Review 9.1-9.4

Geometric Sequences

Exponential Functions

Exponential Growth or Decay

Linear, Quadratic, and Exponential Functions

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An infinite sequence is a function whose domain is the set of positive integers.

a1, a2, a3, a4, . . . , an, . . .

The first three terms of the sequence an = 2n2 are

a1 = 2(1)2 = 2

a2 = 2(2)2 = 8

a3 = 2(3)2 = 18.

finite sequence

terms

Geometric Sequences and Series

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A sequence is geometric if the ratios of consecutive terms are the same.

2, 8, 32, 128, 512, . . .

geometric sequence

The common ratio, r, is 4.

82

4

328

4

12832

4

512128

4

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The nth term of a geometric sequence has the form

an = a1rn - 1

where r is the common ratio of consecutive terms of the sequence.

15, 75, 375, 1875, . . . a1 = 15

The nth term is 15(5n-1).

75 515

r

a2 = 15(5)

a3 = 15(52

)

a4 = 15(53

)

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Example: Find the 9th term of the geometric sequence

7, 21, 63, . . .a1 = 7

The 9th term is 45,927.

21 37

r

an = a1rn – 1 = 7(3)n – 1

a9 = 7(3)9 – 1 = 7(3)8

= 7(6561) = 45,927

Evaluate.

1. 100(1.08)20

2. 100(0.95)25

3. 100(1 – 0.02)10

4. 100(1 + 0.08)–10

≈ 466.1

≈ 27.74

≈ 81.71

≈ 46.32

You can model growth or decay by a constant percent increase or decrease with the following formula:

In the formula, the base of the exponential expression, 1 + r, is called the growth factor. Similarly, 1 – r is the decay factor.

In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth. Use a graph to predict when the population will reach 20,000.

P(t) = a(1 + r)t

Substitute 350 for a and 0.14 for r.

P(t) = 350(1 + 0.14)t

P(t) = 350(1.14)t Simplify.

Exponential growth function.

Check It Out! Example 2

A motor scooter purchased for $1000 depreciates at an annual rate of 15%. Write an exponential function and graph the function. Use the graph to predict when the value will fall below $100.

f(t) = a(1 – r)t

Substitute 1,000 for a and 0.15 for r.

f(t) = 1000(1 – 0.15)t

f(t) = 1000(0.85)t Simplify.

Exponential decay function.

Check It Out! Example 3

Identifying from an equation:

Linear

Has an x with no exponent.

y = 5x + 1

y = ½x

2x + 3y = 6

Quadratic

Has an x2 in the equation.

y = 2x2 + 3x – 5

y = x2 + 9

x2 + 4y = 7

Exponential

Has an x as the

exponent.

y = 3x + 1

y = 52x

4x + y = 13

Examples:

• LINEAR, QUADRATIC or EXPONENTIAL?

a)y = 6x + 3

b)y = 7x2 +5x – 2

c)9x + 3 = y

d)42x = 8

Identifying from a graph:

Linear

Makes a straight line

Quadratic

Makes a parabola

Exponential

Rises or falls quickly in

one direction

LINEAR, QUADRATIC or EXPONENTIAL?

a) b)

c) d)

Is the table linear, quadratic or exponential?

Quadratic

• See same y more than once.

• 2nd difference is the same

Linear

• Never see the same y value twice.

• 1st difference is the same

Exponential

• y changes more quickly than x.

• Never see the same y value twice.

• Common multiplication pattern

Concept

Example 1A. Graph the ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.

(1, 2), (2, 5), (3, 6), (4, 5), (5, 2)

Answer: The ordered pairs appear to represent a quadratic equation.

Example 1

Answer: The ordered pairs appear to represent an exponential function.

B. Graph the ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.

(–1, 6), (0, 2),

Example 1

A. linear

B. quadratic

C. exponential

A. Graph the set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.(–2, –6), (0, –3), (2, 0), (4, 3)

Example 1

A. linear

B. quadratic

C. exponential

B. Graph the set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.(–2, 0), (–1, –3), (0, –4), (1, –3), (2, 0)

Example 2A. Look for a pattern in the table of values to determine which kind of model best describes the data.

–1 1 3 5 7

2 2 2 First differences:

Answer: Since the first differences are all equal, the table of values represents a linear function.

2

Example 2B. Look for a pattern in the table of values to determine which kind of model best describes the data.

–24 –8 First differences:

The first differences are not all equal. So, the table of values does not represent a linear function. Find the second differences and compare.

36 12 4__43

__49

–2 __23

__89–

Example 2

16

First differences:

The second differences are not all equal. So, the table of values does not represent a quadratic function. Find the ratios of the y-values and compare.

1 __79

5 __13Second differences:

36 4 __49

12 __43

__13

__13

Ratios: __13

__13

–24 –8 –2 __23

__89–

Example 2The ratios of successive y-values are equal.

Answer: The table of values can be modeled by an exponential function.

Example 2

A. linear

B. quadratic

C. exponential

D. none of the above

A. Look for a pattern in the table of values to determine which kind of model best describes the data.

Example 2

A. linear

B. quadratic

C. exponential

D. none of the above

B. Look for a pattern in the table of values to determine which kind of model best describes the data.

Example 3Write an Equation

Determine which kind of model best describes the data. Then write an equation for the function that models the data.

Step 1 Determine which model fits the data.

–1 –8 –64 –512 –4096

–7 –56 –448 –3584 First differences:

Example 3Write an Equation

–7 –56 –448 –3584 First differences:

–49 –392 –3136Second differences:

× 8 × 8

The table of values can be modeled by an exponential function.

–1 –8 –64Ratios: –512 –4096

× 8 × 8

Example 3Write an Equation

Step 2 Write an equation for the function that models the data.

The equation has the form y = abx. Find the value of a by choosing one of the ordered pairs from the table of values. Let’s use (1, –8).

y = abx Equation for exponential function

–8 = a(8)1 x = 1, y = –8, b = 8–8 = a(8) Simplify.–1 = a An equation that models the

data is y = –(8)x.Answer: y = –(8)x

Example 3

A. quadratic; y = 3x2

B. linear; y = 6x

C. exponential; y = 3x

D. linear; y = 3x

Determine which model best describes the data. Then write an equation for the function that models the data.

Example 4Write an Equation for a Real-World Situation

KARATE The table shows the number of children enrolled in a beginner’s karate class for four consecutive years. Determine which model best represents the data. Then write a function that models that data.

Example 4Write an Equation for a Real-World Situation

Understand We need to find a model for the data, and then write a function.

Plan Find a pattern using successive differences or ratios. Then use the general form of the equation to write a function.

Solve The first differences are all 3. A linear function of the form y = mx + b models the data.

Example 4Write an Equation for a Real-World Situation

y = mx + b Equation for linear function

8 = 3(0) + b x = 0, y = 8, and m = 3b = 8 Simplify.

Answer: The equation that models the data is y = 3x + 8.

CheckYou used (0, 8) to write the function. Verify that every other ordered pair satisfies the function.

Example 4

A. linear; y = 4x + 4

B. quadratic; y = 8x2

C. exponential; y = 2 ● 4x

D. exponential; y = 4 ● 2x

WILDLIFE The table shows the growth of prairie dogs in a colony over the years. Determine which model best represents the data. Then write a function that models the data.

Identify functions using differences or ratiosEXAMPLE 2

ANSWER

The table of values represents a linear function.

x – 2 – 1 0 1 2

y – 2 1 4 7 10

Differences: 3 3 3 3

b.

Identify functions using differences or ratios

EXAMPLE 2

Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function.

ANSWER

The table of values represents a quadratic function.

x –2 –1 0 1 2

y –6 –6 –4 0 6

First differences: 0 2 4 6

Second differences: 2 2 2

a.

GUIDED PRACTICE for Examples 1 and 2

2. Tell whether the table of values represents a linear function, an exponential function, or a quadratic function.

ANSWER exponential function

0

y 2

x – 2 – 1 1

0.08 0.4 10

x y

0 -5

1 -4

2 -1

3 4

4 11

x y

-2 -2

-1 -4

0 -8

2 -32

5 -256

Is the table linear, quadratic or exponential?

x y1 0

2 -1

3 0

4 3

5 8

x y1 5

2 9

3 13

4 17

5 21

x y1 3

2 9

3 27

4 81

5 243

Write an equation for a functionEXAMPLE 3

Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Then write an equation for the function.

x –2 –1 0 1 2

y 2 0.5 0 0.5 2

SOLUTION

Write an equation for a functionEXAMPLE 3

STEP 1 Determine which type of function the table of values represents.

x –2 –1 0 1 2

y 2 0.5 0 0.5 2

First differences: –1.5 –0.5 0.5 1.5

Second differences: 1 1 1