# High School: Functions

### Linear, Quadratic, and Exponential Models HSF-LE.B.5

5. Interpret the parameters in a linear or exponential function in terms of a context.

Your students have been solving linear and exponential functions for what feels like centuries, and it's about to come to an end. As one final test of everything they've learned, you ask them a simple question. "Okay, class. We have y dollars in revenue for every x packets of gum sold and y = 0.95x. If x equals 20, how much will y equal?"

Jimmy, a brilliant young student of yours, beams up at you and answers, "It'll equal 19 packets of gum." Your heart sinks to the floor, and you realize your mistake all along.

It's not enough to understand how to solve linear and exponential functions. Sure, it's useful, but it just won't cut it. Students should know what equations actually mean when applied to certain contexts, not just how to solve for x.

Understanding an equation's context is important not only so that students know in what units to report their answers (in dollars and not packets of gum), but also in order to take in data and make use of it. This means that given a particular context, students should be able to understand trends, make predictions, and extrapolate from the mathematical functions they're given.

Some students make these connections quickly and effortlessly, and others might find it a little more difficult to do so. One possible way to assist those struggling students is to assign a clear meaning to each variable so that they know that y always means dollars and x always means packets of gum.

In more complex problems such as exponentials and polynomials, it may be useful to break down the problem so that it's clearly understood what is changing by how much for every what. Translating the equation into words or vice versa may help understand the equation in terms of the overall context. (For instance, every additional packet of gum sold, denoted by x, increases the revenue y by 0.95 dollars. That's what the equation y = 0.95x ultimately means.)

#### Drills

1. A pizza bakery notices that the temperature in the oven it uses increases at a rate of f(x) = 1x. What does this really mean?

The temperature doesn't change

Even though f(x) = 1x is an exponential function, this was a bit of trick question. We can multiply 1 by 1 by 1 by 1 as many times as we want, and it'll still be 1; that means the temperature of the oven doesn't change.

2. A sales team models their revenue as a function of time x, in months. If their revenue is given by f(x) = 2x + 30, how would this best be described?

We know just by looking that the function is linear. That eliminates (B) immediately. Since the slope of the line is positive, it can't be declining either. Stagnation just means that it stays the same, which isn't true because it changes as x changes. That means we're left with (A) as the right answer, since the slope of the line is positive and the linear function increases at a constant rate.

3. A bank suspects that something is amiss. Its internal reserves have been mysteriously changing in value, modeled by the equation y = 2x where y is the amount of money in millions left in the bank after a period of x months. This implies most likely which of the following?

Someone is stealing hundred dollar bills

We can see that as the months go by (as x increases), the amount of money in millions declines (y decreases). (A), (C), and (D) don't explain what's happening according to the function. Only (B) fits the bill. Pun intended.

4. Comcast network has noticed that its ad campaign has been paying off. It's been gaining y customers over x days since the ads started as modeled by y = 500x + 5000. How many customers did Comcast have before the ads started playing?

Since x represents the days after the ads started playing, the number of customers that Comcast had before then can be given by the function y = 500x + 5000 when x = 0. If that's the case, y = 5000 customers.

5. The Lettuceheads, a vegetarian rock and roll group, has hired a contractor to build their very own stage. The contractor has estimated that the stress on the stage will increase according to the function f(x) = 3x, where x is the number of people on stage. Assuming each person weighs exactly the same, what does this function indicate?

This is an exponential function, so there's nothing "equal" or "steady" about it. With every additional x, the y value increases more and more. Even though they all weigh the same, every additional person on stage causes the stress levels to increase far more than the previous person. Probably not the best stage to build, Lettuceheads.

6. You're caught in the middle of Life or Death III, a brutal first person shooter, when suddenly you realize your in-game health is currently being modeled as y = 25 – 3x, where x is in minutes. What does this mean?

The game wouldn't need to tell you if (B) was the case, and (A) would be a perk. The function we're given, y = -3x, is linear with a negative slope. That means your health is steadily decreasing by the same amount with every passing minute. Pick up a health pack, quickly!

7. What can you assume when a linear function is involved?

Something will occur in a steady fashion

A linear function indicates the same amount of change in y with every additional x. This means steady change, not exponential change. Answers (B) and (D) are incorrect because they fail to consider that the linear function may also be steadily decreasing.

8. Someone tells you that exponential functions can only increase exponentially. What do you tell them?

"Actually, exponential functions can also decrease or stay the same."

An exponential function can increase, decrease, or stay the same. Examples of these are y = 1x, which stays the same, y = 2x, which increases, and y = -3x, which decreases. Convinced?

9. A function spits out the same dependent variable for every independent variable. In other words, for every x we put into f(x), we always get a constant. What kind of function could f(x) be?

Either

The function f(x) could be either linear or exponential. It's possible to have f(x) = c where c is a constant, but it's also possible to have f(x) = 1x so that even though f(x) includes x as a variable, the value of x doesn't change the outcome of f(x).

10. A function spits out dependent variables that are constantly increasing for every independent variable. In other words, for every x we put into f(x), we get values that are further and further apart. What kind of function could f(x) be?