Unit 4 Lesson 2 Practice Problems IM® Algebra 2 TM authored by Illustrative Mathematics®

number one asks us or tells us that in 1990 the value of home is 170,000 since then its value has increased by 5% per year what is the approximate value of the home in 1993 so we've got the initial value of 170,000 then it's increasing by 5% so it has its original 100% in Value Plus another 5% so this is increasing or the growth factor is going to be 105% but as decimal that's 1.05 so the growth factor is 1.05 and then we've got 1990 to 1993 that's three years so 170,000 * 1.05 to the 3 that that's going to give us an approximate value of $6,795 and 25 cents then it wants us to write an equation and function notation to represent the value of the home as function of time in years since 1990 and call that so I'm going to say we're going to do value okay so the value based on the number of years is going to be equal to the initial value of the house 170,000 times the growth factor to the power so if we wanted after five years we'd plug in five 10 years we'd plug in 10 so in part then we want to figure out what the value of the home will the value of the home be more than 500,000 in 2020 assuming it keeps increasing at that 5% so now remember that the initial year is 1990 so what we want to figure out is how far is 2020 from 1990 so 2020 minus 1990 means that we're looking at 30 years later so we're going to take this function here and we're going to plug in 30 so we're going to do 170,000 * 1.05 to the 30th so we're looking at after 30 years so then the value of this house after 30 years is going to be 34,000 $ 73020 so yes it's going to be over that $500,000 number two the graph of the graph shows the wolf population has been growing exponentially what was the population when it was first measured okay so when it was first measured at zero years this population was here and it looks like each of these is counting by 20s so this is going to be at 100 so the initial wolf population was 100 by what Factor did the population grow in the first year so now we're looking at this oneyear growth so at one year the population is at 120 so we want the factor of growth so we're taking the new number of wolves divided by the original so 120 divided by 100 and we get growth factor of 1.2 so then we'll use that to write the equation right so the wolf population oops they want us to use so the wolf population after certain number of years that they want us to do so have and here is going to equal the initial wolf population of 100 times the growth factor to the exponent of the number of years so 100 time 1.2 to the number three here's the gra graph of an exponential function find an equation defining so very similar to what we just did so we need to come up with the initial amount and then we're going to need the growth factor and then we'll write the equation so here we're looking at this is 200 so this is 100 so it's looking like the initial population is at 50 so then we can take look after one year and see that it goes to 100 so this one was at 50 after one year it's at 100 let's make sure that that keeps going since we're not sure if this is 50 so at two it's at 200 after 3 years it's at 400 so that does appear to be correct because we can see that this is doubling each year so 50 * 2 is 100 * 2 is 200 * 2 is 400 so our growth factor is 2 so then we can write our function as function of so ofx equals our initial amount times our growth factor and then in this case to the since our input is number four the equation of equal 24,500 * 88 to the represents the value of car in dollars years after it was purchased so what do these two numbers represent well this represents the purchase price or what it was worth initially when it was bought and then 88 is the growth factor or in this case the dec Factor right because it's less than one so this means that this car is only keeping 88 % of its value each year so it's decreasing by 12% so what does of 9 represent it represents the value of the car after 9 years or nine years after it was purchased didn't ask us to actually find it in part sketch graph that represents the function for these values so let's plug these in so of 0 okay so 88 to the 0 is 1 * 24,500 is just 24,500 because that's how much it was worth in the first year or at year zero so now we'll do 24,000 * 88 because we're doing it to the one so after the first year so times 88 so this one's going to give us 2,560 and then we'll multiply by 88 again for this next one and that's giving us 18,9 72.8 so this just wants us to sketch this so at 0 1 and two and so now we need to get up to about 25,000 so if just count by 10 so this is 10,000 20,000 so then these would each be about 5,000 so I'll just label this one at 20,000 so at zero this was at 24,500 after one year it's at 21,500 and after two years it's at like 18 19,000 so then this would be just kind of sketch of that number five the first two terms of an exponential sequence are 18 and six what are the next three terms so remember that an exponential sequence is also geometric sequence it means it's growing or decaying by the same factor so in this case we're dividing by three so then we'll do 6 / 3 is two and we need the next three terms so then divide by two again so 2/3 and then divide by 3 again so that's going to be 2 9ths because remember dividing by three is the same as multiplying by 1/3 so then this would be 2 * one on top and 3 * three on bottom to get us that number six bacteria population has been doubling each day for the last 5 days so we know doubling is growth factor of times by two it is currently at 1,00 100,000 so what was the bacterial population 5 days ago so now we have 100,000 up here and then we want to look at one one day prior two days prior 3 days prior 4 days prior and then five days prior so what did it start at so now if the growth factor if we would take this and it's going to multiply by two each time to be growing then if we want to go backwards we're going to have to divide by two so 100,000 divided by 2 is going to be 50,000 and then 50,000 ided 2 is 25,000 and then 25,000 / 2 is 12,500 divided by 2 is 6,250 ided 2 again is 3,125 is where it started 5 days ago number seven select all Expressions that are equivalent to 27 to the 1/3 power so couple different ways we can write this before we get started remember 1/3 power is like cube root so cube root of 27 and then remember that 27 is 3 to the 3 power so the cube roet of 27 is just three so those are couple different ways to write that so let's take look it's definitely not 9 because 9 to the 3 power is not 27 3 to the 3 power is 27 so is good it's not the square root of 27 it's the cube Ro of 27 so is bad but is good it is is because that's 27 is 3 cubed so the so this part in here is the same as 27 so this is equivalent to the cube root of 27 127th no because it's equal to three and 3 and 127th are not the same thing and neither is 1 over 27 to the 3 power so and