Naming and Graphing Functions
Date: 06/16/97 at 09:50:30 From: Perfect Distributors Subject: Functions Dr. Math, I have been given an assignment on functions and I have no idea where to start. In the first part of the assignment, we have to name seven functions: a) y = a(x-b)^2+c b) y = ax^2+bx+c c) y = ab^cx+d d) y = (a/cx)+b e) y = (a)(sin)(bx+c)+d f) y = (a)(cos)(bx+c)+d g) y = (a)(tan)(bx+c)+d Then we have to describe the effect on the graphs of these functions when selected constants a, b, c or d are altered. Can you help point me in the right direction? - Ben
Date: 06/16/97 at 12:52:00 From: Doctor Mike Subject: Re: Functions G'Day Ben, In this country we have an expression to "separate the wheat from the chaff", which means to ignore the distractions and get down to the important stuff. We all speak English, but I never know for sure about a particular word or phrase. In each case you have y as a function of x, and the constants give many many different variations on the basic theme. To see the main idea, look what happens when the constant multipliers are equal to one and the constants that are added or subtracted are equal to zero. In the case of the function in (a), you get y = 1(x-0)^2+0 = x^2, or y equals x squared. So, the name of the basic family of functions in (a) is the "squaring" function. The one in (b) is related to it, but is a bit more general, namely the "quadratic" functions. The function in (c) is a bit puzzling. The normal convention is to give exponentials a higher priority than multiplication, so this would mean "y = a(b^c)x+d" which would be variations on "y = x". However, I suspect your source meant "y = ab^(cx)+d" which would be variations on "y = b^x". So, the name for this family could be "identity" or maybe "exponential". For (d) the basic version is "y = 1/x" which is the "reciprocal" function. The last three are the trigonometric "sine" "cosine" and "tangent" functions. Okay, now that we have names for the function families we are dealing with, let's see what kind of variety we can find in these families. I'll give just a few details on one of them to start you off. You will need to spend a lot of time with this to really get a feel for what is happening (and go through most of a pad of graph paper to boot!) Theme y = sin(x) for x = 0 to x = 360 degrees Var. 1 y = 2*sin(x) for x = 0 to x = 360 degrees Var. 2 y = sin(2*x) for x = 0 to x = 360 degrees Var. 3 y = sin(x)+1 for x = 0 to x = 360 degrees Var. 4 y = sin(x-90) for x = 0 to x = 360 degrees Var. 5 y = 2*sin(2*x-90)+1 for x = 0 to x = 360 degrees I assume you are familiar with the basic shape of the sine function. It starts with sin(0) = 0 and the graph makes a "hill" and then a "valley" and winds up again with sin(360) = 0. The high point is at sin(90) = 1 and the low point is sin(270) = -1. With me so far? Variation 1 changes the height of the hill (twice as high) and the depth of the valley at 270 degrees (twice as deep) by using the multiplicative constant. Graph it. This is a "hands on" math lab. This is sometimes called an "amplitude" variation. Variation 2 is a little tricky. As x goes from 0 to 360, 2*x goes from 0 to 720, or twice around the circle. The graph, then, has a "hill", a "valley", "another hill" and "another valley" before x gets to 360. This is sometimes called a "frequency" variation. In this case, the hills and valleys are "more frequent" for your trip. Definitely, you want to carefully sketch the graph of this one. Variation 3 just adds one (1) to each and every y-value. This raises the graph vertically by 1 unit. The result here is that it now looks like the bottom of the "valley" rests on the horizontal axis. Variation 4 changes the angle you take the sine of, e.g. if x = 0 you take the sine of -90 degrees. If x = 100 you take the sine of 10 degrees. If you go out to the maximum value of x, namely x = 360, for particular function, then you are to evaluate y = sin(270). You are using the same basic "sine" function, but the "-90" part shifts the range of angles you are using. Sometimes this kind of thing for a trig function is called a "phase shift" variation. Variation 5 is a combination of all the others. It has amplitude, frequency, and phase shift variations, and an increase in value. Thus it has all kinds of variation applied at the same time. I hopes this helps, but you will have to try a lot of variations on your own before it really sinks in. -Doctor Mike, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 06/18/97 at 02:22:53 From: Perfect Distributors Subject: Functions (again) Dr. Math, Thanks for your help before. I have attempted to draw graphs of the first two functions to find what each variable does. I have discovered what most variables do, but was wondering if you could help me name what each variable does (such as phase shift in the case of trig functions). I am still unclear on two: y = ab^(cx)+d and y = (a/cx)+b. I couldn't isolate the indervidual variables to see their results on the graphs of these two functions. Once again, Thanks from Ben
Date: 06/20/97 at 20:31:34 From: Doctor Mike Subject: Re: Functions (again) Hi again, Let's handle the easy part first. The "+d" or "+b" at the end just moves the graph up or down. Think of putting a piece of graph paper on the table and a transparent sheet on top of it. If you then draw the graph, it will appear that you are drawing on the graph paper, but what you drew is really on the transparent sheet. You can then see the effect of adding/subtracting a constant by sliding the transparent sheet up/down. (Replacing "x" by "x+k" or "x-k" is like sliding the sheet left/right.) Next, the "a" multipliers are like the amplitude effects I mentioned last time. Larger multipliers make any hills higher and the valleys deeper, and smaller multipliers make them not so high and not so deep. However, the word "amplitude" is not commonly used except for the "wavy" functions like sine. The two types of functions do not have hills or valleys, but the multipliers do tend to stretch or collapse their shapes up and down. Note that in "y = (a/cx)+b" it could also be written "y =(a/c)*(1/x)+b" so the parameter c is really part of the multiplier. The larger c gets, the smaller the (a/c) multiplier gets. If we eliminate the things mentioned above from the exponential function, we get y = b^(cx). These two constants, the base "b" and the coefficient "c" of x, affect the size of the graph, but the result will always be sort of a "swoosh" in one direction on another. You can see the two basic shapes by doing the graphs for "y = 2^x" and "y = (1/2)^x. As you compute larger and larger "x" exponents of 2, the results get larger and larger (2, 4, 8, 16, 32, 64, 128, .....). As you compute larger and larger "x" exponents of 1/2, the results get smaller and smaller (1/2, 1/4, 1/8, 1/16, .....). What happens on the left end (negative x) of these two graphs? You can see where the graphs cross the vertical y-axis by setting x = 0 and solving for y. What do you get? It's difficult to talk about these things without drawing a few, so you will have to do that. Pick out some particular members of these families of functions and graph them to see what they look like. Good luck. I hope this helps some. -Doctor Mike, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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