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 number 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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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
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56 Unit 1: Relationships between Quantities and Reasoning with Equations Connect A colony of bacteria 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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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
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0 12 3 4 5 6 7 98 10 © Fermium–253 (in grams) Fermium–253
Days Every three days, the amount of fermium . law
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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 function 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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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
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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
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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