How are rational functions similar to linear, quadratic, or exponential functions? How are they different? When are these similarities or differences important when looking for intersections between rational functions and other types of functions? Be sure to cite examples as you explain your ideas.
Think about this as you answer: What can occur or exist in Rational Functions that may not appear in other types of functions? What implications does this have when possibly finding intersections of rational functions and other functions?
Answers
It's evaluation day at work! Don't be nervous. You'll be the one doing the evaluating! At Company XYZ, you are asked to observe and record the productivity of three workers (Sean, Billy, and Tom) for one eight-hour day to evaluate their work and to determine which of these employees the company should promote to train other employees.
As you observe them, you see that Sean's productivity rate increases at a constant rate over the eight-hour period. Billy took a little while to get going, so his productivity rate started off fairly slow and then increased more and more quickly. Lastly, Tom had a small accident and hurt his hand after about four hours of work. Because of this, his productivity rate started off increasing then peaked, but because of his accident, it slowed down over the next four hours.
You decide to graph the worker's productivity rates over the eight-hour time period.
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Notice that the graphs all take on a different shape based on the patterns that each worker's day took. Here's where things get interesting! You see, each of these functions that model the worker's productivity are specific mathematical models. These include linear, exponential, and quadratic functions. Let's take a look at each of these functions and see how they compare with one another.
Linear Functions
Linear functions are used to model phenomena that increase or decrease at a constant rate. These types of functions are polynomial functions with a highest exponent of one on the variable. The graphs of these functions are in the shape of a line. Which of the worker's productivity seem to take on this pattern? If you're thinking that Sean's productivity seems to take on this pattern, then you are correct! We see that Sean's graph is the graph of a line, and we know that his rate of productivity increases at a constant rate. It also just so happens that the function corresponding to this graph is:
s(x) = 12.5x
We see that the highest exponent of the variable, x, is one.
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The function representing Sean's productivity rate is a linear function, and we see the patterns that linear functions take on. Also, notice that Sean reaches a productivity rate of 100% by the end of the day. You make a note that he is doing an excellent job! Moving on!
Exponential Functions
Exponential functions are functions that have the variable in the exponent. They increase or decrease slowly then quickly or quickly then slowly. Therefore, they are used to model phenomena that take on this pattern, just like Billy's productivity rate! We can model Billy's productivity rate with the exponential function:
b(x) = 1.75x - 1
Notice that the variable, x, is in the exponent of 1.75 like we just said happens in an exponential function. We also see that the graph of Billy's function starts off increasing slowly, but then increases much more quickly as the day goes on.
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We now know that the function representing Billy's productivity rate is an exponential function. We see that at the end of the day, Billy isn't quite at 100% productivity, but he is pretty close. You record him as doing a great job and move onto Tom.
Quadratic Functions
Okay, last but not least, you take a look at Tom. You notice that Tom's productivity rate increases, peaks, and then decreases due to his accident. This is the pattern that a quadratic function takes on. A quadratic function is a polynomial function with a highest exponent of two. These types of functions are used to model phenomena that increase and hit a maximum then decrease, or decrease and hit a minimum then increase. Their graphs look like a U or an upside down U.
The function modeling Tom's productivity rate is:
t(x) = -4x2 + 32x
which is a polynomial function with highest exponent equal to two, exactly the definition of a quadratic function. We also observe that the graph of the function increases, hits a maximum, then decreases, solidifying the fact that this is a quadratic function.