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Exponents and Exponential Functions
Zero and Negative Exponents
Objective: To simplify expression involving zero and negative exponents.
Objectives
• I can simplify powers.
• I can simplify exponential expressions.
• I can evaluate an exponential expression.
• I can use an exponential expression.
Vocabulary
• You can extend the idea of exponents to include zero and negative exponents.
• Consider 33, 32, 𝑎𝑛𝑑 31. Decreasing the exponents by 1 is the same as dividing by
3. If you continue the pattern, 30 = 1 𝑎𝑛𝑑 3−1 =1
3.
• Zero and Negative exponents: Zero as an exponent: For every nonzero number a, 𝑎0 = 1.
40 = 1(−3)0= 1
(5.14)0= 1
Negative Exponent: For every nonzero number a and an integer
n, 𝑎−𝑛 =1
𝑎𝑛.
7−3 =1
73
(−5)−2=1
(−5)2
Vocabulary
• Why can’t you use 0 as a base with zero exponents?
The following properties imply the pattern:
30 = 1 20 = 1 10 = 1 00 = 1
• However, consider the following pattern.
03 = 0 02 = 0 01 = 0 00 = 0
• It is not possible for 00 to equal both 1 and 0. Therefore 00 is undefined.
• Why can’t you use 0 as a base with a negative exponent? Using 0 as a base with a negative exponent will result in division by zero, which is undefined.
Simplifying Powers
What is the simplified form of each expression?1. (–4.25)0
2. t0
3. –30
4. b3pc0
5.1
2
0
6. 4wz0
Practice
What is the simplified form of each expression?1. 4ab0
2. -90
3. K5j0
4. b0v0
5. 120r8
6. x0
Vocabulary
• An algebraic expression is in simplest form when powers with a variable base are written with only positive exponents.
Simplifying Exponential Expressions
What is the simplified form of the expression?1. 9-2
2. 4-3
3. 3-2
4. 6-1
5. (-4)-2
Practice
What is the simplified form of each expression?1. 5a3b-2
2. x-9
3.1
n−3
4. 4c-3b
5.2
a−3
6.n−5
m2
Practice
What is the simplified form of the expression?1. 120t7u-11
2.7ab−2
3w
3. c-5d-7
4. (-5)-3
5.6a−1c−3
d0
Vocabulary
• When you evaluate an exponential expression, you can simplify the expression before substituting values for the variables.
Evaluating an Exponential Expression
What is the value of 3s3t-2 for s = 2 and t = –3?
What is the value of each expression for n = –2 and w = 5?1. 𝑛−4𝑤0
2.𝑛−1
𝑤2
3.𝑛0
𝑤6
4.1
𝑛𝑤−1
Practice
Evaluate each expression for r = –3 and s = 5.1. r−3
2. 4s−1
3. s−3
4. r0s−2
5.3r
s−2
6. r−4s2
7.s0
r−2
8. 2−4r3s−2
Using an Exponential Expression
A population of marine bacteria doubles every hour under controlled laboratory conditions. The number of bacteria is modeled by the expression 1000 × 2ℎ, where h is the number of hours after a scientist measures the population size. Evaluate the expression for h = 0 and h = –3. What does each value of the expression represent in the situation?
Using an Exponential Expression
A population of insects triples every week. The number of insects is modeled by the expression 5400 × 3𝑤, where w is the number of weeks after the population was measured. Evaluate the expression for w = –2, w = 0, and w = 1. What does each value of the expression represent in the situation?
Practice
• The number of visitors to a certain Web site triples every month. The number of visitors is modeled by the expression 8100 × 3𝑚, where m is the number of months after the number of visitors was measured. Evaluate the expression m = –4. What does the value of the expression represent in the situation?
• A Galapagoes cactus finch population increases by half every decade. The number of finches is modeled by the expression 45 × 1.5𝑑, where d is the number of decades after the population was measured. Evaluate the expression for d = –2, d = 0, and d = 1. What does each value of the expression represent in the situation?
Multiplying Powers with the Same Base
Objective: To multiply powers with the same base.
Objectives
• I can multiply powers.
• I can multiply powers in algebraic expressions.
• I can multiply with scientific notation.
• I can simplify expressions with rational exponents.
• I can simplify expressions with rational exponents.
Vocabulary
• You can use a property of exponents to multiply powers with the same base.
• You can write a product of powers with the same base, such as 34 x 32, using one exponent.
• 34 x 32 = (3 x 3 x 3 x 3) x (3 x 3) = 36
• Notice that the sum of the exponents in the expression 34 x 32 equals the exponent of 36. In general, an equation such as 34 × 32 = 36 can be written using variables: 𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛.
• Here’s Why It works: You can use repeated multiplication to rewrite a product of powers.• 𝑎𝑚 × 𝑎𝑛 = 𝑎 × 𝑎 × ⋯ × 𝑎 × 𝑎 × 𝑎 × ⋯ × 𝑎 = 𝑎 × 𝑎 × ⋯ × 𝑎 = 𝑎𝑚+𝑛
m factors of a n factors of a m + n factors of a
Multiplying Powers
What is each expression written using each base only once?
1. 124 x 123
2. (–5)–2 (–5)7
3. 83 x 86
4. (0.5)–3 (0.5)–8
5. 9–3 x 92 x 96
Practice
Rewrite each expression using each base only once.1. 73 × 74
2. 22 × 27 × 20
3. (−6)12× (−6)5× (−6)2
4. 5−2 × 5−4 × 58
5. 96 × 9−4 × 9−2
6. (−8)5× (−8)−5
Vocabulary
• When variable factors have more than one base, be careful to combine only those powers with the same base.
Multiplying Powers in Algebraic Expressions
What is the simplified form of each expression?
1. 4z5 x 9z–12
2. 2a x 9b4 x 3a2
3. 5r4 x r9 x 3r
4. –4c3 x 7d2 x 2c–2
5. j2 x k–2 x 12j
Practice
Simplify each expression.1. 𝑚3𝑚4
2. 5𝑐4 × 𝑐6
3. 4𝑡−5 × 2𝑡−3
4. (𝑥5𝑦2) 𝑥−6𝑦
5. (5𝑥5)(3𝑦6)(3𝑥2)
6. −𝑚2 × 4𝑟3 × 12𝑟−4 × 5𝑚
Vocabulary
• You can use the property for multiplying powers with the same base to multiply two numbers written in scientific notation.
• Recall that you can use powers of 10 to make writing very large and very small numbers more convenient. In scientific notation, you can write any number as 𝑎 × 10𝑏, where 1 ≤ 𝑎 < 10. • Example: 256,000 is written in scientific notation as 2.56 × 105
• Example: 0.0045 is written in scientific notation as 4.5 × 10−3
Multiplying Numbers in Scientific Notation
What is the simplified form? Write your answer in scientific notation.
1. (3 x 105)(5 x 10–12)
2. (7 x 108)(4 x 105)
3. (1.13 x 10–7)(9.98 x 105)(3.34 x 1022)
4. (3 x 105)(8 x 104)
Multiplying Numbers in Scientific Notation
• At 20°C, one cubic meter of water has a mass of about 9.88 × 105g. Each gram of water contains about 3.34 × 1022 molecules of water. About how many molecules of water does the droplet of water contain? 𝑉 = 1.13 × 10−7𝑚3
• About how many molecules of water are in a swimming pool that holds 200𝑚3 of water? Write your answer in scientific notation.
Practice
• A human body contains about 2.7 × 104 microliters of blood for each pound of body weight. Each microliter of blood contains about 7 × 104
white blood cells. About how many white blood cells are in the body of a 140-pound person?
• The distance light travels in one second (one light-second) is about 1.86 × 105 mi. Saturn is about 475 light seconds from the sun. About how many miles from the sun is Saturn?
Vocabulary• Exponents can be expressed as fractions. Fractional exponents are called rational
exponents.
• Recall that 32 means 3 x 3, which equals 9. You can write the same expression using rational
exponents: 91
2. The equation 91
2 = b indicates that b is the positive number that when multiplied by itself, equals 9.
• In general, 𝑎1
𝑚 = 𝑏 means that b multiplied as a factor m times equals a.
• You can also have expressions like 93
2, which means 91
2 × 91
2 × 91
2. Consider each factor
individually. Because 91
2 = 3, you know 91
2 × 91
2 × 91
2 = 3 × 3 × 3 = 27. So 93
2 = 27.
Simplifying Expressions with Rational ExponentsSimplify the expression.
1. 811
4
2. 161
4
3. 271
3
4. 641
2
5. 643
2
6. 253
2
7. 272
3
8. 163
4
Practice
Simplify the expression.
1. 81
3
2. 6251
4
3. 10001
3
4. 163
4
5. 95
2
6. 647
3
Vocabulary
• You can use the properties of multiplying powers with the same base to simplify expressions with rational exponents.
• Multiplying Powers with the same base• Words: To multiply powers with the same base, add the exponents.
• Algebra: am x an = am+n, where a ≠ 0 and m and n are integers.
• Examples:
• 43 x 45 = 43+5 = 48
• B7 x B–4 = B7+(-4) = B3
Simplifying Expressions with Rational ExponentsSimplfiy the expression.
1. 2𝑎2
3 × 3𝑏1
4 𝑎1
3 × 5𝑏1
2
2. 2𝑐3
5 × 2𝑐1
5
3. 𝑛1
3 × 𝑛4
3
4. 𝑏2
3 × 𝑐2
5 𝑏4
9 × 𝑐9
10
5. 3𝑗2
3 × 7𝑚1
4 3𝑗1
6 × 7𝑚3
2
Practice
Simplify each expression.
1. 8𝑏2
3 × 9𝑡1
5 8𝑏5
3 × 9𝑡3
5
2. 7𝑑3
2 × 2𝑔5
6 2𝑔3
2 × 7𝑑5
6
3. 4𝑟2
5 × 5𝑠2
7 5𝑠5
7 × 4𝑟3
5
More Multiplication Properties of Exponents
Objective: To raise a power to a power. To raise a product to a power.
Objectives
• I can simplify a power raised to a power.
• I can simplify an expression with powers.
• I can simplify a product raised to a power.
• I can simplify an expression with products.
• I can raise a number in scientific notation to a power.
Vocabulary
• You can use properties of exponents to simplify a power raised to a power or a product raised to a power.
• You can use repeated multiplication to simplify a power raised to a power.
(t5)2 = t5 x t5 = t5+5 = t5x2 = t10
• Notice that (t5)2 = t5x2. Raising a power to a power is the same as raising the base to the product of the exponents.
Vocabulary
• Raising a Power to a Power• Words: To raise a power to a power, multiply the exponents.
• Algebra: (am)n = amn, where a ≠ 0 and m and n are integers.
• Examples:
• (𝑎3
2)3 = 𝑎3
2×3 = 𝑎
9
2
• (𝑥1
2)3
5 = 𝑥1
2×
3
5 = 𝑥3
10
• (54)2 = 54×2 = 58
• (𝑚3)5 = 𝑚3×5 = 𝑚15
Simplifying a Power Raised to a Power
What is the simplified form of:
1. (𝑛4)7
2. (𝑝5)4
3. (𝑝4)5
4. (𝑥2
3)1
2
5. (𝑝1
2)1
4
6. (𝑝1
4)1
2
Practice
Simplify each expression.
1. (𝑛8)4
2. (𝑛4)8
3. (𝑐2)1
2
4. (𝑥2
5)10
5. (𝑤7)−1
6. (𝑦3
5)−1
2
Simplifying an Expression with Powers
What is the simplified form of:
1. 𝑦3(𝑦5
2)−2
2. 𝑥2(𝑥6)−4
3. 𝑤−2(𝑤5
3)3
4. (𝑠−5)−1
2(𝑠3
2)
5. (𝑎−5)−2(𝑎)1
2
6. 𝑡2(𝑡−3)1
5
Practice
Simplify each expression.
1. 𝑑(𝑑−2)−9
2. (𝑧8)0(𝑧1
2)
3. (𝑎2
3)3𝑐4
4. (𝑐3)1
9(𝑑3)0
5. (𝑡2)−2(𝑡2)−5
6. (𝑚3)−1(𝑥1
3)1
4
Vocabulary
• You can use repeated multiplication to simplify an expression.
• Example: (4𝑚1
2)3 = 4𝑚1
2 × 4𝑚1
2 × 4𝑚1
2 = 4 × 4 × 4 × 𝑚1
2 × 𝑚1
2 × 𝑚1
2 = 64𝑚3
2
• Notice that (4𝑚1
2)3 = 43 × 𝑚3
2. This illustrates another property of exponents.
• Raising a Product to Power:• Words: To raise a product to a power, raise each factor to the power and multiply.
• Algebra: (ab)n = anbn, where a ≠ 0, b ≠ 0, and n is an integer.
• Example:
• (4𝑏)3
2= 43
2𝑏3
2 = 8𝑏3
2
• (3𝑟)4= 34𝑟4 = 81𝑟4
Simplifying a Product Raised to a Power
What is the simplified form of each expression?
1. (5x3)2
2. (7m9)3
3. (2z)–4
4. (36g4)−1
2
5. (3g4)–2
Practice
Simplify each expression.
1. (12g4)–1
2. (xy)0
3. (3n–6)–1
4. (r2s)5
5. (7a)−2
6. (5y1
2)4
Simplifying an Expression with Products
What is the simplified form of each expression?
1. (n1
2)10(4mn−2
3)3
2. (x−2)2(3xy5)4
3. (3c5
2)4(c2)3
4. (6ab)3(5a−3)2
5. (n5)2(3mn–2)3
Practice
Simplify each expression.
1. (2x1
6)3x2
2. (y2z−3)1
6(y3)2
3. (mg4)–1(mg4)
4. (3b−2)2(a2b4)3
5. c–12(c–2d)3d5
6. 4j2k6(2j11)3k5
Raising a Number in Scientific Notation to a Power
The expression 1
2𝑚𝑣2 gives the kinetic energy in joules, of an object
with a mass of m kilograms traveling at a speed of v meters per second. What is the kinetic energy of an experimental unmanned jet with a mass of 1.3 × 103 kg traveling at a speed of about 3.1 × 103 m/s?
What is the kinetic energy of an aircraft with a mass of 2.5 × 105 kg traveling at a speed of 3 × 102 m/s?
Practice
What is the following in scientific notation?
1. (3 x 105)2
2. (4 x 102)5
3. (3.5 x 10–4)3
4. (7.4 x 104)2
5. (2 x 10–3)3
6. (2.37 x 108)3
7. (2 x 10–10)3
8. (6.25 x 10–12)–2
Exit Ticket
• (2m3)0(3m6)–1
• (9 x 10–3)2
• (m3)–1(x2)5
Division Properties of Exponents
Objective: To divide powers with the same base. To raise a quotient to a power.
Objectives
• I can divide algebraic expressions.
• I can divide numbers in scientific notation.
• I can raise a quotient to a power.
• I can simplify an exponential expression.
Vocabulary
• You can use properties of exponents to divide powers with the same base.
• You can use repeated multiplication to simplify quotients of powers with the same base.
• Expand the numerator and the denominator.
• Then divide out the common factors.45
43 =4×4×4×4×4
4×4×4= 42
Vocabulary
• Dividing Powers with the Same Base:• Words: To divide powers with the same base, subtract the exponents.
• Algebra: 𝑎𝑚
𝑎𝑛 = 𝑎𝑚−𝑛, where a ≠ 0 and m and n are integers.
• Examples:
•26
22 = 26−2 = 24
•𝑥4
𝑥7 = 𝑥4−7 = 𝑥−3 =1
𝑥3
•𝑠
34
𝑠12
= 𝑠3
4−
1
2 = 𝑠3
4−
2
4 = 𝑠1
4
Dividing Algebraic Expressions
What is the simplified form of each expression?
1.𝑥
52
𝑥2
2.𝑚2𝑛4
𝑚5𝑛3
3.𝑦
34
𝑦12
4.𝑑
72
𝑑3
5.𝑘6𝑗2
𝑘𝑗5
6.𝑎−3𝑏7
𝑎5𝑏2
7.𝑥4𝑦−1𝑧8
𝑥4𝑦−5𝑧
Practice
What is the simplified form of each expression?
1.38
36
2.𝑑14
𝑑17
3.𝑥11𝑦3
𝑥11𝑦
4.10𝑚6𝑛3
5𝑚2𝑛7
5.32𝑚5𝑡6
35𝑚7𝑡−5
6.12𝑎−1𝑏6𝑐−3
4𝑎5𝑏−1𝑐5
Vocabulary
• You can use the property of dividing powers with the same base to divide numbers in scientific notation.
Dividing Numbers in Scientific Notation
• Population density describes the number of people per unit area. During one year, the population of Angola was 1.21 × 107 people. The area of Angola is 4.81 × 105mi2. What was the population density of Angola that year?
• During one year, Honduras has a population of 7.33 × 106 people. The area of Honduras is 4.33 × 104mi2. What was the population density of Honduras that year?
Practice
What is the simplified answer in scientific notation?
1. 5.2 x 1013 ÷ 1.3 x 107
2. 3.6 x 10–10 ÷ 9 x 10–6
3. 6.5 x 104 ÷ 5 x 106
4. 8.4 x 10–5 ÷ 2 x 10–8
5. 4.65 x 10–4 ÷ 3.1 x 102
6. 3.5 x 106 ÷ 5 x 108
Vocabulary
• You can use repeated multiplication to simplify a quotient raised to a power.
(𝑥
𝑦)3=
𝑥
𝑦×
𝑥
𝑦×
𝑥
𝑦=
𝑥∙𝑥∙𝑥
𝑦∙𝑦∙𝑦=
𝑥3
𝑦3 This suggests another property of exponents.
• Raising a Quotient to a Power:• Words: To raise a quotient to a power, raise the numerator and the denominator to
the power and simplify.
• Algebra: (𝑎
𝑏)𝑛=
𝑎𝑛
𝑏𝑛, where a ≠ 0, b ≠ 0, and n is a rational number.
• Examples: (3
5)3=
33
53 =27
125(
𝑥
𝑦)5=
𝑥5
𝑦5 (𝑎
𝑏)
1
2=𝑎
12
𝑏12
Raising a Quotient to a Power
• What is the simplified form?
1. (𝑧
23
5)3
2.4
𝑥3
2
3.𝑎
34
𝑎5
4
4.3
8
2
Practice
• What is the simplified form?
1.1
𝑎
3
2.3𝑥
4
4
3.2𝑥
3𝑦
5
4.6
52
3
5.22
23
5
6.8
𝑛5
6
7.2𝑝
9
3
Vocabulary
• You can write an expression of the form 𝑎
𝑏
−𝑛using positive exponents.
𝑎
𝑏
−𝑛=
1
𝑎
𝑏
𝑛 Use the definition of negative exponents
=1
𝑎𝑛
𝑏𝑛
Raise the quotient to a power.
= 1 ∙𝑏𝑛
𝑎𝑛 Multiply by the reciprocal of 𝑎𝑛
𝑏𝑛 which is 𝑏𝑛
𝑎𝑛.
=𝑏𝑛
𝑎𝑛 =𝑏
𝑎
𝑛Simplify. Write the quotient using one exponent.
So, 𝑎
𝑏
−𝑛=
𝑏
𝑎
𝑛for all nonzero numbers a and b and positive integers n.
Simplifying an Exponential Expression
• What is the simplified form of?
1.2𝑥6
𝑦4
−3
2.𝑎
5𝑏
−2
3.3𝑥2
5𝑦4
−4
4.2
5
−1
5. −7𝑥
32
5𝑦4
−2
Practice
• What is the simplified form of?
1.5
4
−4
2. −2𝑥
16
3𝑦4
−3
3.3𝑥
12
15
2
4.6𝑛2
3𝑛
−3
5.𝑏
45
𝑏7
−5
6.3
5𝑐2
0
Exit Ticket
• Simplify• x5y2
xy2
• 5.4 x 1012
1.2 x 103
• (6x4 / 3x2)–3
Rational Exponents and Radicals
Objective: To rewrite expressions involving radicals and rational exponents.
Objectives
• I can find roots.
• I can convert to radical form.
• I can convert to exponential form.
• I can use a radical expression.
Vocabulary
• You can use rational exponents to represent radicals.
• In a radical expression, the number under the radical sign is the radicand.
• The number in the crook of the radical sign is the index.
• The index gives the degree of the root.
• For a cube root, the degree is 3. If there is no index, the degree is 2, which means square root.
Vocabulary
• Recall what you know about square roots. Since 52 = 25, you know
that 25 = 5. You also know that 251
2 = 5. Using the transitive property of equality, you
can conclude that 25 = 251
2. Similarly, 3
8 = 81
3.
• You can simplify radical expressions by finding like factors, just as when simplifying powers with rational exponents.
Finding Roots
What is the simplified form of each expression?1.
3125
2.4
16
3.3
27
4.5
32
5.3
64
6.2
36
Practice
What is the value of each expression?1.
249
2.5
1
3.4
625
4.2
81
5.3
216
6.4
81
Vocabulary
• You can also write expressions that have rational exponents like 2
3in radical
form.
• 82
3 = 82∙1
3 = (82)1
3 =3
82
• 82
3 = 81
3∙2 = (8
1
3)2 = (3
8)2
• So, 82
3 =3
82 = (3
8)2
• Equivalence of Radicals and Rational Exponents:• If the nth root of a is a real number and m and n are positive integers, then
𝑎1
𝑛 = 𝑛 𝑎 𝑎𝑛𝑑 𝑎𝑚
𝑛 =𝑛
𝑎𝑚 = (𝑛 𝑎)𝑚.
Converting to Radical Form
What is the radical form?
1. 12𝑎2
3
2. (64𝑎)4
5
3. 𝑎5
6
4. 5𝑥1
3
5. (54𝑦)2
3
6. 25𝑥1
2
7. 𝑧3
4
Practice
Write each expression in radical form.
1. 𝑎2
3
2. (64𝑏)3
4
3. (25𝑥)1
2
4. 27𝑎2
3
5. (98𝑑)1
2
6. 18𝑏1
4
7. (24𝑐)2
3
Converting to Exponential Form
What is the exponential form?
1.5
𝑏3
2.3
27𝑑5
3.3
𝑠2
4.3
𝑥4
5. (4𝑦)5
6.4
256𝑎8
7.5
𝑎3
Practice
Write each expression in exponential form.
1. (2𝑐)4
2.4
256𝑎3
3. 3(8𝑥)2
4.3
27𝑐2
5. 4625𝑦3
6. 36𝑥
7.4
𝑥3
8.3
8𝑏5
Using a Radical Expression
• You can estimate the metabolic rate of living organisms based on body mass using Kleiber’s law. The formula 𝑅 = 73.3
4𝑀3 relates
metabolic rate R measured in Calories per day to body mass M measured in kilograms. What is the metabolic rate of a dog with a body mass of 18 kg?
• What is the metabolic rate of a man with a body mass of 75 kg?
Practice
• A company that manufactures memory chips for digital cameras uses the formula 𝑐 = 120
3𝑛2 + 1300 to determine the cost c, in dollars,
of producing n chips. How much of it cost to produce 250 chips?
Practice
• Carbon-14 is present in all living organisms and decays at a predictable rate. To estimate the age of an organism, archaeologists measure the amount of carbon-14 left in its remains. The approximate amount of carbon-14 remaining after 5000
years can be found using the formula 𝐴 = 𝐴0(2.7)−3
5, where 𝐴0 is the initial amount of carbon-14 in the sample that is tested. How much carbon-14 is left in a sample that is 5000 years old and originally contained 7.0 × 10−12 grams of carbon-14?
Exponential Functions
Objective: To evaluate and graph exponential functions.
Objectives
• I can identify linear and exponential functions.
• I can evaluate an exponential function.
• I can graph an exponential function.
• I can graph an exponential model.
• I can solve one-variable equations.
Vocabulary
• Some functions model an initial amount that is repeatedly multiplied by the same positive number. In the rules for these functions, the independent variable is an exponent.
• Exponential Function: An exponential function is a function of the form 𝑦 = 𝑎 ∙ 𝑏𝑥, where a ≠ 0, b > 0, b ≠ 1, and x is a real number.
Vocabulary
• Suppose all the x-values in a table have a common difference. If all the y-values have a common difference, then the table represents a linear function. If all of the y-values have a common ratio, then the table represents an exponential function.
Identifying Linear and Exponential Functions
• Does the table represent a linear or an exponential function?
𝑦 = 3𝑥
• Does the table or rule represent a linear or an exponential function?
𝑦 = 3 ∙ 6𝑥
X Y
0 –1
1 –3
2 –9
3 –27
X Y
1 –1
2 1
3 3
4 5
Practice
Determine whether each table or rule represents a linear or an exponential function.
• 𝑦 = 4 ∙ 5𝑥
• 𝑦 = 12 ∙ 𝑥
• 𝑦 = −5 ∙ 0.25𝑥
• 𝑦 = 7𝑥 + 3
X Y
1 2
2 8
3 32
4 128
X Y
0 6
1 9
2 12
3 15
Evaluating an Exponential Function
• Suppose 30 flour beetles are left undisturbed in a warehouse bin. The beetle population doubles each week. The function 𝑦 = 30 ∙ 2𝑥 gives the population after x weeks. How many beetles will there be after 56 days?
• An initial population of 20 rabbits triples every half year. The function 𝑓 𝑥 = 20 ∙ 3𝑥 gives the population after x half-year periods. How many rabbits will there be after 3 years?
Practice
• An investment of $5000 doubles in value every decade. The function 𝑓 𝑥 = 5000 ∙ 2𝑥, where x is the number of decades, models the growth of the value of the investment. How much is the investment worth after 30 years?
• A population of 75 foxes in a wildlife preserve quadruples in size every 15 years. The function 𝑦 = 75 ∙ 4𝑥, x is the number of 15-year periods, models the population growth. How many foxes will there be after 45 years?
• Evaluate each function for the given value.𝑓 𝑥 = 6𝑥 for 𝑥 = 2 𝑔 𝑡 = 2 ∙ 0.4𝑡 for 𝑡 = −2
𝑦 = 20 ∙ 0.5𝑥 for 𝑥 = 3 ℎ 𝑤 = −0.5 ∙ 4𝑤 for 𝑤 = 18
Graphing an Exponential Function
Graph each of the equations.1. 𝑦 = 3 ∙ 2𝑥
2. 𝑦 = 0.5 ∙ 3𝑥
3. 𝑦 = −0.5 ∙ 3𝑥
4. 𝑦 = 4𝑥
5. 𝑦 = 10 ∙3
2
𝑥
Practice
Graph each exponential function.1. 𝑦 = −4𝑥
2. 𝑦 = 0.1 ∙ 2𝑥
3. 𝑦 =1
3
𝑥
4. 𝑦 =1
4∙ 2𝑥
5. 𝑦 = −1
3
𝑥
6. 𝑦 = 1.25𝑥
Graphing an Exponential Model
• Computer mapping software allows you to zoom in on an area to view it in more detail. The function 𝑓 𝑥 = 100 ∙ 0.25𝑥 models the percent of the original area the map shows after zooming in x times. Graph the function.
• You can also zoom out to view a larger area on the map. The function 𝑓 𝑥 = 100 ∙ 4𝑥 models the percent of the original area the map shows after zooming out x times. Graph the function.
Practice
• A new museum had 7500 visitors this year. The museum curators expect the number of visitors to grow my 5% each year. The function 𝑦 = 7500 ∙1.05𝑥 models the predicted number of visitors each year after x years. Graph the function.
• A solid waste disposal plan proposes to reduce the amount of garbage each person throws out by 2% each year. This year, each person threw out an average of 1500 pounds of garbage. The function 𝑦 = 1500 ∙ 0.98𝑥 models the average amount of garbage each person will throw out each year after x years. Graph the function.
Vocabulary
• You have solved one-variable linear equations using graphs and a graphing calculator.
• In the next problem, you will write each side of the equation as a function and graph the functions.
• The x-value where the functions intersect is a solution.
Solving One-Variable Equations
What is the solution?1. 2𝑥 = 0.5𝑥 + 2
2. 0.3𝑥 = 5
3. 1.25𝑥 = −2𝑥
4. − 2 𝑥 =3
4𝑥 − 4
5. 4𝑥 =3
2𝑥 + 5
6. 𝑥 + 3 = 3𝑥
Exponential Growth and Decay
Objective: To model exponential growth and decay.
Objectives
• I can model exponential growth.
• I can find compound interest.
• I can model exponential decay.
Vocabulary
• An exponential function can model growth or decay of an initial amount.
• Exponential Growth:
• Exponential growth can be modeled by the function 𝑦 = 𝑎 ∙ 𝑏𝑥, where a > 0 and b > 1.
• The base b is the growth factor, which equals 1 plus the percent rate of change expressed as a decimal.
• Algebra: 𝑦 = 𝑎 ∙ 𝑏𝑥
• a = initial amount (when x = 0)
• b = the base, which is greater than 1, is the growth factor
• x = exponent
Modeling Exponential Growth
• Since 2005, the amount of money spent at restaurants in the United States has increased about 7% each year. In 2005, about $360 billion was spent at restaurants.
a. If the trend continues, about how much will be spent at restaurants in 2015?
b. What is a expression that represents the equivalent monthly increase of spending at U.S. restaurants in 2005?
• Suppose that in 1985, there were 285 cell phone subscribers in a small town. The number of subscribers increased by 75% each year after 1985. How many cell phone subscribers were in the small town in 1994? Write an expression to represent the equivalent monthly cell phone subscription increase.
Practice
• Identify the initial amount a and the growth factor b in each exponential function.
• 𝑔 𝑥 = 14 ∙ 2𝑥
• 𝑓 𝑡 = 1.4𝑡
• 𝑦 = 150 ∙ 1.0894𝑥
• 𝑦 = 25,600 ∙ 1.01𝑥
• A population of 100 frogs increases at an annual rate of 22%. How many frogs will there be in 5 years? Write an expression to represent the equivalent monthly population increase rate.
Practice
The number of college students enrolled at a college is 15,000 and grows 4% each year.
a. The initial amount a is __________.
b. The percent rate of change is 4%, so the growth factor b is 1 + _____ = _____.
c. To find the number of student enrolled after one year, you calculate 15,000 ∙ ______.
d. Complete the equation y = __________ ∙ __________ to find the number of students enrolled after x years.
e. Use your equation to predict the number of students enrolled after 25 years.
Vocabulary
• When a bank pays interest on both the principal and the interest an account has already earned, the bank is paying compound interest.
• Compound interest is an example of exponential growth.
• You can use the following formula to find the balance of an account that earns compound interest.
𝐴 = 𝑃 1 +𝑟
𝑛
𝑛𝑡
• A = the balance • P = the principal (the initial deposit)• r = the annual interest rate (expressed as a decimal)• n = the number of times interest is compounded per year• t = the time in years
Compound Interest
• Suppose that when your friend was born, your friend’s parents deposited $2000 in an account paying 4.5% interest compounded quarterly. What will the account balance be after 18 years?
• Suppose the account pays interest compounded monthly. What will the account balance be after 18 years?
Practice
Find the balance in each account after the given period.1. $4000 principal earning 6% compounded annually, after 5 years
2. $12,000 principal earning 4.8% compounded annually, after 7 years
3. $500 principal earning 4% compounded quarterly, after 6 years
4. $20,000 deposit earning 3.5% compounded monthly, after 10 years
5. $5000 deposit earning 1.5% compounded quarterly, after 3 years
6. $13,500 deposit earning 3.3% compounded annually, after 1 year
7. $775 deposit earning 4.25% compounded annually, after 12 years
8. $3500 deposit earning 6.75% compounded monthly, after 6 months
Vocabulary
• The function 𝑦 = 𝑎 ∙ 𝑏𝑥 can model exponential decay as well as exponential growth.
• In both cases, b is determined by the percent rate of change.
• The value of b tells if the equation models exponential growth or decay.
• Exponential Decay:• Exponential decay can be modeled by the function 𝑦 = 𝑎 ∙ 𝑏𝑥, where a > 0 and 0 < b < 1.• The base b is the decay factor, which equals 1 minus the percent rate of change expressed as a decimal.• Algebra: 𝑦 = 𝑎 ∙ 𝑏𝑥
• a = initial amount (when x = 0)• b = the base is the decay factor• x = exponent
Modeling Exponential Decay
• The kilopascal is a unit of measure for atmospheric pressure. The atmospheric pressure at sea level is about 101 kilopascals. For every 1000-m increase in altitude, the pressure decreases about 11.5%. What is the approximate pressure at an altitude of 3000 m?
• What is the atmospheric pressure at an altitude of 5000 m?
Practice
• Identify the initial amount a and the decay factor b in each exponential function.
1. 𝑦 = 5 ∙ 0.5𝑥
2. 𝑓 𝑥 = 10 ∙ 0.1𝑥
3. 𝑔 𝑥 = 1002
3
𝑥
4. 𝑦 = 0.1 ∙ 0.9𝑥
• The population of a city is 45,000 and decreases 2% each year. If the trend continues, what will the population be after 15 years?
Geometric Sequences
Objective: To write and use recursive formulas for geometric sequences.
Objectives
• I can identify geometric sequences.
• I can find recursive and explicit formulas.
• I can use sequences.
• I can write geometric sequences as functions.
Vocabulary
• In a geometric sequence, the ratio of any term to its preceding term is a constant value.
• Geometric Sequence:
• A geometric sequence what a starting value a and a common ratio r is a sequence of the form 𝑎, 𝑎𝑟, 𝑎𝑟2, 𝑎𝑟3 …
• A recursive definition for the sequence has two parts:
• 𝑎1 = 𝑎 Initial Condition
• 𝑎𝑛 = 𝑎𝑛−1 ∙ 𝑟, for 𝑟 ≥ 2 Recursive formula
• An explicit definition for this sequence is a single formula: 𝑎𝑛 = 𝑎1 ∙ 𝑟𝑛−1, for 𝑛 ≥ 1
• Every geometric sequence has a starting value and a common ratio. The starting value and common ratio define a unique geometric sequence.
Identifying Geometric Sequences
Determine if the following are geometric sequences. If not, is it arithmetic?
1. 20, 200, 2000, 20,000, 200,000, …
2. 2, 4, 6, 8, 10, …
3. 5, –5, 5, –5, 5, …
4. 3, 6, 12, 24, 48, …
5.1
3,
1
9,
1
27,
1
81, …
6. 3, 6, 9, 12, 15, …
7. 4, 7, 11, 16, 22, …
Practice
Determine whether the sequence is a geometric sequence.1. 2, 8, 32, 128, …
2. 5, 10, 15, 20, …
3. 162, 54, 18, 6, …
4. 256, 192, 144, 108, …
5. 6, –12, 24, –48, …
6. 10, 20, 40, 80, …
Practice
Find the common ratio for each geometric sequence.1. 3, 6, 12, 24, …
2. 81, 27, 9, 3, …
3. 128, 96, 72, 54, …
4. 5, 20, 80, 320, …
5. 7, –7, 7, –7, …
6. 686, 98, 14, 2, …
Vocabulary
• Any geometric sequence can be written with both an explicit and a recursive formula.
• The recursive formula is useful for finding the next term in the sequence.
• The explicit formula is more convenient when finding the nth term.
Finding Recursive and Explicit Formulas
Find the recursive and explicit formulas.1. 7, 21, 63, 189, …
2. 2, 4, 8, 16, …
3. 40, 20, 10, 5, …
4. 5, 20, 80, 320, …
5. 4, –8, 16, –32, …
6. 162, 108, 72, 48, …
7. 3, 6, 12, 24, …
Practice
Write the explicit formula for each geometric sequence.1. 2, 6, 18, 54, …
2. 3, 6, 12, 24, …
3. 200, 40, 8, 13
5, …
4. 3, –12, 48, –192, …
5. 8, –8, 8, –8, …
6. 686, 98, 14, 2, …
Practice
Write the recursive formula for each geometric sequence.1. 4, 8, 16, 32, …
2. 1, 5, 25, 125, …
3. 100, 50, 25, 12.5, …
4. 2, –8, 32, –64, …
5. −1
36,
1
12, −
1
4,
3
4, …
6. 192, 128, 851
3, 56
8
9, …
Using Sequences
• Two managers at a clothing store created sequences to show the original price and the marked-down prices of an item. Write a recursive formula and an explicit formula for each sequence. What will the price of the item be after the 6th markdown?
1. $60, $51, $43.35, $36.85, …
2. $60, $52, $44, $36, …
• Write a recursive formula and an explicit formula for each sequence. Find the 8th term of each sequence.
1. 14, 84, 504, 3024, …
2. 648, 324, 162, 81, …
Practice
When a radioactive substance decays, measurements of the amount remaining over constant intervals of time form a geometric sequence. The table shows the amount of Fl-18, remaining after different constant intervals. Write the explicit and recursive formulas for the geometric sequence formed by the amount of Fl-18 remaining.
Time (minutes) Fl-18 (picograms)
0 260
110 130
220 65
330 32.5
Practice
• A store manager plans to offer discounts on some sweaters according to this sequence: $48, $36, $27, $20.25, … Write the explicit and recursive formulas for the sequence.
Vocabulary
• You can also represent a sequence by using function notation. This allows you to plot the sequence using the points 𝑛, 𝑎𝑛 , where n is the term number and 𝑎𝑛 is the term.
Writing Geometric Sequences as Functions
• A geometric sequence has an initial value of 6 and a common ration of 2. Write a function to represent the sequence. Graph the function.
• A geometric sequence has an initial value of 2 and a common ratio of 3. Write a function to represent the sequence. Graph the function.
Practice
• A geometric sequence has an initial value of 18 and
a common ratio of 1
2. Write a function to represent
this sequence. Graph the function.
• Write and graph the function that represents the sequence in the table.
X F(x)
1 8
2 16
3 32
4 64
