CHAPTER 5 – Exponential and Logarithmic Functions and Equations Section 5.1 – Exponential Functions Objectives • Determine whether a function is exponential. • Identify the characteristics of exponential functions of the form 𝑓(𝑥) = 𝑏 𝑥 , including the domain, range, intercept, asymptote, end behavior, and general graphs. • Determine the formula for an exponential function given its graph. • Sketch the graph of exponential functions using transformations. • Solve exponential equations by relating the bases. • Solve compound interest application problems. • Determine the present value of an investment. • Solve exponential application problems Preliminaries An exponential function is a function of the form 𝑓(𝑥) = real number and 𝑏 has the properties 𝑏 is called the

_______

, where 𝑥 is any

and

.

of the exponential function.

List the properties of the exponential function 𝑓(𝑥) = 𝑏 𝑥 , where 𝑏 > 0 and 𝑏 ≠ 1. Domain:

Range:

y-intercept:

Asymptote: End behavior if 𝑏 > 1: and End behavior if 0 < 𝑏 < 1: and

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Graph if 𝑏 > 1:

Graph if 0 < 𝑏 < 1:

Examples?

Periodic compound interest can be calculated by using the formula 𝑟 𝑛𝑡 𝐴 = 𝑃 (1 + ) 𝑛 Write down the meaning of each value in the formula. 𝐴: 𝑃: 𝑟: 𝑛: 𝑡:

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Warm-up 1. Solve the following equations. 1 (A) 2𝑥 = 8 (B) 3𝑥 = 9

(C) 4𝑥 = 1

2. How many times per year would interest be calculated in each of the situations described below? Annually:

Semi-annually:

Quarterly:

Monthly:

Weekly:

Daily:

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Class Notes and Examples 5.1.1 Sketch the graphs of the following exponential functions. 𝑦 = 2𝑥

𝑦 = 3𝑥

𝑦 = 4𝑥

List the domain, range, intercept, asymptote, and end behavior for the above graphs.

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5.1.2

Sketch the graphs of the following exponential functions. 1 𝑥

𝑦 = (2)

1 𝑥

𝑦 = (3)

1 𝑥

𝑦 = (4)

List the domain, range, intercept, asymptote, and end behavior for the above graphs.

Summarize the properties that you found in examples 5.1.1 and 5.1.2 for exponential functions of the form 𝑓(𝑥) = 𝑏 𝑥 where 𝑏 > 1 and 𝑔(𝑥) = 𝑏 𝑥 , where 0 < 𝑏 < 1

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5.1.3

Determine a formula of the form 𝑦 = 𝑏 𝑥 for each of the exponential functions graphed below. (A) 8 7 6 5 4 3 2 1 -4

-3

-2

-1-1

(2, 4)

1

2

3

4

1

2

3

4

(B) 4 3 9

2

(−2, ) 4

1

-4

-3

-2

-1 -1

-2

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5.1.4

Each of the following functions were created using transformations of a base exponential function. State the base function and the transformations that were performed. List the domain, horizontal asymptote, range, and 𝑦-intercept. Graph the given function. (A) 𝐻(𝑥) = 2𝑥+3 Base function:

Transformation(s):

Domain:

Horizontal Asymptote:

Range:

y-intercept:

Graph:

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1 𝑥

(B) 𝐽(𝑥) = (3) + 2 Base function:

Transformation(s):

Domain:

Horizontal Asymptote:

Range:

y-intercept:

Graph:

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(C) 𝐿(𝑥) = −2 ∙ 4𝑥−3 Base function:

Transformation(s):

Domain:

Horizontal Asymptote:

Range:

y-intercept:

Graph:

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5.1.5

Determine a formula of the form 𝑦 = 𝐶 ∙ 𝑏 𝑥 for the exponential functions graphed below. Check by graphing on your calculator. (A)

24 20 16 (−1, 12)

12 8 4

-4

-3

-2

-1 -4

1

2

3

4

-4

-3

-2

-1 -4

1

2

3

4

(B)

(−1, − 4)

-8 -12 -16 -20

-24

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5.1.6

Determine a function of the form 𝑦 = 𝐶 ∙ 𝑏 𝑥 that passes through the points (1, 12) and (3, 192). Check your answer.

5.1.7

The table below gives values for a function of the form 𝑦 = 𝐶 ∙ 𝑏 𝑥 . Determine the values of 𝐶 and 𝑏. 𝑥

1

2

3

4

𝑦

8

12

18

27

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What strategy can you use to solve an exponential equation of the form 𝑏 𝑢 = 𝑏 𝑣 ?

5.1.8

Solve the following equations by rewriting in the form 𝑏 𝑢 = 𝑏 𝑣 . (A) 8−𝑥 = 64

1 2𝑥+1 3

(B) 27𝑥 = ( )

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5.1.9

Eric has started a new weightlifting routine. The amount of weight, in pounds, he is able to lift at the end of t weeks can be modeled by the following function. 𝑤(𝑡) = 260 − 140(2.6)−0.2𝑡 (A)

How much was Eric able to lift at the start of his weightlifting routine?

(B)

How much will Eric be able to lift at the end of 10 weeks? (Round to the nearest pound.)

5.1.10 Suppose Melissa invests $9400 into a high-yield savings account that pays 5.7% interest compounded quarterly. Her brother, Billy, invests $10,200 into a different account that pays 4.8% compounded monthly. If no other investments are made, who will have more money in their account at the end of 10 years? How much more money will that person have?

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5.1.11 Phillip wants to have $10,000 in 6 years, so he will place money into a savings account that pays 3.2% interest compounded weekly. How much should Phillip invest now to have $10,000 in 6 years? Check your answer.

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Section 5.1 Self-Assessment (Answers on page 257) 1.

(Multiple Choice) Determine the equation of the asymptote for 𝑦 = 2 ∙ 3𝑥−7 − 8. (A) 𝑥 = 7

2.

3.

(B) 𝑦 = 7

(C) 𝑦 = −8

(D) 𝑦 = −16 (E) 𝑥 = −8

(Multiple Choice) Determine the y-intercept for 𝑦 = 4 ∙ 2𝑥 − 9. (A) (0, −1)

(B) (0, −8)

(D) (0, −5)

(E) There is no y-intercept

(C) (0, −9)

Solve the equation algebraically by rewriting in the form 𝑏 𝑢 = 𝑏 𝑣 . 1 2𝑥+3 64 = ( ) 4 𝑥

4.

Determine a function of the form 𝑦 = 𝐶 ∙ 𝑏 𝑥 that passes through the points (−1, 12) 3

and (2, 2).

5.

(Multiple Choice) How much money should you invest at 3.6% compounded quarterly so that you have $10,000 after 6 years? The minimum amount of money that you need to invest is: (A) (B) (C) (D) (E)

Less than $8400 Between $8400 and $8900 Between $8900 and $9400 Between $9400 and $9900 More than $9900

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