Math 1050 - Precalculus - New Graphs from Old
Math 1050 - MAPLE LAB 2

Translations:
Vertical - The function f(x) is vertically translated through the operation f(x)+c or f(x)-c for c>0. The operation f(x)+c slides the original graph up c units, while f(x)-c slides the graph down c units. For example look at the function f(x)=x3. How will this compare to f(x)+4=x3+4  and f(x)-2=x3-2? To see the changes we can graph all three on the same axis. Start by defining the function f(x) in MAPLE.

> f:=x->x^3;

> plot({f(x),f(x)+4,f(x)-2},x=-3..3,y=-10..10,color=[blue,red,tan]);

The top line represents the function f(x)+4, the middle line is the original function f(x)=x3, and the bottom line is the function f(x)-2. Notice they all have the same shape, they are just shifted up and down the y axis by c units. Notice that MAPLE was directed to use the colors blue, red and tan in the graph, this was done only because the default colors include yellow when there are more than two equations per graph, and the yellow line is difficult to see on the printout. However, be careful, the first function graphed is not the blue one.

If you would like to see the algebraic representation of your functions, you can find them using MAPLE, and then simplify if necessary. To have MAPLE find the algebraic representation of f(x)+4 the command would be as follows.

> f(x)+4;

Horizontal - The function f(x) is horizontally translated through the operation f(x+c) or f(x-c) for c>0. However, the operation f(x+c) slides the graph c units to the left along the x axis, while f(x-c) slides the graph c units to the right. Let's look at the same function f(x)= x3 and compare it to f(x+2)=(x+2)3  and f(x-4)=(x-4)3.

> plot({f(x),f(x+2),f(x-4)},x=-6..6,y=-10..10,color=[blue,red,tan]);

The graph on the left represents the function f(x+2). The original graph of f(x) (found in the middle) has been moved 2 units to the left. The graph on the right represents f(x-4), the original graph has been moved 4 units to the right. So f(x+c) moves the graph back c units in the x direction. Again to see the algebraic representation of f(x+2) use MAPLE.

> f(x+2);

If we had wanted Maple to multiply this out for us, we would use the expand command.

Stretching and Shrinking:
Vertical - The function g(x) is vertically stretched or shrunk through when g(x) is multiplied by a . If a>1 the graph of a*g(x) will be stretched away from the x axis by a factor of a, while if 02+4 to see how 3g(x)=3(-x2+4) and 1/2g(x)=1/2(-x2+4) will compare to the original. First define the function g(x) in MAPLE, then plot the three functions on the same axis.

> g:=x->-x^2+4;

> plot({g(x),3*g(x),.5*g(x)},x=-3..3,y=-2..12,color=[blue,red,tan]);

Notice that all three graphs still have the same x intercepts, but the graphs have changed their shape this time. The original function, if graphed alone can be seen as the middle line. The top one is stretched out from the x axis and represents the function 3g(x). Notice at each x value the y value for the new function is three times as high as the original. The bottom line represents the function 1/2g(x), this one is shrunk down towards the x axis by a factor of 1/2, and each y value is half as high as the orignal function's y values. An algebraic representation of 3g(x) follows.

> 3*g(x);

> plot({g(x),g(4*x),g(1/3*x)},x=-7..7,y=-2..5,color=[blue,red,tan]);

Again the original funciton g(x) is in the middle, but what happened to the others? Notice the y intercepts remain the same this time, but the x intercepts changed along with the shapes. The graph of g(1/3x) is represented as the outside (or top) graph which is 3 times as wide as the original in the x direction. The graph of g(4x) is the narrow inside graph, and the distance from the y axis to the graph is now 1/4 as wide as the original graph. The algebraic representation of g(4x) is found below.

> g(4*x);

Reflecting:
Across the y axis - The function h(x) can be reflected across the y axis through the operation h(-x), that is by multiplying by a -1 inside the function.  To visualize the effects of this operation graph h(x)= , and compare this to h(-x)= .

> h:=x->(x)^(1/2);

> plot({h(x),h(-x)},x=-5..5,y=-1..3);

The original graph of h(x) is located on the right and its reflection across the y axis is the graphs mirror image on the left. Notice that the shape of the graph did not change, but it has been flipped over the y axis.

Across the x axis - The function h(x) can also be reflected across the x axis through the operation -h(x), multiplying by -1 on the outside of the function. Now all the y values will change sign and be on the opposite side of the x axis of the original. To see this graphically plot -h(x)= - .

> plot({h(x),-h(x)},x=-1..5,y=-3..3);

The orignal graph can now be seen as the top graph and the graph of -h(x) is the bottom one. Notice the shape remains the same, as well as the distance of each graph from the x axis. To form the graph of -h(x) each point of the original graph has been moved to the opposite side of the x axis, we call this a reflection across the x axis.

Complete then print out a copy of all the exercises to hand in to your instructor.

Exercises: