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).

Symmetry

We begin with even and odd symmetry.

 A function f(x) is called even if its value at x is the same as its value at -x. That is, if f(x) = f(-x). An even function has a graph that is symmetric about the y-axis: take the graph for positive values of x, and flip it over the y-axis to get the graph for negative values of x. A function f(x) is called odd if its value at -x is the negative of its value at x. That is, if f(-x) = - f(x). An odd function has a graph that is symmetric about the origin: take the graph for positive values of x, flip it over the y-axis, then over the x-axis, and you have the graph for negative values of x! Equivalently, you can reflect this part of the graph about the origin, point by point, as is shown in the sketch below.

Check that the functions y = f(x) = x 2 and y = g(x) = x 3 - 3x and their graphs demonstrate even and odd symmetries respectively. That is, show that f(-x) = f(x) and g(-x) = -g(x) for all x. Their graphs are given below.

 y = x 2 y = x3 - 2x

Four example graphs

These four functions are used in the examples and exercises to follow. Their graphs and basic properties should be memorized.

Shifting and Scaling

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.

Effect of absolute value

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|.

Examples and Exercises

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

• starting with the graph of y = |x|,
• move this graph right 1,
• make it steeper since it has been scaled by 2 and
• move the graph down 3.
By positioning the corner of the graph at the point (1,-3) and noting that the y-intercept is -1 (that's another point on the graph we could plot) we get the correct steepness displayed.

In the following activity, you will see examples of graphs and the effects we have described above for the function y = |x|.

Examples of Graphing Techniques
• Click on one of the effects "Horizontal", "Vertical", "Scale/Flip", or "All 3" to pick the type of effect you want to see.
• Click on "New" to generate a problem with this type of effect. You will now see a the formula of a new function; it will be altered from y = |x| by the type of effect/s you have chosen.
• Click on "Graph it" to see the graph of the new function.

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.

Practice Graphing Techniques
• Choose the type of function you want to graph by clicking on one of the four "Function type" buttons.
• To move the graph horizontally or vertically, click on the "Horizontal" or the "Vertical" button, then use the mouse to drag the graph to the right place.
• To flip the graph, click on "Flip"
• To scale the graph, click on one of the four Scaling factors to apply. This is cumulative so scaling by 2 and then by 3 scales the original function by 6.
• When you're done, click on "Check it".
• To get a new function click again on one of the four "Function type" buttons.
• If you want to give up, you can click on "Graph it" to see what the graph looks like.