## Using Slider Applets to Study Function Families

You are all familiar with linear functionsf(x) = mx + band the corresponding graphs ofy = mx + b, which are, of course, straight lines. The two constantsm, andb, which give the slope and the y-intercept of the graph, are calledof theparametersfamily of linear functions. We will study many other such families of functions in precalculus and calculus, and we want to understand how changing the values of the parameters changes the function and its graph. Slider Applets are very convenient for this purpose.The Slider Applet on the left will draw the graph of

y = mx +bfor values ofmandbthat you can select (between -5 and 5) by moving the sliders or clicking the arrows at the ends of the sliders.

Note that we have already entered the formulamx +b. in the little text window. The actual values ofmandbare determined by the position of the corresponding sliders. As you change these values by moving the slider or clicking the arrows at the ends of the sliders, the new value will appear at the right side of the slider. NOTE: You can get better control of the values by clicking on the arrows!

For the rest of this exerciseExercise 1.don't change anything in the function window!Use only the top two sliders. Ignore the two crosses on the graph until part d)a) The values of

mandbare initially both set to the value 0. Produce the graph ofy = x + 2by moving the m-slider and clicking on the arrows so thatm =1and similarly adjusting the b-slider so thatb = 2. You should recognize the resulting line as having slope 1 and y-intercept 2.b) Now produce the graph of

y = -2.5x - 1and check that the slope and y-intercept look like what you expect.c) Now just move the b-slider from -5 to 5 and describe the functions whose graphs you see. What changes and what remains the same about the linear functions whose graphs you see? Next set the b-slider to 0 and move just the m-slider. What changes and what remains the same about the linear functions whose graphs you see?

d) Now try to adjust the parameters

mandbso that the resulting line goes through the two crosses. (Remember, you get better control with the arrows.) What is the slope and y-intercept of this line?

Now let's look at a family of functions with 3 parameters. Use the mouse and keyboard to add the termExercise 2.c/xto the expressionmx+bthat's already in the function box of the applet. The function box should now read m*x+b+c/x. You can see from the c-slider that the parametercis initially set to 0, so that if you don't change it, everything will be as in Exercise 1.(Notice that if

cis not zero, the function

f(x) = mx + b + c/x

does not have a value atx = 0.)When

cis not zero, we no longer have a linear function, so m and b no longer denote slope and y-intercept.a) Use the sliders to set

m = 1,b =2, andc = 1so that you will see the graph ofy = x + 2 + 1/x.

Now use the c-slider to change the value ofcto 2. Write down the equation of the graph you see. (Your equation should not contain the lettersm, borc! Now use the c-slider (and the arrows) to change the value ofcto some other values, say -3 and 4 and note the appearance of the graphs.b) Leaving

c=2unchanged, describe in a sentence the effect of changingmandb.c) Set the parameter

cequal to 1. Can you find values of the parametersm,band so that the graph of

y = mx+b + c/x

passes through the two crosses? Answer the same question whencis set equal to -2.

Precalculus Materials by B. Kaskosz and L. Pakula, University of Rhode Island, Copyright 2002.