Shifty Graphs

Prerequisites: Function notation, graphs of functions.

Goal: To understand how adding or multiplying by a constant changes the graph of a function.

Today we begin by starting Maple, then copying and pasting the following code into our Maple worksheet. Note that in Maple, the code a^b means ab.

with(plots):

xMin := -3:    #  left endpoint (domain)
xMax :=  3:    # right endpoint (domain)
yMin := -2:    # lower endpoint (range)
yMax :=  9:    # upper endpoint (range)

graph1 := plot( (x+0)^2, x=xMin..xMax, y=yMin..yMax, color=cyan):
graph2 := plot( (x+1)^2, x=xMin..xMax, y=yMin..yMax, color=blue):

display(graph1,graph2);
  1. After pasting this code into Maple, press <enter> (or <return>). Two graphs should appear on the same axes, one in cyan (greeenish-blue) and the other in blue. Which is which?
  2. The parts in purple are parts that we will be editing. First, edit the above code so that you obtain a graph of the functions x2 (same as (x+0)2) and (x-1)2 on the same axes. How has the graph changed?
  3. What will happen if you graph x2 and (x+7)2 on the same axes? Try it. Probably you will need to change something else to get an idea of how the two graphs are related. What needs to be changed (in addition to one of the functions)?
  4. Next, let's compare the graphs of (x+0)2 and x2+3. Is there any difference between (x+3)2 and x2+3? Predict the appearance of x2-35, then graph it together with x2 and (x-35)2.
  5. Predict the appearance of the graph of (x-1.5)2-4, then graph it. How accurate was your guess? Try the following: (x+7)2-12 and (x-4)2+17.
  6. Let's start over now with a new function; compare the graphs of cos(x) and cos(x+1). Make predictions about the appearance of the graphs of cos(x-h), cos(x+h), cos(x)-h, and cos(x)+h (assume that "h" is some number that I will tell you later).
  7. How about cos(x) compared to 3*cos(x), or cos(x) compared to cos(x)/2? What do you expect 5*cos(x-2)+1 to look like?
  8. Lastly, how does cos(x) compare to cos(2*x) or cos(x/3)? Can you predict, without graphing it first, what 4*cos(2*(x-1))+3 will look like?
Last Update: 2007/06/21