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LESSON Writing Exponential Equations in 9 Two Variables

UNDERSTAND An exponential equation is an equation in which the independent variable is an exponent . In the exponential equation y 5 a ? bx, y is the dependent variable, x is the independent variable, and a and b are constants . The base, b, can be any positive real other than 1 . The equation y 5 32x is an exponential equation because the variable, x, is in the exponent . The equation y 5 7x3 is not an exponential equation because the variable, x, is the base and the exponent is the constant number 3 . In y 5 a ? bx, the constant a represents the starting value of a quantity being measured, such as the number of living things in an area . The base b shows how that quantity changes as the variable x changes .

UNDERSTAND The graph of an exponential equation is not a straight line . It is a curve that is either always increasing or always decreasing . For the equation y 5 a ? bx, when the base b . 1 and a . 0, y

the equation models exponential growth . The graph 8 x on the right shows the equation y 5 2 . At first, the curve 6

rises slowly above the x-axis, but it goes up sharply as the 4 y ϭ 2x x-values increase . The equation models exponential growth 2 because the base is greater than 1 . The value of the base, 0 x 2, means that every time x increases by 1, the value of y –8 –6 –4 –2 2 4 6 8 –2 doubles, or is multiplied by 2 . –4

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rning, Lea When a . 0 and 0 , b , 1, the equation models y umph 8 Tri

exponential decay . The graph on the right shows the

x 4 __1 6 201 equation y 5 (​​ ​ 3 ​ )​​​ . At first, the curve goes down sharply ©

4 . and then gets closer and closer to zero as the x-values __1 x law y ϭ ( 3 ) 2 by increase . The equation models exponential decay because –8 –6 –4 –2 0 2 4 6 8 x 1

__ –2 hibited the base, ​ 3 ​, is less than 1 . The base tells us that every time pro 1 –4 __ is

x increases by 1, the value of y is multiplied by ​ 3 . e –6 pag

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56 Unit 1: Relationships between Quantities and Reasoning with Equations Connect A colony of starts out with 150 cells and triples in population every hour . Write an equation that models the number of cells, y, after x hours . Then, graph the equation .

1 Determine the values of a and b .

The initial population of bacteria is 150, so a 5 150 . Every hour, the population is tripling, which is the same as being multiplied by 3 . This means that the base, b, is 3 . 2 Write the equation by substituting for a and b in y 5 a ? bx .

▸ The equation that gives the number of bacteria cells, y, after x hours is y 5 150 ? 3x . 3 Use the equation to make a table of values .

x y 5 150  3x y 0 y 5 150  30 5 150  1 5 150 150

1 y 5 150  31 5 150  3 5 450 450 4 2 2 y 5 150  3 5 150  9 5 1,350 1,350 Graph the equation .

Plot the points from the table and connect them with a smooth curve . LLC ▸

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Hours hibited

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s Is this model an example of exponential thi growth or exponential decay? How can you tell? Duplicating

Lesson 9: Writing Exponential Equations in Two Variables 57 EXAMPLE A The half-life of a substance is the time it takes for half of that substance to break down or decay . The half-life of fermium-253 is 3 days . Write an equation for the amount remaining from a sample of 700 grams of fermium-253 after x days . Make a graph that models the decay of fermium-253 .

1 Determine the values of a and b in the equation y 5 a ? bx .

The initial amount of the sample is 700 grams . So, a 5 700 . After 3 days, there will be half as much fermium, or 350 grams . 350 5 700 ? b3 __1 3 ​ 2 ​ 5 b __ 3 __1 ​ ​ ​ ​ 5 b √2 0 79.  b 2 Write the equation .

▸ y 5 700 ? (0 79). x

3 Plot points and connect them to graph the equation .

Use the equation to find the value of y at several times . LLC ▸ 700 600 500 rning, Lea 400 300 200 umph Tri

100 C 4 CHE K 201

0 12 3 4 5 6 7 98 10 © Fermium–253 (in grams) Fermium–253

Days Every three days, the amount of fermium . law

by

is cut in half . Use this information to complete the table below and then compare it to your graph . hibited pro

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Days 0 3 6 9 pag

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58 Unit 1: Relationships between Quantities and Reasoning with Equations __r nt EXAMPLE B Compound interest is calculated by using the exponential A 5 P(​​ 1 1 ​ n ​ )​​ ​, where A is the accumulated amount after t years, P is the principal (the amount invested), r is the annual interest rate expressed as a decimal, and n is the number of times that the interest is compounded per year . Write an equation to find the amount that is accumulated when $500 is invested in an account with a 3% annual interest rate, compounded monthly . Make a graph that shows how the account grows over time .

1 Identify the values of P, r, and n .

The principal, P, is the original amount that is invested, $500 . So, P 5 500 . The annual interest rate is 3% . Expressed as a decimal, r 5 0 03. . The interest for this account is compounded monthly, and there are 12 months in a year . So, n 5 12 . 2 Substitute the values of P, r, and n into the compound interest equation and simplify .

__r nt A 5 P ​​( 1 1 ​ n ​ )​​ ​ ____0 03. 12t A 5 500(​​ 1 1 ​ 12 ​ )​​ ​ 12t 3 ▸ A 5 500(1 0025). Plot points and connect them to graph the equation .

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(in dollars)

4 500 201 Account Balance

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Lesson 9: Writing Exponential Equations in Two Variables 59 Practice

Determine whether each equation is an exponential equation. x 1 x 10 1. 5 __ 2. 5 p 3. 5 2  y 6 ? (​ 8)​ y y 12 x

Graph the following exponential equations. x __1 x __1 4. y 5 ​ 6 ​ ? 3 5. y 5 6 ? (​​ ​ 4 ​ )​​​ y y

10 10

8 8

6 6

4 4

2 2

–2–4–6–8–10 0 2 4 6 8 10 x –2–4–6–8–10 0 2 4 6 8 10 x

–2 –2

–4 –4

–6 –6

–8 –8

–10 –10

Find the base, b, for the exponential equation of the form y 5 a ? bx that describes each situation.

6. A colony of fruit flies doubles in population every day . The variable y gives the number of fruit flies after x days . b 5 7. Enrollment at a preschool has dropped 8. A savings account earns 3% annual by 4 .5% each year . The variable y gives interest, compounded annually . The the number of students at the school after variable y gives the amount of money in LLC x years . b 5 the account after x years . b 5 rning, Lea The base is equal to 1 2 r, “Annually” means once HINT HINT per year.

where r is the percent decrease umph Tri

expressed as a decimal. 4 201

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Write an equation for each situation. by

9. Sanjay bought a car for $18,500 . 10. A colony of bacteria doubles in population hibited pro According to his insurance company, the every 24 hours . If there were 20 bacteria is

value of the car depreciates 5% each year . initially, how many bacteria will there be e pag

What will the value of the car be x years after x days? s thi after Sanjay purchased it? Duplicating

60 Unit 1: Relationships between Quantities and Reasoning with Equations Graph the relationship in each situation.

11. Membership in the Parents’ Association at 12. A scientist has a sample of 1,500 grams an elementary school has increased each of flerovium-289, a radioisotope with a year for the past 5 years by an average of half-life of 30 seconds . Graph the amount 6% . This year, there were 211 members . of flerovium-289 in the sample over the Make a graph to represent the number of next 3 minutes . people in the Parents’ Association over the next 10 years, assuming this trend 1500 1350 continues . 1200 1050 450 900 400 750 350 600 300 450 250 300

Members 200 150 Flerovium–289 (in grams) Flerovium–289 0 12 3 4 5 6 7 98 10 0 1 2 3 Year Time (in minutes)

Solve.

13. a m PREDICT When a coffee shop opened at 6 . ., there 50 were 4 customers . At 7 a.m . there were 6 customers, and at 8 a.m. there were 9 customers . The number of customers 40

continues to increase exponentially . Graph the number of 30 customers in the coffee shop from 6 a.m. to noon . When will there be more than 30 customers in the coffee shop? 20 Customers 10

0 6 A.M. 7 A.M.8 A.M. 9 A.M. 10 A.M.11 A.M. 12 P.M. Time LLC

rning, 14. COMPARE Javier plans to invest $2,000 in a certificate of 3,200 Lea deposit, or CD, for a period of 10 years . His bank offers two 3,000 2,800 umph types of CDs . The Super Saver has an annual interest rate of Tri

2,600 4 4% compounded quarterly . The Thrifty Thriver has an annual 2,400 201 interest rate of 4 .5% compounded annually . Graph the 2,200 ©

. amount of money that would be in each account for the 2,000 law by

next 10 years . Which option will give Javier more for his (in dollars) CD Value 0 2 4 6 8 10 investment? Time (in years) hibited pro is e pag s thi

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Lesson 9: Writing Exponential Equations in Two Variables 61