Ch. 5 Properties of Exponents & Power FunctionsUnit 3

Unit 3 Evidence #1 - Exponents

5.1 Exponential Functions

5.2 Properties of Exponents & Power Functions

5.3 Rational Exponents & Roots

5.4 Applications of Exponential & Power Equations

5.5 Building Inverses of Functions

Unit 3 Evidence #2 - Logarithms

5.6 Logarithmic Functions

5.7 Properties of Logarithms

5.8 Applications of Logarithms

ACT StandardsN 605. Apply properties of rational exponents.F 703. Exhibit knowledge of geometric sequences.

Ch. 5.1 Exponential FunctionsLearning Intentions:

Write explicit equations for geometric sequences. Define exponential function & recognize y = a𝒃𝒙 as the parent function. See real-world growth & decay situations & recognize that the

exponential function models growth when b > 1decay when b < 1

Evaluate an exponential function using either the explicit equations or graphical methods.

Discover point-ratio form. Learn about half-life & doubling time.

Submerged in a storage tank in La Hague, France,this radioactive waste glows blue. The blue light is known as the “Cherenkov glow.”

𝒏 𝒖𝒏

0 22

1 20

2 17

3 14

4 10

5 8

6 8

7 7

8 6

9 5

10 3

11 3

12 1

Algebra II

Ch. 5.1RadioactiveDecayInvestigation

𝒏 𝒖𝒏

0 23

1 19

2 17

3 15

4 9

5 8

6 8

7 7

8 7

9 6

10 5

11 4

12 3

13 1

Block: B-2 Block: B-4 Block: B-5𝒏 𝒖𝒏

0 30

1 27

2 24

3 21

4 18

5 17

6 15

7 15

8 13

9 10

10 4

11 4

12 3

13 3

14 3

15 3

16 3

17 2

18 1

𝒏 𝒖𝒏0 18

1 16

2 16

3 14

4 10

5 7

6 5

7 3

8 3

9 3

10 3

11 2

12 1

Block:

Geometric Sequence -> Explicit Equation -> Exponential Model

ቊ𝑢0 = 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑢𝑛 = 𝑟 ∙ 𝑢𝑛−1

𝑢𝑛 = 𝑢0∙ 𝑟𝑛 y = a𝒃𝒙

Exponential function: a function with a variable in the exponent, typically used to model growth or decay.

General form (parent) of an exponential function: y = a𝒃𝒙

where a = 𝒖𝟎 = y-intercept ; (0, 𝒖𝟎) = (0, a)

b = r = base of power & common growth/decay rate

Ch. 5.1 Investigation - Step 3

𝑛 𝑢𝑛 𝒖𝟎 ∙ 𝒓𝒏

Given y = a𝑏𝑥, it is convenient when you know the y-intercept (0, a).If you are given the growth rate & another point (𝑥1, 𝑦1) on the curve, you can use Point-Ratio Form to write an equation.

Ex.) Write 3 equations given the points ‘not hidden’ below.

Is it possible to write the equation for this curve in terms of y = 𝒂𝒃𝒙 if you are not given the y-interceptof (0, 4)?

SOLUTION: Ex.) Write 3 equations given the points ‘not hidden’ below.

Is it possible to write the equation in terms of y = a𝒃𝒙 if you are not given the y-intercept?

YES. Solve for (0, a) & re-write the equation.

Let x = 0 in any one of point-ratio equations

y = 12.96(𝟏. 𝟖)𝟎−𝟐

y = 12.96 (𝟏. 𝟖)−𝟐

y = 4Thus, (0, a) = (0, 4) and

y = 4(𝟏. 𝟖)𝒙

y = 𝒚𝟏(𝐛)𝒙−𝒙𝟏

Find b. (See below)

b = 1.8Let (𝑥1, 𝑦1) = (2, 12.96)

y = 12.96(𝟏. 𝟖)𝒙−𝟐

Recall: r =𝒖𝒏

𝒖𝒏−𝟏(i.e. r =

𝑢1

𝑢0=𝑢2

𝑢1=𝑢3

𝑢2=𝑢4

𝑢3= ….)

Therefore:

b = 𝑦𝑥

𝑦𝑥−1= 𝑦1

𝑦0=... or b =

𝒇(𝒙)

𝒇(𝒙−𝟏)= 𝒇(𝟏)

𝒇(𝟎)= 𝒇(𝟐)

𝒇(𝟏)

b = 𝒇(𝟑)

𝒇(𝟐)= 𝒇(𝟒)

𝒇(𝟑)

b = 23.328

12.96= 41.9904

23.328

b = 1.8

y = 𝒚𝟏(𝐛)𝒙−𝒙𝟏

Find b. (See below)

b = 1.8Let (𝑥1, 𝑦1) = (3, 23.328)

y = 23.328(𝟏. 𝟖)𝒙−𝟑

y = 𝒚𝟏(𝐛)𝒙−𝒙𝟏

Find b. (See below)

b = 1.8Let (𝑥1, 𝑦1) = (4, 41.9904)

y = 41.9904(𝟏. 𝟖)𝒙−𝟒

Half-Life vs. Doubling Time

Half-life: the time needed for a Doubling time: the time needed for a

given amount (y) to decrease by 𝟏

𝟐. given amount (y) to double.

p. 256 #8.) Each of the red curves is a transformation of y = 𝟐𝒙 (shown in black).Focus on how the two marked points on the black curve are transformed tobecome the corresponding points on the red curve. Write an equation for the red’s.

a.) b.)

c.) d.)

Hint:

Let y = 𝒚−𝒌

𝒃

and

x = 𝒙−𝒉

𝒂

Given: y = a𝑏𝑥

(𝑦−𝑘

𝑏) = a𝑏(

𝑥−ℎ

𝑎)

Given: y = 𝟐𝒙

( ) = 𝟐( )

SOLUTIONS: p. 256 #8.) Each of the red curves is a transformation of y = 𝟐𝒙 (shown in black). Write an equation for the red curves.

a.) b.)

c.)

Hint:

Let y -> 𝑦−𝑘

𝑏

x -> 𝑥−ℎ

𝑎

Given: y = 2𝑥

(𝒚−𝑘

𝑏) = 𝟐(

𝒙−ℎ

𝑎)

h = 3

y = 𝟐𝒙−𝟑

Optional form: (𝑥1, 𝑦1) = (3, 1)Solve for (0, a)y = 𝟐𝟎−𝟑

y = 𝟐−𝟑 = 𝟏

𝟖

Thus,

y =𝟏

𝟖(𝟐)𝒙

T(0, 1) = (0, 3)T(1, 2) = (1, 6)T(x, y) = (x, 3y)

b = 3

y = 3∙ 𝟐𝒙

T(0, 1) = (0, 1)T(1, 2) = (3, 2)T(x, y) = (3x, y)

a = 3

y = 𝟐𝒙

𝟑 or

y = 𝟑𝟐𝒙

k = 2

y = 𝟐𝒙 + 2

d.)

p. 256 #10.) Each of the red curves is a transformation of y = 𝟎. 𝟓𝒙 (shown in black).Focus on how the two marked points on the black curve are transformed tobecome the corresponding points on the red curve. Write an equation for the red’s.

Note:

y = 𝟎. 𝟓𝒙 = (𝟏

𝟐)𝒙

y = 𝟏𝒙

𝟐𝒙=

𝟏

𝟐𝒙

y = 𝟐−𝒙

Thus, y = 𝟎. 𝟓𝒙

or y = 𝟐−𝒙

SOLUTIONS: p. 256 #10.) Each of the red curves is a transformation of y = 𝟎. 𝟓𝒙

(shown in black). Focus on how the two marked points on the black curve are transformed to become the corresponding points on the red curve. Write an equation for the red’s.

T(-1, 2) = (0, 0)T(0, 1) = (1, -1)T(x, y) = (x+1, y+2)

h = 1 k = 2y – 2 = (𝟎. 𝟓)𝒙−𝟏

or

y = (𝟎. 𝟓)𝒙−𝟏+ 𝟐

T(-1, 2) = (-2, 6)T(0, 1) = (0, 3)T(x, y) = (2x, 3y)

a = 2 b = 3𝒚

𝟑= (𝟎. 𝟓)

𝒙

𝟐 or

𝒚 = 3(𝟎. 𝟓)𝒙

𝟐 or

𝒚 = 3·𝟐𝟎. 𝟓𝒙

T(-1, 2) = (-1, 3)T(0, 1) = (0, 4)T(x, y) = (x, -1y + 5)

b = -1 k = 5𝒚−𝟓

−𝟏= (𝟎. 𝟓)𝒙 or

y = –(𝟎. 𝟓)𝒙 + 5

If b = -1 vertical reflection over x-axisTHEN, shift curve up 5 units (k = 5)

A formula for a geometric sequence generates a set of discrete points. Now, you will learn how to find the equation of the continuous function that passes through the points.

A rare coin in Jo’s coin collection is worth $450. The value has been & will continue to increase by 15% each year. At this rate, how much will the coin be worth in 11.5 years?

Discrete Sequences Formulas:

Recursive: ቊ𝒖𝟎 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡𝒖𝒏 = 𝒓 ∙ 𝒖𝒏−𝟏 𝑠. 𝑡. 𝑛 ≥ 1

Explicit: 𝒖𝒏 = 𝑢0·(𝑟)𝑛

Discrete Explicit Formula:

𝒖𝒏 = ___________

Domain: 𝒏 = {0, 1, 2, 3, ...}

Continuous Function:

𝒇(𝒙) = __________

Domain: {𝒙ȁ𝒙 ≥ 𝟎}

SOLUTIONS: A rare coin in Jo’s coin collection is worth $450. The value has been & will continue to increase by 15% each year.

Discrete Sequences Formulas:

Recursive: ቊ𝒖𝟎 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡𝒖𝒏 = 𝒓 ∙ 𝒖𝒏−𝟏 𝑠. 𝑡. 𝑛 ≥ 1

Explicit: 𝒖𝒏 = 𝑢0·(𝑟)𝑛

Continuous Function: 𝒇(𝒙) = 𝑎 ∙ 𝑏𝑥 Domain: {𝒙ȁ𝒙 ≥ 𝟎}

𝒇(𝒙) = 450 ∙ (1.15)𝑥

ቊ𝒖𝟎 = 𝟒𝟓𝟎𝒖𝒏 = 𝟏. 𝟏𝟓 ∙ 𝒖𝒏−𝟏 𝑠. 𝑡. 𝑛 ≥ 1

𝒖𝒏 = 𝟒𝟓𝟎·(𝟏. 𝟏𝟓)𝑛

Domain: 𝒏 = {0, 1, 2, 3, ...}

At this rate, how much will the coin be worth in 11.5 years?

Let 𝑥 = 11.5

𝒇(𝒙) = 450 ∙ (1.15)𝑥

𝒇(𝟏𝟏. 𝟓) = 450 ∙ (1.15)𝟏𝟏.𝟓

𝒇(𝟏𝟏. 𝟓) = 2245.110277...

𝒇(𝟏𝟏. 𝟓) ≈ 2245.11 The coin will be worth $2,245.11 in 11.5 years