## Exponential Functions And Sequences Cumulative Assessment

Question 1.Fill in the exponent of x with a number to simplify the expression.

Question 2.The graph of the exponential function f is shown. Find f.

Question 3.Student A claims he can form a linear system from the equations shown that has infinitely many solutions. Student B claims she can form a linear system from the equations shown that has one solution. Student C claims he can form a linear system from the equations shown that has no solution.a. Select two equations to support Student As claim.b. Select two equations to support Student Bs claim.c. Select two equations to support Student Cs claim.

Question 4.Fill in the inequality with < , , > , or so that the system of linear inequalities has no solution.

Question 5.The second term of a sequence is 7. Each term of the sequence is 10 more than the preceding term. Fill in values to write a recursive rule and an explicit rule for the sequence.

Question 7.Select all the functions whose x-value is an integer when f = 10.

Question 8.Place each function into one of the three categories. For exponential functions, state whether the function represents exponential growth, exponential decay, or neither.

Question 9.

## Properties Of Exponents 61 Exercises

**Vocabulary and Core Concept Check**

Question 1.Which definitions or properties would you use to simplify the expression -2? Explain.Answer:

)-2Answer:

Question 45.Which of the expressions represent the volume of the sphere? Explain.Answer:

Question 46.**MODELING WITH MATHEMATICS**Diffusion is the movement of molecules from one location to another. The time t it takes molecules to diffuse a distance of x centimeters is given by t = \, where D is the diffusion coefficient. The diffusion coefficient for a drop of ink in water is about 10-5 square centimeters per second. How long will it take the ink to diffuse 1 micrometer ?Answer:

**In Exercises 4750, simplify the expression. Write your answer using only positive exponents.**

Question 47.

Question 55.**PROBLEM SOLVING**In 2012, on average, about 9.46 × 10-1 pound of potatoes was produced for every 2.3 × 10-5 acre harvested. How many pounds of potatoes on average were produced for each acre harvested? Write your answer in scientific notation and in standard form.Answer:

Question 56.**PROBLEM SOLVING**The speed of light is approximately 3 × 105 kilometers per second. How long does it take sunlight to reach Jupiter? Write your answer in scientific notation and in standard form.Answer:

**In Exercises 5962, rewrite the expression as a power of a product.**

questions 59.

Question 65.**REASONING**Find x and y when \ = b9 and \ = b13. Explain how you found your answer.Answer:

f. Volume = \ mm3

**Communicate Your Answer**

**6.2 Lesson**

Question 1.

2563 / 4

## Exponential Functions And Sequences Chapter Test

**Evaluate the expression.**

256x + 2 = 163x 1

Question 13.Graph f = 2x. Compare the graph to the graph of g = 6x. Describe the domain and range of f.

**Use the equation to complete the statement with the symbol < , > , or =. Do not attempt to solve the equation.**

Question 14.\

Question 15.9a 9-b

Question 16.The first two terms of a sequence are a1 = 3 and a2 = -12. Let a3 be the third term when the sequence is arithmetic and let b3 be the third term when the sequence is geometric. Find a3 b3.

Question 17.At sea level, Earths atmosphere exerts a pressure of 1 atmosphere. Atmospheric pressure P decreases with altitude. It can be modeled by P =a, where a is the altitude .a. Identify the initial amount, decay factor, and decay rate.b. Use a graphing calculator to graph the function. Use the graph to estimate the atmospheric pressure at an altitude of 5000 feet.

Question 18.You follow the training schedule from your coach.a. Write an explicit rule and a recursive rule for the geometric sequence.b. On what day do you run approximately 3 kilometers?

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## Lesson 65 Solving Exponential Functions

**Essential QuestionHow can you solve an exponential equation graphically?EXPLORATION 1Solving an Exponential Equation GraphicallyWork with a partner.** Use a graphing calculator to solve the exponential equation 2.5x 3 = 6.25 graphically. Describe your process and explain how you determined the solution.

**EXPLORATION 2The Number of Solutions of an Exponential EquationWork with a partner.**a. Use a graphing calculator to graph the equation y = 2x.b. In the same viewing window, graph a linear equation that does not intersect the graph of y = 2x.c. In the same viewing window, graph a linear equation that intersects the graph of y = 2x in more than one point.d. Is it possible for an exponential equation to have no solution? more than one solution? Explain your reasoning.

**EXPLORATION 3Work with a partner. **Use a graphing calculator to solve each equation.a. 2x = \b. 2x + 1 = 0

Question 41.**MODELING WITH MATHEMATICS**You scan a photo into a computer at four times its original size. You continue to increase its size repeatedly by 100% using the computer. The new size of the photo y in comparison to its original size after x enlargements on the computer is represented by y = 2x + 2. How many times must the photo be enlarged on the computer so the new photo is 32 times the original size?Answer:

**In Exercises 4346, solve the equation.**

Question 43.

Explain how you can use mental math to solve the equation 8x 4 = 1.Answer:

**In Exercises 5358, solve the equation.**

Question 53.

**6.6 Lesson**

## Recursively Defined Sequences 67 Exercises

Question 1.**COMPLETE THE SENTENCE** A recursive rule gives the beginning term of a sequence and a_____________ that tells how an is related to one or more preceding terms.Answer:

Question 2.**WHICH ONE DOESNT BELONG?** Which rule does not belong with the other three? Explain your reasoning.Answer:

**Monitoring Progress and Modeling with Mathematics**

**Vocabulary and Core Concept Check**

**In Exercises 36, determine whether the recursive rule represents an arithmetic sequence or a geometric sequence.**

Question 3.a1 = 2, an = 7an 1Answer:

a1 = 18, an = an 1 + 1Answer:

a1 = 5, an = an 1 4Answer:

a1 = 3, an = -6an 1Answer:

**In Exercises 712, write the first six terms of the sequence. Then graph the sequence.**

Question 7.a1 = 0, an = an 1 + 2Answer:

a1 = 10, an = an 1 5Answer:

a1 = 2, an = 3an 1Answer:

a1 = 8, an = 1.5an 1Answer:

a1 = 80, an = \an 1Answer:

a1 = -7, an = -4an 1Answer:

**In Exercises 1320, write a recursive rule for the sequence.**

Question 13.

**MODELING WITH MATHEMATICS **Write a recursive rule for the number of bacterial cells over time.Answer:

Question 22.**MODELING WITH MATHEMATICS** Write a recursive rule for the length of the deer antler over time.Answer:

**In Exercises 2328, write an explicit rule for the recursive rule.**

Question 23.a1 = -3, an = an 1 + 3Answer:

a1 = 8, an = an 1 12Answer:

a1 = 16, an = 0.5an 1Answer:

a1 = -2, an = 9an 1Answer:

a1 = 4, an = an 1 + 17Answer:

a1 = 5, an = -5an 1Answer:

Question 29.

an = -81)n 1Answer:

Answer:

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## Exponential Growth And Decay 64 Exercises

**Vocabulary and Core Concept Check**

Question 1.In the exponential growth function y = at, the quantity r is called the ________.Answer:

**In Exercises 1316, write a function that represents the situation.**

Question 13.Sales of $10,000 increase by 65% each year.Answer:

Your starting annual salary of $35,000 increases by 4% each year.Answer:

A population of 210,000 increases by 12.5% each year.Answer:

An item costs $4.50, and its price increases by 3.5% each year.Answer:

Question 17.**MODELING WITH MATHEMATICS**The population of a city has been increasing by 2% annually. The sign shown is from the year 2000.a. Write an exponential growth function that represents the population t years after 2000.b. What will the population be in 2020? Round your answer to the nearest thousand.Answer:

Question 18.**MODELING WITH MATHEMATICS**A young channel catfish weighs about 0.1 pound. During the next 8 weeks, its weight increases by about 23% each week.a. Write an exponential growth function that represents the weight of the catfish after t weeks during the 8-week period.b. About how much will the catfish weigh after 4 weeks? Round your answer to the nearest thousandth.Answer:

**In Exercises 1926, identify the initial amount a and the rate of decay r of the exponential function. Evaluate the function when t = 3. Round your answer to the nearest tenth.**

Question 19.

y = )tAnswer:

y = 0.5 )tAnswer:

**In Exercises 2730, write a function that represents the situation.**

Question 33.

Question 41.

Question 73.

## Big Ideas Math Book Algebra 1 Solution Key Chapter 6 Exponential Functions And Sequences

Make use of this provided Big Ideas Math Book Algebra 1 Ch 6 Exponential Functions and Sequences Answer Key topic-wise and clear all your subject doubts within seconds. **BIM Algebra 1 Textbook Solutions for Chapter 6 Exponential Functions and Sequences** concepts are explained in a step-wise manner so they are easy to understand while preparation. Also, BIM Answer Key for Algebra 1 Ch 6 is free to access online and offline at any time. Kickstart your preparation with the ultimate Ch 6 Exponential Functions and Sequences Big Ideas Math Algebra 1 Answers guide & score highest marks in various exams.

Question 11.**ABSTRACT REASONING**Recall that a perfect square is a number with integers as its square roots. Is the product of two perfect squares always a perfect square? Is the quotient of two perfect squares always a perfect square? Explain your reasoning.

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## Lesson 67 Recursively Defined Sequences

**Essential Question**

How can you define a sequence recursively?A recursive rule gives the beginning term of a sequence and a recursive equation that tells how an is related to one or more preceding terms.

**EXPLORATION 1Describing a PatternWork with a partner.** Consider a hypothetical population of rabbits. Start with one breeding pair. After each month, each breeding pair produces another breeding pair. The total number of rabbits each month follows the exponential pattern 2, 4, 8, 16, 32,. . .. Now suppose that in the first month after each pair is born, the pair is too young to reproduce. Each pair produces another pair after it is 2 months old. Find the total number of pairs in months 6, 7, and 8.

**EXPLORATION 2Work with a partner.** Consider the following recursive equation.Each term in the sequence is the sum of the two preceding terms.Copy and complete the table. Compare the results with the sequence of the number of pairs in Exploration 1.

**Communicate Your Answer**

How can you define a sequence recursively?

Question 4.Use the Internet or some other reference to determine the mathematician who first described the sequences in Explorations 1 and 2.

**6.7 Lesson**

**Write a recursive rule for the sequence.**

Question 5.8, 3, -2, -7, -12, . . .

Question 6.1.3, 2.6, 3.9, 5.2, 6.5, . . .

Question 7.4, 20, 100, 500, 2500, . . .

Question 8.128, -32, 8, -2, 0.5, . . .

Question 9.Write a recursive rule for the height of the sunflower over time.

**Monitoring Progress**

Question 12.

an = -2.5n 1

## Lesson 61 Properties Of Exponents

**Essential Question**How can you write general rules involving properties of exponents?

**EXPLORATION 1Writing Rules for Properties of ExponentsWork with a partner.**a. What happens when you multiply two powers with the same base? Write the product of the two powers as a single power. Then write a general rule for finding the product of two powers with the same base.b. What happens when you divide two powers with the same base? Write the quotient of the two powers as a single power. Then write a general rule for finding the quotient of two powers with the same base.c. What happens when you find a power of a power? Write the expression as a single power. Then write a general rule for finding a power of a power.d. What happens when you find a power of a product? Write the expression as the product of two powers. Then write a general rule for finding a power of a product.e. What happens when you find a power of a quotient? Write the expression as the quotient of two powers. Then write a general rule for finding a power of a quotient.

**Communicate Your Answer**

How can you write general rules involving properties of exponents?

Question 3.There are 33 small cubes in the cube below. Write an expression for the number of small cubes in the large cube at the right.

**6.1 Lesson**

Question 14.)-2

**Monitoring Progress**

Question 15.Write two expressions that represent the area of a base of the cylinder in Example 5.

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## Lesson 64 Exponential Growth And Decay

**Essential Question**What are some of the characteristics of exponential growth and exponential decay functions?

**EXPLORATION 1Predicting a Future EventWork with a partner.** It is estimated, that in 1782, there were about 100,000 nesting pairs of bald eagles in the United States. By the 1960s, this number had dropped to about 500 nesting pairs. In 1967, the bald eagle was declared an endangered species in the United States. With protection, the nesting pair population began to increase. Finally, in 2007, the bald eagle was removed from the list of endangered and threatened species.Describe the pattern shown in the graph. Is it exponential growth? Assume the pattern continues. When will the population return to that of the late 1700s? Explain your reasoning.

**EXPLORATION 2Describing a Decay PatternWork with a partner. **A forensic pathologist was called to estimate the time of death of a person. At midnight, the body temperature was 80.5°F and the room temperature was a constant 60°F. One hour later, the body temperature was 78.5°F.a. By what percent did the difference between the body temperature and the room temperature drop during the hour?b. Assume that the original body temperature was 98.6°F. Use the percent decrease found in part to make a table showing the decreases in body temperature. Use the table to estimate the time of death.

**Communicate Your Answer**

What are some of the characteristics of exponential growth and exponential decay functions?

y = t + 2

## Exponential Functions And Sequences Chapter Review

Simplify the expression. Write your answer using only positive exponents.

Question 1.

You deposit $750 in a savings account that earns 5% annual interest compounded quarterly. Write a function that represents the balance after t years. What is the balance of the account after 4 years?

Question 19.The value of a TV is $1500. Its value decreases by 14% each year. Write a function that represents the value y of the TV after t years. Find the approximate monthly percent decrease in value. Graph the function from part . Use the graph to estimate the value of the TV after 3 years.

**Solve the equation.**

**Write the first six terms of the sequence. Then graph the sequence.**

Question 32.a1 = 4, an = an 1 + 5

Question 33.a1 = -4, an = -3an 1

Question 34.a1 = 32, an = \an 1

**Write a recursive rule for the sequence.**

Question 35.3, 8, 13, 18, 23, . . .

Question 36.3, 6, 12, 24, 48, . . .

Question 37.7, 6, 13, 19, 32, . .

Question 38.The first term of a sequence is 8. Each term of the sequence is 5 times the preceding term. Graph the first four terms of the sequence. Write a recursive rule and an explicit rule for the sequence.

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## Exponential Functions And Sequences Mathematical Practices

**Monitoring Progress**

Question 1.A rabbit population over 8 consecutive years is given by 50, 80, 128, 205, 328, 524, 839, 1342. Find the population in the tenth year.

Question 2.The sums of the numbers in the first eight rows of Pascals Triangle are 1, 2, 4, 8, 16, 32, 64, 128. Find the sum of the numbers in the tenth row.

## Exponential Functions 61 64 Quiz

**Simplify the expression. Write your answer using only positive exponents.**

Question 1.

f = \t

Question 16.f = 80)t

Question 17.The table shows several units of mass. a. How many times larger is a kilogram than a nanogram? Write your answer using only positive exponents.b. How many times smaller is a milligram than a hectogram? Write your answer using only positive exponents.c. Which is greater, 10,000 milligrams or 1000 decigrams? Explain your reasoning.

Question 18.You store blankets in a cedar chest. What is the volume of the cedar chest?

Question 19.The function f = 5t represents the number of frogs in a pond after t years. a. Does the function represent exponential growth or exponential decay? Explain.b. Graph the function. Describe the domain and range.c. What is the yearly percent change? the approximate monthly percent change?d. How many frogs are in the pond after 4 years?

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## Big Ideas Math Algebra 1 Answers Chapter 6 Exponential Functions And Sequences

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## Exponential Functions 63 Exercises

**Vocabulary and Core Concept Check**

Question 1.Sketch an increasing exponential function whose graph has a y-intercept of 2.Answer:

Describe and correct the error in finding the domain and range of the function.Answer:

**In Exercises 43 and 44, graph the function with the given description. Compare the function to f = 0.5x over the interval x = 0 to x = 2.**

Question 43.An exponential function g models a relationship in which the dependent variable is multiplied by 2.5 for every 1 unit the independent variable x increases. The value of the function at 0 is 8.Answer:

Question 44.An exponential function h models a relationship in which the dependent variable is multiplied by \ for every 1 unit the independent variable x increases. The value of the function at 0 is 32.Answer:

Question 45.**MODELING WITH MATHEMATICS**You graph an exponential function on a calculator. You zoom in repeatedly to 25% of the screen size. The function y = 0.25x represents the percent of the original screen display that you see, where x is the number of times you zoom in.a. Graph the function. Describe the domain and range.b. Find and interpret the y-intercept.c. You zoom in twice. What percent of the original screen do you see?Answer:

**In Exercises 4750, write an exponential function represented by the table or graph.**

Question 47.

Question 53.**WRITING**Graph the function f = -2x. Then graph g = -2x 3. How are the y-intercept, domain, and range affected by the translation?Answer:

**Write the percent as a decimal.**

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