Using Slider Applets to Study Function FamiliesYou are all familiar with linear functions f(x) = mx + b and the corresponding graphs of y = mx + b, which are, of course, straight lines. The two constants m, and b, which give the slope and the y-intercept of the graph, are called parameters of the family 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 +b for values of m and b that 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 formula mx +b . in the little text window. The actual values of m and b are 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!
Exercise 1. For the rest of this exercise 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 m and b are initially both set to the value 0. Produce the graph of y = x + 2 by moving the m-slider and clicking on the arrows so that m =1 and similarly adjusting the b-slider so that b = 2. You should recognize the resulting line as having slope 1 and y-intercept 2.
b) Now produce the graph of y = -2.5x - 1 and 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 m and b so 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?
Exercise 2. Now let's look at a family of functions with 3 parameters. Use the mouse and keyboard to add the term c/x to the expression mx+b that'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 parameter c is initially set to 0, so that if you don't change it, everything will be as in Exercise 1.
(Notice that if c is not zero, the function
f(x) = mx + b + c/x
does not have a value at x = 0.)
When c is 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, and c = 1 so that you will see the graph of
y = x + 2 + 1/x .
Now use the c-slider to change the value of c to 2. Write down the equation of the graph you see. (Your equation should not contain the letters m, b or c! Now use the c-slider (and the arrows) to change the value of c to some other values, say -3 and 4 and note the appearance of the graphs.
b) Leaving c=2 unchanged, describe in a sentence the effect of changing m and b.
c) Set the parameter c equal to 1. Can you find values of the parameters m,b and so that the graph of
y = mx+b + c/x
passes through the two crosses? Answer the same question when c is set equal to -2.
Precalculus Materials by B. Kaskosz and L. Pakula, University of Rhode Island, Copyright 2002.