a) !! 2x4( )2 = b) !!2 x

2( )3 =

c) !!8g32 5g–4( ) = d) !!6 2n( )–3 =

e) !!3a107

⎛

⎝⎜⎞

⎠⎟

2

= f) !!6x3 y89xy5 =


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Polynomials and Polynomial Multiplication: Variables raised to exponents and joined by addition form the building blocks, called terms, of algebraic expressions known as polynomials. Some examples are below. Note that the coefficient is the number in front of the variable part of a term.

Expression Terms Name Coefficients in Order

!!2x3 1 term:!!2x3 Monomial 2

!!4x − 7 2 terms:!!4x , −7 Binomial 4, !−7

!!x2 + 5 2 terms: !!x

2 , 5 Binomial 1, 5

!!2x2 + 3x − 5 3 terms:

!!2x2 , 3x , −5

Trinomial 2, 3,!−5

Polynomial multiplication requires the use of the exponent properties learned previously in this lesson. Some examples are below. Problem 16 MEDIA EXAMPLE – Multiplying Polynomials

Multiply and simplify. a) !!3x x −4( ) = b) !!−2x x

3 − x + 5( ) =

c) !!3x x − 4( ) + 2 x − 4( ) = Problem 17 WORKED EXAMPLE – Multiplying Polynomials

Multiply and simplify. a)

!!

2x x − 4( ) =2x x( ) + 2x −4( )=2x2 − 8x

b)

!!

−x x4 − 3x + 6( ) = −x( ) x4( )+ −x( ) −3x( )+ −x( ) 6( )= −x5 + 3x2( )+ −6x( )= −x5 + 3x2 − 6x


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Multiplying Binomials and the FOIL Method:

A binomial as a two-term polynomial. When we multiply two binomials, we can use the FOIL method (meaning First, Outside, Inside, Last) to help us keep track of our multiplications. Here is the general form:

When we multiply two binomials together, we initially obtain four terms. Usually, two of these will combine resulting in three terms, or, a trinomial. Problem 18 MEDIA EXAMPLE – Multiplying Binomials/Higher Order Polynomials

Multiply and simplify.

a) !! x + !3( ) x + !4( )!= b) !! 2d − !4( ) 3d + !5( )!=

c) !! x − 2( ) x2 + !2x − !4( )!= d) !! 3!− !2a( )2 =


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Problem 19 WORKED EXAMPLE – Multiplying Binomials/Higher Order

Polynomials

Multiply each set of polynomials below and combine like terms to simplify.

a)

!!

3x –!2( ) 4x + !3( )!= 3x( ) 4x( )+ 3x( ) 3( )+ −2( ) 4x( )+ −2( ) 3( )= 12x2( )+ 9x( )+ −8x( )+ −6( ) !!!!!!!!!!!!!!!!!!!!!!=12x2 +9x −8x −6 !!!!!!!!!!!!!!!!!!!!!!!!=12x2 + x − 6

b)

!!

x +1( ) −2x2 + x −5( )= x( ) −2x2( )+ x( ) x( )+ x( ) −5( )+ 1( ) −2x2( )+ 1( ) x( )+ 1( ) −5( )= −2x3( )+ 1x2( )+ −5x( )+ −2x2( )+ 1x( )+ −5( )= −2x3( )+ 1x2 −2x2( )+ −5x +1x( )+ −5( )= −2x3( )+ −x2( )+ −4x( )+ −5( )= −2x3 − x2 −4x −5

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!


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Problem 20 YOU TRY – Multiplying Binomials/Higher Order Polynomials

Multiply each set of polynomials below and combine like terms to simplify.

a) !! x − !1( ) x + !4( ) = b) !! 3x − !4( ) 5x + !2( ) =

c) !! x + !5( )2 = d) !! x − 5( )2 =

e) !! x −5( ) x +5( ) = f) !! x +2( ) x −2( ) =

g) !! x −4( ) x2 + 2x + 1( ) = h) !! 2x −1( ) x2 − 2x + 1( ) =


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Section 2.2 – Using Properties of Exponents to Evaluate Functions Problem 21 MEDIA EXAMPLE – Function Evaluation Given the function, !!f (x)=2x

2 , evaluate each of the following. Show your work. Write final results as ordered pairs.

a) !!f −5( ) = b) !!f 56

⎛⎝⎜

⎞⎠⎟=

c) !!f 10x + 1( ) = d) !!f −2x3( ) = !

Problem 22 WORKED EXAMPLE – Function Evaluation

Given !!f x( ) = 5

x2, evaluate each of the following. Show your work. Write final results as ordered

pairs. a)

!!

f −5( ) = 5−5( )2

[Replace!x !with!−5!in!f x( )]

!!!!!!!!!!! = 525 [ −5( )2 = −5( ) −5( ) =25]

!!!!!!!!!!! = 15 [Reduce!fraction]

−5, 15⎛⎝⎜

⎞⎠⎟

b)

!!

f 5x( ) = 55x( )2

[Replace!x !with!5x !in!f x( )]

!!!!!!!!!!! = 525x2 [ 5x( )2 = 5x( ) 5x( ) =25x2]

!!!!!!!!!!! = 15x2 [Reduce!fraction]

5x , 15x2⎛⎝⎜

⎞⎠⎟


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Problem 23 You Try – Function Evaluation Given !!f x( ) ! = !3x2 , evaluate each of the following. Show your work. Write final results as ordered pairs.

a) !!f 23

⎛⎝⎜

⎞⎠⎟=

b) !!f −5x3( ) ! =

c) !!f 2x − 5( ) ! =


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Section 2.3 – Combining Functions

Basic Mathematical Operations The basic mathematical operations are: addition, subtraction, multiplication, and division. When working with function notation, these operations will look like this:

Addition Subtraction Multiplication Division

!f x( )+ g x( )

!f x( )− g x( )

!f x( )⋅ g x( )

!

f x( )g x( ) !!g x( )≠0

Problem 24 WORKED EXAMPLE – Adding and Subtracting Functions Given !!f x( ) =2x2 +3x −5 and !!g x( ) = −x2 +5x +1 , determine each of the following. a)

!!

f x( )+ g x( ) = 2x2 +3x −5( )+ −x2 +5x +1( )=2x2 + 3x − 5 − x2 + 5x + 1=2x2 − x2 + 3x + 5x − 5 + 1= x2 + 8x − 4

b)

!!

f x( )− g x( ) = 2x2 + 3x − 5( ) − −x2 + 5x + 1( )=2x2 + 3x − 5 + x2 − 5x − 1=2x2 + x2 + 3x − 5x − 5 − 1=3x2 − 2x − 6

Problem 25 MEDIA EXAMPLE – Adding and Subtracting Functions Given !!f x( ) =3x2 + 2x − 1 and !!g x( ) = −x2 − 2x + 5 , determine each of the following. a) !

f x( ) + g x( ) = b) !f x( ) − g x( ) =

Problem 26 YOU TRY – Adding and Subtracting Functions Given !!f x( ) = x2 + 4 and !!g x( ) = −2x2 − 3x + 1 , determine each of the following. a) !f x( ) + g x( ) = b) !

f x( ) − g x( ) =


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Function Multiplication and the Multiplication Property of Exponents

When multiplying functions, you will often need to work with exponents. Try to recognize the examples below as being similar to ones completed earlier in the lesson. Problem 27 WORKED EXAMPLE – Function Multiplication

For each set of functions below, show all work to determine !f x( )⋅ g x( ) .

a)

!!

Given!f x( ) = −8x4 and g x( ) =5x3 ,f x( )⋅ g x( ) = −8x4( ) 5x3( )

= −8⋅5( ) x4 ⋅x3( )= −40x7

b)

!!

Given!f x( ) =3x + 2and g x( ) =2x − 5,f x( )⋅ g x( ) = 3x +2( ) 2x − 5( )

= 3x( ) 2x( )+ 3x( ) −5( )+ 2( ) 2x( )+ 2( ) −5( )=6x2 − 15x + 4x − 10=6x2 − 11x − 10

Problem 28 MEDIA EXAMPLE – Function Multiplication

Given !!f x( ) =3x +2 and !!g x( ) =2x2 +3x −1 , show all work to determine . Problem 29 YOU TRY – Function Multiplication

For each of the following, show all work to determine!f x( )⋅ g x( ) .

a)!!f x( ) =3x −2and g x( ) =3x +2 b) !!f x( ) =2x2 and g x( ) = x3 −4x +5

!f x( )⋅ g x( )


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Function Division and the Division Property of Exponents When dividing functions, you will also need to work with exponents. Try to recognize the examples below as being similar to ones completed earlier in the lesson. Problem 30 WORKED EXAMPLE – Function Division

For each of the following, determine !

f x( )g x( ) . Use only positive exponents in your final answer.

!!a)!f x( ) =15x15 and g x( ) =3x9

!!

f x( )g x( ) =

15x153x9

=5x15−9=5x6

!!b) f x( ) = −4x5 and g x( ) =2x8

!!

f x( )g x( ) =

−4x52x8

= −2x5 − 8= −2x −3

= − 2x3

Problem 31 MEDIA EXAMPLE – Function Division



a)!!f x( ) =10x4 + 3x2 and g x( ) =2x2 b)!!f x( ) = −12x5 + 8x2 + 5 and g x( ) = −4x2 Problem 32 YOU TRY – Function Division



a) !!f x( ) =25x5 − 4x7 and g x( ) = −5x4 b) !!f x( ) =20x6 − 16x3 + 8 and g x( ) = −4x3


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Working with Functions in Different Forms: Tables and Graphs Functions can be presented in multiple ways including: equations, data sets, graphs and applications. If you understand function notation, then the process for working with functions is the same no matter how the information if presented. Problem 33 MEDIA EXAMPLE – Functions in Table Form Functions f (x) and g(x) are defined in the tables below. Find a – e below using the tables.

x –3 –2 0 1 4 5 8 10 12

!f x( ) 8 6 3 2 5 8 11 15 20

x 0 2 3 4 5 8 9 11 15

!g x( ) 1 3 5 10 4 2 0 –2 –5

a) !!f 1( ) = b) !!g 9( ) = c) !!f 0( )+ g 0( )= d) !!g 5( )− f 8( ) = e) !!f 0( )⋅ g 3( )= Problem 34 YOU TRY – Functions in Table Form Given the following two tables, complete the third table. Show work in the table cell for each column. The first one is done for you. Show your work the same way as the sample.

x 0 1 2 3 4

!f x( ) 4 3 –2 0 1

x 0 1 2 3 4

!g x( ) 6 –3 4 –2 2

x 0 1 2 3 4

!f x( )+ g x( )

!!

f 0( )+ g 0( )= 4 + 6= 10


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Problem 35 YOU TRY –Functions in Graph Form Use the graph to determine each of the following. Assume integer answers.

a) !!g 4( ) = _______ b) !!f 2( ) = _______ c) !!g 0( ) = _______

d) !!If f x( ) =0, x = _______ e) !!If g x( ) =0, x = _______

f) !!f −1( )+ g −1( ) =_______ g) !!g −6( )− f −7( ) = _______

h) !!f 1( )⋅ g −2( ) = _______ i) !!g 0( )⋅ f 1( ) = _______

j) !!

f −1( )g −6( ) = _______ k)

!!

g −6( )f −1( ) = _______


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Section 2.4 – Applications of Function Operations One of the classic applications of function operations is the forming of the profit function, !

P x( ) ,

by subtracting the cost function,!C x( ) , from the revenue function,!

R x( ) , as shown below.

Profit = Revenue – Cost

Given functions !P x( ) = Profit,!

R x( ) = Revenue, and !C x( ) = Cost:

!P x( ) = R x( )−C x( )

Problem 36 MEDIA EXAMPLE – Cost, Revenue, Profit A local courier service estimates its monthly operating costs to be $1500 plus $0.85 per delivery. The revenues are $6 for each delivery. Let x = the number of deliveries in a given month. a) Write a function, !

C x( ) , to represent the monthly costs for making x deliveries per month. b) Write a function, !

R x( ) , to represent the revenue for making x deliveries per month. c) Write a function, !

P x( ) , that represents the monthly profits for making x deliveries per month. d) Determine algebraically the break-even point for the function !

P x( ) and how many deliveries you must make each month to begin making money. e) Solve the equation !!P x( ) =0 graphically to confirm the break-even point.


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Problem 37 YOU TRY – Cost, Revenue, Profit Charlie’s Chocolate Shoppe sells their chocolates for $1.80 per piece. The fixed costs to run the Chocolate Shoppe total $450 for the week, and Charlie estimates that each chocolate costs about $0.60 to produce. Charlie estimates that he can produce up to 3,000 chocolates in one week. a) Write a function, !

C n( ) , to model Charlie’s total weekly costs if he makes !n chocolates. b) Write a function, !

R n( ) , to represent the revenue from the sale of !n chocolates. c) Write a function, !

P n( ) , that represents Charlie’s profit from selling !n chocolates.

d) Interpret the meaning of the statement !!P 300( ) = −90. Write your answer as a complete sentence. e) Determine the practical domain and practical range for !

P n( ) , then use that information to define an appropriate viewing window. Sketch the graph from your calculator.

Practical Domain: Practical Range:

f) How many chocolates must Charlie sell in order to break even? Show complete work. Write your final answer as a complete sentence. Mark the break-even point on the graph above.

lesson 2 - mini-lesson · lesson 2 - mini-lesson section 2.1 – properties of exponents what is an...

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