|2.2 Graphing Techniques|
What is the difference between the graphs of the functions y = x2 and y = x2 + 4? How about the differences between y = x2 and y = (x + 4)2? Instinctively it seems as though their graphs should be very similar, and indeed they are. However, in order to sketch the graphs we need a precise correlation between the algebraic differences and the graphical differences.
There are a number of concepts we will discuss in this section in order to learn how to sketch graphs: symmetry (even and odd functions), horizontal and vertical shifts, and vertical scaling (stretching and shrinking).
We begin with even and odd symmetry.
Check that the functions
|y = x3 - 2x|
These four functions are used in the examples and exercises to follow. Their graphs and basic properties should be memorized.
The three main transformations we will consider, horizontal shifts, vertical scalings and vertical shifts, are described here with examples. Understanding these examples will help in the exercises to follow.
|Horizontal shift: The graph of
f(x - c) is the graph of f(x) shifted horizontally
|c| units. The graph is shifted right if c > 0 and left if
c < 0.|
For c = 2, move right 2.
For c = -3, move left 3.
|Vertical scaling: The graph of
kf(x) is the graph of f(x) scaled vertically.
If |k| > 1 the graph is stretched out from the x-axis and
becomes steeper. If |k| < 1 the graph is shrunk or compressed
towards the x-axis and becomes less steep. In addition, if
k < 0 the graph is also flipped over the x axis.|
For k = 1/3, less steep.
For k = -2, steeper and flipped over the x axis.
|Vertical shift: The graph of
f(x) + c is the graph of f(x) shifted vertically
|c| units. The graph is shifted up if c > 0 and down if
c < 0.|
For c = 4, move up 4.
For c = -3, move down 3.
When a function is contained within absolute value symbols, first consider the graph of the same function without the absolute value, and then flip the parts of the graph that are below the x axis. This is due to the fact that if f(x) > 0 then the absolute value bars have no effect and those points above the x axis. If f(x) < 0 then the absolute value bars make that y value positive so those points below the x axis are flipped above the x axis. For example, here are the the graphs of f(x) = x2 - 3 and f(x) = |x2 - 3|.
When the actions described above are combined in one function we produce the resulting graph by determining the cumulative effect of all actions. For example, the graph of y = 2|x - 1| - 3 is found by
In the following activity, you will see examples of graphs and the effects we have described above for the function y = |x|.
Now do it yourself with a variety of functions and their graphs. Notice that you only need to know how to graph the original function in order to make the changes we have described here. The effects are cumulative so order is important. For example, a scaling followed by a vertical shift is not equivalent to the vertical shift followed by the scaling.