**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 *a ^{b}*.

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);

- 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?
- 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
*x*(same as^{2}*(x+0)*) and^{2}*(x-1)*on the same axes. How has the graph changed?^{2} - What will happen if you graph
*x*and^{2}*(x+7)*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)?^{2} - Next, let's compare the graphs of
*(x+0)*and^{2}*x*. Is there any difference between^{2}+3*(x+3)*and^{2}*x*? Predict the appearance of^{2}+3*x*, then graph it together with^{2}-35*x*and^{2}*(x-35)*.^{2} - Predict the appearance of the graph of
*(x-1.5)*-4, then graph it. How accurate was your guess? Try the following:^{2}*(x+7)*-12 and^{2}*(x-4)*+17.^{2} - 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). - 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? - 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?