# High School: Functions

### Linear, Quadratic, and Exponential Models HSF-LE.A.1

1. Distinguish between situations that can be modeled with linear functions and with exponential functions.

Students should be able to actually see the differences between a linear and exponential function, whether it's through graphs, tables, or other media. Then, they should be able to apply these functions to specific situations, or at minimum, understand when each function may be of use.

#### Drills

1. A senior executive at a local bank wants you to prove to a customer out in the lobby that her bank account, with \$100,000 in it, will be growing every year, thanks to a 5% interest paid out once a year. The woman will not be withdrawing any money from the account. The woman however, wants to know whether or not the difference between each year will be steady or not. Explain why or why not.

No, this is an exponential growth problem

 Year Principal + Interest 5% Compound Interest Interval 0 100,000 5000 1 105,000 5250 250 2 110,250 5512.5 262.5 3 115,763 5788.125 275.625 4 121,551 6077.53125 289.40625 5 127,628 6381.407812 303.8765625 6 134,010 6700.478203 319.0703906 7 140,710 7035.502113 335.0239102 8 147,746 7387.277219 351.7751057 9 155,133 7756.64108 369.3638609

This is a conceptual problem. While the interest rate increase stays the same at 1.05x (where x is the starting amount each year), the problem states that the woman won't withdraw any of her money. This means that each year, the interest will be paid out into her account and increase the total amount: 5% of the principal amount the first year, then 5% of the principal amount plus the interest gained the first year, and so on. The compound interest grows this woman's bank account exponentially, not linearly. We can make a table to convince ourselves (since in linear growth, the interval would always be the same).

2. You're put in charge of a new facility in a company that produces more food to feed more people in hopes to extinguish world hunger. You've designed a new type of corn that grows at a rate of f(x) = 3x, where x is the number of months of the harvest. The old type of corn grew at a linear rate of f(x) = 3x. Which of the following is true?

Compared to the old corn, the new corn grows slower at first and then substantially faster

At first, it may seem as though new corn is a bit slower because after one month, they will produce the same amount of corn. After two months, the new corn will have produced 33% more than the old corn. Eventually, the new corn will pull ahead substantially and leave the old corn in the dust. We can compare the relative slopes of the functions to see that this is true.

3. A sugar processing plant is able to process 10 tons of sugar per month (that's more than even we can eat!). Assuming that this process stays steady, which of the following is the best equation to model the sugar production?

y = 10x

The "steadiness" implied in the question means that a linear equation is the best function to use here. The answers (A) and (C) imply exponential growth, and (D) might look better, but it's quadratic, not linear.

4. Exercise biologists have discovered that, to reduce soreness, people should start biceps curls at 10 pounds. Then, progress weekly to 11 pounds, 14 pounds, 20 pounds, 32 pounds, 56 pounds, and so on. This implies what kind of growth?

Exponential

The weekly interval is not linear. That means this progress implies exponential growth. As a side note, eventually plateaus are reached when working out and growth may not be modeled exponentially later on. You'll still be ripped, though.

5. A biologist works in a lab where bacteria produce an awful smelling gas called ammonia. Even though the amount of bacteria remains the same, the amount of ammonia produced increases with time. The scientist believes this can be modeled with which kind of function?

An exponential function

The biologist has noticed that the ammonia increases in output with every constant interval of time. This means that our difference with every interval isn't the same, meaning it can't be a linear function. As a result, our function has to be exponential.

6. The Choo-Choo Express Train Company has trains that go from 0 to 60 miles/hour in 10 seconds. Once up to speed, they remain at 60 mph until about 1,000 feet from the next train station. The speed of the trains after they accelerate and before they begin to slow down is best modeled by which of the following?

Linear functions

Since we have a steady speed and we're measuring distance covered, the difference for every interval of time will remain constant. The speed of the trains is best modeled by a simple linear function.

7. Your boss at the local meat packing plant is no math genius (he makes a mean barbecue tri-tip, though). He's not sure that a new system of packing all the hamburgers is going to increase daily output. In fact, he tried to implement the new system but didn't see immediate results. The new system was modeled as y = 1.5x and the old one was y = 1.5x, with y as the boxes of hamburgers packed after x days after implementing the system. What would you tell your boss?

It will take almost a week for the new system to overtake the old one

Graphing the two functions will clearly show that the new system will overtake the old one. The only question is when? We can plug in x = 7 for both functions and the old system will give us y = 1.5 × 7 = 10.5 while the new system will give y = 1.57 = 17.1. That means that by day 7, the new system will have overtaken the old one. Your boss will thank you with his barbecue tri-tip.

8. An exponential function will always surpass a linear function's dependent variables (y values) eventually. Is this statement true or false?

False

It is very easily disproven, as certain negative exponential functions will never even have positive y values. For example, y = -1.2x will never have y values that surpass the x-axis. With the exception of horizontal lines y ≤ 0, any line will surpass that exponential function in y values.

9. A linear function is one by which the difference between intervals does which of the following?

Stays the same

The difference between each interval in a linear function is always the same. If the difference between each interval is increasing or decreasing you should suspect an exponential function.

10. An exponential function is one whose interval does which of the following?

Grows by an equal factor