Flirting with Slopes
by Noureddine El Alam, Sage Hill School
20402 Newport Coast Drive
Newport Coast, CA 92657
(949) 219-0100 (1400)
If you thought that the concept of "Multiplier" only applies to linear functions, think twice! The power of pattern recognition in mathematics will be tested in this interactive presentation.
Note: Feel free to use any part of this manuscript for educational purposes. I would love to get your feedback, suggestions, or questions. Thank you in advance.
The letter “m” that is often associated with slope is more than just a dummy constant that measures the steepness of a line. It is called the multiplier, of course. The concept of slope makes perfect sense when one works with straight curves. Very quickly we find ourselves using Differential Calculus to talk about slopes of the tangent lines since slope makes no sense for non-straight curves. In this paper, we shall take a hard look at the multiplier from a unique perspective, and address practically any curve of elementary functions. Let us begin simply!
Linear Functions
The equation of the line that passes through two given pointsandis:
We can write the equation as follows:
The numerator in the second factor forces the quotient to equal the multiplicative and the additive unitswhen and respectively. On the other hand, the first factor adjusts the entire equation so that the function maps and maps simultaneously. This multiplier not only measures the slope of the line in question, but more important, it is a unit converter.
Variation of Linear Functions
Let us investigate the absolute value function at two points. Call the point areferencepoint.This point is, in essence, the vertex of the function’s graph.
Using the earlier technique, we get:
, where the multiplier is .
Example. Let us look at an example using a standard approach. Given two points and, then clearly the point is symmetric towith respect to . So we have two rays emanating from the point , and hence the piecewise defined function:
As expected, the latter equation coincides with .
Quadratic Functions
Consider the quadratic function written in the vertex form: where is the vertex. Think of as your reference point . Solving for the constant , which is the multiplier in this case, gives us . This looks like but not quite. The denominator is squared. Now, we can look at the quadratic function as follows:
.
Applications. Throw a marker straight up in the air from a height of 3 feet. The marker reaches a maximum height of 11 feet in 2 seconds. Find the function that models this activity. Obviously, the function must be a quadratic function with the point and a reference point .
,
,
where the multiplier is .
Radical Functions
Guess what happens if you want to generate a radical function whose graph passes through the reference point and the point ?
,
where the multiplier equals .
Check it. If , we get and if , we get .
Cubic Functions
The tricky part with cubic functions is to figure out the meaning of the reference point. It turns out that the reference point is the point of inflection. This requires a separate presentation. Hint: have you ever thought of completing the cube?
We expect to get:
,
where the multiplier equals. Check and test with various functions!
Let us consider, for example, the points . The inflection point occurs at
We get:
Exponential Functions
Now let us investigate some more serious functions. Can you guess the general form of an exponential function whose graph passes through and?
This may not come as easily, but there is, of course, a way to figure it out. It is something students do all the time. Sadly, students usually substitute for the given points right away, and hence miss out on the insight and the beauty of the process. Simply write the general form of an exponential function
, where . Since the two points must satisfy the equation, substitute and solve for the constantsand . After a few minutes of algebraic manipulations, and depending on your approach, you should be able to get one of the following equations:
Verify the given points in both equations. They should work!
Both equations are equivalent. Let us stick with the top one, but now suppose that the exponential function has a horizontal asymptote . The graph experiences a vertical shift this time. We get:
.
,
where is the multiplier.
Notice that the original equation can be written as follows:
Applications.The temperature of a cup of coffee was 95 degrees Fahrenheit when first made. The temperature dropped to 82 degrees Fahrenheit 9 minutes later. Assume that the ambient temperature is 70 degrees Fahrenheit. Find the function that models this situation and compute the temperature of the cup of coffee one and a half hours later.
The multiplier is
The temperature of the cup of coffee one and a half hours later is
Here is a good question to ask students: what is the temperature of the cup of coffee after so many hours? Students should be able to intuitively think that the temperature should get infinitely closer to the ambient temperature. The question boils down to finding .
Inverse Functions
We have already worked with inverse functions. Radical and quadratic functions illustrated an example. We could have found the inverse of the generic quadratic function and compared it to the radical function we came up with. One thing you must keep in mind if you would like to take this route is that the original function must be one to one, for otherwise, the inverse relation will not be a function. So, we would have to restrict the domain to, where . Let exchange roles and solve for . You should get .
Logarithmic Functions
Similarly, we can work on logarithmic functions. We can either use the previous exponential model and find its inverse or use the technique of the multiplier found earlier.
Now suppose that the logarithmic function has a vertical asymptote . The graph experiences a vertical shift, and we get:
Can you guess the value of the multiplier in the original equation with base? Try it! Note that the multiplier is a constant, and hence, is independent of the variables
The original function is a function whose graph has a vertical asymptote with equation .
Applications. Using the earlier application, find the time for which the cup of coffee reaches a temperature of .
A similar, but a bit more difficult approach, works for periodic functions modeled by trigonometric functions. Try it on your own and have fun with your students!
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