What is the difference between linear exponential and quadratic functions?
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In this way, what is the difference between exponential and linear functions?
Linear functions are graphed as straight lines while exponential functions are curved. Linear functions are typically in the form y = mx + b, which is used to discover the slope, or simply the change in y divided by the change in x, while exponential functions are typically in the form y = (1 + r) x.
are all exponential models linear? This tells us this is a linear model, so its graph should be a line. An exponential model is a mathematical model in which the variable is in the exponent. The graph of an exponential model increases slowly at first, then more quickly, or decreases quickly at first and then more slowly.
Just so, how do you tell if an equation is linear quadratic or exponential?
To recognize if a function is linear, quadratic (a parabola), or exponential without an equation or graph, look at the differences of the y-values between successive integral x-values. If the difference is constant, the graph is linear.
Why is an exponential graph not linear?
In exponential functions, a little means a lot. In this function, the independent variable is an exponent in the equation. In Function 1, the variables show a constant rate of change, so this is not an exponential function. In Function 2, the graph is a straight line, so it is a linear function.
Linear functions are functions that 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 seems to take on this pattern? If you're thinking that Sean's productivity seems to take on this pattern, then you're 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.
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's doing an excellent job! Moving on!
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.
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.
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.
Even though Tom had an accident mid-day, you can see that his productivity rate was good until then. You write down that he is doing a good job, understanding that accidents happen.
Your observations show that all three of the employees are doing a really great job, making your recommendation for who to promote a hard one to make. Thankfully, you have your functions to help you out. The employee that actually hits a 100% productivity rate is Sean, while the other two don't quite get there during the day. Based on this, you decide to recommend Sean for the promotion since his daily work pattern seems to result in the highest productivity rate. Pretty neat that you were able to figure all of this out using mathematics, huh?
Linear, exponential, and quadratic functions can be used to model real-world phenomena. Because each type of function has different characteristics, we figure out which one to use based on the pattern that the phenomenon takes on.
For instance, graphs of linear functions increase or decrease at a constant rate, graphs of exponential functions increase or decrease slowly at first and then more quickly or quickly at first and then more slowly, and graphs of quadratic functions increase, hit a maximum, then decrease or decrease, hit a minimum, then increase. Algebraically, linear functions are polynomial functions with a highest exponent of one, exponential functions have a variable in the exponent, and quadratic functions are polynomial functions with a highest exponent of two.
By comparing and analyzing these types of functions, as we did for Company XYZ's employees, we can better understand which one to use to model a given scenario. Isn't math great?
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