Exponential Functions and Sequences

Exponential Functions and Sequences
Chapter 6

Properties of Exponents
I can use the properties of exponents.

Core Concepts (pages 170 and 171 in Student Journal) Zero Exponent Property For any nonzero number a, a0 = 1. Negative Exponent Property For any integer n and any nonzero number a, a-n = 1/an Product of Powers Property For any real number a and integers m and n, aman = am+n Quotient of Powers Property For any nonzero real number a and integers m and n, am/an = am-n

Power of a Power Property For any real number a and integers m and n, (am)n = amn Power of a Product Property For any real numbers a and b and integer m, (ab)m = ambm Power of a Quotient Property For any real number a, nonzero number b, and integer m, (a/b)m = am/bm

Examples (space on pages 170 and 171 in Student Journal) Evaluate the expression. 2.50 (-3)-2

Solutions 1 1/32 = 1/9

Simplify the expression using only positive exponents. c) 20x2/y-2 d) (23)(25) e) (-7)2/(-7)4 f) (d-3)5

Solutions c) x2y2 d) 28 = 256 e) 1/(-7)2 = 1/49 f) d-15 = 1/d15

Simplify each expression using only positive exponents. g) (-3x)3 h) (b/-3)4 i) (2x/5)3 j) (3c/4)-2

Solutions g) (-3)3x3 = -27x3 h) b4/(-3)4 = b4/81 i) 23x3/53 = 8x3/125 j) 3-2c-2/4-2 = 42/32c2 = 16/9c2

k) A laptop uses 8.4 X 10-3 kilowatt-hours of energy in 3.5 X 10-1 hours. How many kilowatt-hours of energy does the computer use per hour?

Solution k) 8.4 X 10-3/3.5 X 10-1 = 2.4 X 10-2 = 2.4 X 1/102 = 2.4 X 1/100 = .024

Radicals and Rational Exponents
I can find nth roots and evaluate expressions with rational exponents.

Vocabulary (page 175 in Student Journal) nth root: the value of b that makes the equation bn = a true radical: an operation used to find the root of an expression index: the degree of the root

Core Concepts (pages 175 and 176 in Student Journal) Real nth Roots of a For any real number a and integer n > 1: If n is odd, then a has 1 real root If n is even and a > 0, then a has 2 real nth roots If n is even and a = 0, then a has 1 real nth root If n is even and a < 0, then a has 0 real nth roots Rational Exponents For any real number a and positive integers m and n, am/n = (n√a)m

Examples (space on pages 175 and 176 in Student Journal) Find the real roots. 3√-64 √81

Solutions -4 ±9

Evaluate the expression. c) 4√81 d) 4√-16 e) 251/2 f) (-8)1/3 g) 163/2

Solutions c) ±3 d) no real roots e) ±5 f) -2 g) 64

h) The radius of a sphere is given by the equation r = (3V/4π)1/3. If a basketball has a volume of 463 cubic inches, what is it’s radius?

Solution h) r = (3(463)/4(3.14))1/3 = 5 inches

Exponential Functions
I can identify, evaluate, and graph exponential functions.

Vocabulary (page 180 in Student Journal) exponential function: a function of the form y = abx, where a is not equal to 0, b is greater than 0 and not equal to 1 and x is a real number

Examples (space on page 180 in Student Journal) Determine if the table represents a linear or exponential function. a) b)

Solutions linear exponential

Evaluate the function for the given value of x. c) y = -3(4)x ; x = 2 d) y = 2(0.25)x ; x = -3

Solutions c) y = -48 d) y = 128

Graph. Describe the domain and range. e) f(x) = 5(2)x f) f(x) = -(0.25)x g) f(x) = 3(2)x-1 - 3

Solutions e) f) domain: all real domain: all real range: y > 0 range: y < 0

Solutions (continued) g) domain: all real range: y > -3

The graph represents a bacterial population y after x days. h) Write an exponential function for the graph. i) Find the population after 12 hours. j) Find the population after 6 days.

Solutions h) y = 3(3)x i) y = 3(3)1/2 = 5 bacteria j) y = 3(3)6 = 2187 bacteria

Exponential Growth and Decay
I can use and write exponential growth and decay functions.

Vocabulary (page 185 in Student Journal) exponential growth: modeled by an exponential function where the base is a number greater than 1, which occurs when the quantity increases by the same factor over time exponential decay: modeled by an exponential function where the base is a number between 0 and 1, which occurs when the quantity decreases by the same factor over time compound interest: modeled by an exponential function when the interest earned is based on the principal and previously earned interest

Core Concepts (pages 185 and 186 in Student Journal) Exponential Growth Function y = a(1 + r)t, where y is the final amount, a is the initial amount, r is the growth rate as a decimal, and t is the time Exponential Decay Function y = a(1 - r)t, where y is the final amount, a is the initial amount, r is the decay rate as a decimal, and t is the time

Compound Interest y = P(1 + r/n)nt, where y is the final amount, P is the principal (initial amount), r is the annual interest rate as a decimal, t is the time in years, and n is the number of times interest is compounded per year

Examples (space on pages 185 and 186 in Student Journal) The inaugural attendance at an annual art festival was 80,000. The attendance increases by 6% each year. Write an exponential growth function that models this situation. How many people are expected to attend the festival in its 5th year?

Solutions y = 80000( )t y = 80000(1.06)5 = 101,000 people

Determine whether the table represents an exponential growth function, exponential decay function, or neither. c) d)

Solutions c) decay d) growth

Determine whether each function represents an exponential growth or decay function. Identify the percent rate of change. e) y = 4(1.09)t f) y = 24(0.93)t

Solutions e) growth, 9% increase f) decay, 7% decrease

Rewrite each function to determine whether it represents exponential growth or decay. g) y = 20(0.81)t/2 h) y = (2.5)t-2

Solutions g) y = 20(0.811/2)t, y = 20(0.9)t, exponential decay h) y = (2.5)t/(2.5)2, y = 0.16(2.5)t, exponential growth

The table shows the balance of money in an account over time. i) Write a function that represents the balance.

Solution i) y = 300(1.04)t

The value of a car is $15,000. It loses 10% of its value every year. j) Write a function that models the situation. k) Graph the function and use the graph to estimate the value of the car after 6 years.

Solutions j) y = 15000(.9)t k) $8,000 after 6 years

Solving Exponential Equations
I can solve exponential equations.

Core Concepts (page 190 in Student Journal) Properties of Equality for Exponential Equations Two powers with the same positive base b, where b is not equal to 1, are equal if and only if their exponents are equal.

Examples (space page 190 in Student Journal) Solve the equation. 4x+3 = 49 2 = 23x-2 83x-4 = 82x+1

Solutions 6 1 5

Solve. d) 6x = 1296 e) 9x = 3x+2 f) 42x = 8x-2 g) (1/6)x = 216 h) 5x+3 = 1/25

Solutions d) 4 e) 2 f) -6 g) -3 h) -5

Solve by graphing. i) (1/4)x+1 = 5 j) 2x+3 = x - 3

Solutions i) j) no solution

Geometric Sequences I can identify, write and graph geometric sequences.

Geometric Sequences Vocabulary (page 195 in Student Journal)
geometric sequence: the ratio of any term to its preceding term is a constant value common ratio: the ratio is a geometric sequence

Geometric Sequences Core Concepts (page 195 in Student Journal)
Equation for a Geometric Sequence Let an be the nth term of a geometric sequence with the first term a1 and the common ratio r. The nth term is given by an = a1rn-1

Geometric Sequences Examples (space page 195 in Student Journal)
Determine whether the sequence is geometric, arithmetic, or neither. 8, 24, 72, 216, … 8, 3, -2, -7, …

Geometric Sequences Solutions geometric arithmetic

Geometric Sequences Write the next 3 terms of the geometric sequence.
d) 7, -14, 28, -56, …

Geometric Sequences Solutions c) 12, 4, 4/3 d) 112, -224, 448

Geometric Sequences Consider the geometric sequence 3, 12, 48, 192, …
e) Write an equation for the nth term of the geometric sequence. f) Find a10.

Geometric Sequences Solutions e) an = 3(4)n-1 f) 786,432

Geometric Sequences g) Clicking the zoom-out button on a website doubles the side length of the map. Use the table to determine how many clicks on the zoom-out button the side length of the map is 5120 miles.

Geometric Sequences Solution g) 11 clicks

Recursively Defined Sequences
I can write terms for recursively defined sequences and write recursive rules for sequences.

Vocabulary (page 200 in Student Journal) explicit rule: gives an as a function of the term’s position number in the sequence recursive rule: gives the beginning term(s) of a sequence and an equation that tells how an is related to 1 or more preceding term

Core Concepts (page 200 in Student Journal) Recursive Equation for an Arithmetic Sequence an = an-1 + d, where d is the common difference Recursive Equation for a Geometric Sequence an = ran-1, where r is the common ratio

Examples (space on page 200 in Student Journal) Write the first 6 terms of each sequence. a) a1 = 3, an = an-1 + 5 b) a1 = 0.5, an = 2an-1

Solutions 3, 8, 13, 18, 23, 28 0.5, 1, 2, 4, 8, 16

Write a recursive rule for the sequence. c) -8, -1, 6, 13, 20, … d) 405, 135, 45, 15, 5, …

Solutions c) a1 = -8, an = an-1 + 7 d) a1 = 405, an = 1/3an-1

Write an explicit rule for each recursive rule. e) a1 = 9, an = an-1 – 4 f) a1 = 4.5, an = 3an-1

Solutions e) an = -4n +13 f) an = 4.5(3)n-1

Write a recursive rule for each explicit rule. g) an = 6n -5 h) an = 4(3)n-1

Solutions g) a1 = 1, an = an-1 + 6 h) a1 = 4, an = 3an-1

Consider the sequence -4, -5, -9, -14, -23, -37, … i) Write a recursive rule for the sequence. j) Find the next 3 terms in the sequence.

Solutions i) a1 = -4, a2 = -5, an = an-2 + an-1 j) -60, -97, -157

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