1. Review 9.1-9.4 Geometric Sequences Exponential Functions
Exponential Growth or Decay
Linear, Quadratic, and Exponential Functions

2. Definition of Sequence
Geometric Sequences and Series
An infinite sequence is a function whose domain is the set of positive integers.
a1, a2, a3, a4, , an, . . .
terms
The first three terms of the sequence an = 2n2 are
a1 = 2(1)2 = 2
a2 = 2(2)2 = 8
finite sequence
a3 = 2(3)2 = 18.
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Definition of Sequence

3. Definition of Geometric Sequence
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.
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Definition of Geometric Sequence

4. The nth Term of a Geometric Sequence
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.
a1 = 15
15, 75, 375, , . . .
a2 = 15(5)
a3 = 15(52)
a4 = 15(53)
The nth term is 15(5n-1).
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The nth Term of a Geometric Sequence

5. Example: Finding the nth Term
Example: Find the 9th term of the geometric sequence
7, 21, 63, . . .
a1 = 7
an = a1rn – 1 = 7(3)n – 1
a9 = 7(3)9 – 1 = 7(3)8
= 7(6561) = 45,927
The 9th term is 45,927.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Finding the nth Term

6. Evaluate.
1. 100(1.08)20
2. 100(0.95)25
3. 100(1 – 0.02)10
4. 100( )–10
≈ 466.1
≈ 27.74
≈ 81.71
≈ 46.32

7. 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.

8. Check It Out! Example 2
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
Exponential growth function.
P(t) = 350( )t
Substitute 350 for a and 0.14 for r.
P(t) = 350(1.14)t
Simplify.

9. Check It Out! Example 3
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
Exponential decay function.
f(t) = 1000(1 – 0.15)t
Substitute 1,000 for a and 0.15 for r.
f(t) = 1000(0.85)t
Simplify.

10. 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

11. Examples: LINEAR, QUADRATIC or EXPONENTIAL? y = 6x + 3 y = 7x2 +5x – 2

12. Identifying from a graph:
Linear Makes a straight line
Quadratic Makes a parabola
Exponential Rises or falls quickly in one direction

13. LINEAR, QUADRATIC or EXPONENTIAL?
a) b)
c) d)

14. Is the table linear, quadratic or exponential?
y changes more quickly than x.
Never see the same y value twice.
Common multiplication pattern
Linear
Never see the same y value twice.
1st difference is the same
Quadratic
See same y more than once.
2nd difference is the same

15. Concept

16. Example 1
A. 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.

17. Example 1
B. Graph the ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.
(–1, 6), (0, 2),
Answer: The ordered pairs appear to represent an exponential function.

18. Example 1
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)
A. linear
B. quadratic
C. exponential

19. Example 1
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)
A. linear
B. quadratic
C. exponential

20. Example 2
A. Look for a pattern in the table of values to determine which kind of model best describes the data. – 2 2 2 2
First differences:
Answer: Since the first differences are all equal, the table of values represents a linear function.

21. Example 2
B. Look for a pattern in the table of values to determine which kind of model best describes the data. __ 4 3 9
–24 –8 –2 __ 2 3 __ 8 9 –
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.

22. Example 2 –24 –8 –2 2 3 8 9 – First differences: 16 5 1 3 1 7 9__ 2 3 8 9 –
First differences: 16 5 __ 1 3 1 __ 7 9
Second 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. 36 4 __ 9 12 3 __ 1 3 __ 1 3 __ 1 3 __ 1 3
Ratios:

23. Example 2 The ratios of successive y-values are equal.
Answer: The table of values can be modeled by an exponential function.

24. Example 2
A. Look for a pattern in the table of values to determine which kind of model best describes the data.
A. linear
B. quadratic
C. exponential
D. none of the above

25. Example 2
B. Look for a pattern in the table of values to determine which kind of model best describes the data.
A. linear
B. quadratic
C. exponential
D. none of the above

26. Example 3
Write 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.
– – – – –4096 –7
–56
–448
–3584
First differences:

27. Example 3 –7 –56 –448 –3584 First differences: –49 –392 –3136
Write an Equation –7
–56
–448
–3584
First differences:
–49
–392
–3136
Second differences: –1 –8
–64
Ratios:
–512
–4096
× 8
× 8
× 8
× 8
The table of values can be modeled by an exponential function.

28. Example 3
Write 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

29. Example 3
Determine which model best describes the data. Then write an equation for the function that models the data.
A. quadratic; y = 3x2
B. linear; y = 6x
C. exponential; y = 3x
D. linear; y = 3x

30. Write an Equation for a Real-World Situation
Example 4
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.

31. Write an Equation for a Real-World Situation
Example 4
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.

32. Example 4 y = mx + b Equation for linear function
Write an Equation for a Real-World Situation
Example 4
y = mx + b Equation for linear function
8 = 3(0) + b x = 0, y = 8, and m = 3
b = 8 Simplify.
Answer: The equation that models the data is y = 3x + 8.
Check You used (0, 8) to write the function. Verify that every other ordered pair satisfies the function.

33. Example 4
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.
A. linear; y = 4x + 4
B. quadratic; y = 8x2
C. exponential; y = 2 ● 4x
D. exponential; y = 4 ● 2x

34. EXAMPLE 2
Identify functions using differences or ratios b. x
– 2
– 1 1 2 y 4 7 10
Differences:
ANSWER
The table of values represents a linear function.

35. EXAMPLE 2
Identify functions using differences or ratios
Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function. a. x –2 –1 1 2 y –6 –4 6
First differences:
Second differences:
ANSWER
The table of values represents a quadratic function.

36. 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. y 2 x
– 2
– 1 1
0.08
0.4 10
ANSWER
exponential function

37. x y -5 1 -4 2 -1 3 4 11 x y -2 -1 -4 -8 2
-32 5
-256

38. Is the table linear, quadratic or exponential?y 1 2 -1 3 4 5 8 x y 1 3 2 9 27 4 81 5
243 x y 1 5 2 9 3 13 4 17 21

39. EXAMPLE 3
Write an equation for a function
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 1 2 y
0.5

40. EXAMPLE 3
Write an equation for a function
SOLUTION
STEP 1
Determine which type of function the table of values represents. x –2 –1 1 2 y
0.5
First differences: – –
Second differences: