In the previous section, we learned how to use the XFunctions program and, in the last two Exercises, we began to explore the effect of basic arithmetic operations, such as multiplication, on the graph of a function. In this section, we will focus attention on three particular classes of functions:
and their corresponding inverse operations:
Note: Negation is its own inverse, that is, h -1(x) = -x, since h -1(h(x)) = -(-x) = x. By combining multiplication or division with negation, this will cover addition, subtraction, multiplication, or division by any real number.
Since we are all very familiar with how these functions work algebraically and numerically, it only remains to focus on their geometric effect on the number line. This is most clearly visualized using arrow diagrams. Since we usually draw the number line horizontally, we will adapt our arrow diagrams a bit. For example, we can picture f, "addition by 1", as:
Geometrically, we can describe this as a "shift to the right by 1 unit". More generally:
Addition by a positive number, c, shifts the number line in the positive direction by c units; the standard mathematical term for this is a translation.
We can then reason that f -1 (i.e., subtraction by 1 or addition by -1) would be a translation by 1 in the opposite (i.e., negative) direction. In general:
Addition by a any number, c, shifts the number line in the direction of c by |c| units.
In the same way, we can picture g, "multiplication by 2", as:
Geometrically, we can describe this as a "stretch by a factor of 2", so that all points move away from the origin. Note: You have already seen this effect in a previous Exercise. More generally:
Multiplication by a positive number, c > 1, stretches the number line (away from the origin) by a factor of c.
Since the opposite of "stretch" is "shrink", g -1(x) = x/2 should be a shrink by a factor of 2, so that all the points move towards the origin:
Notice that, since division can also be written as multiplication by a reciprocal, g -1(x) = (1/2)x, a "shrink" is simply a "stretch" by a positive number less than 1. To avoid this strange use of the English language, we say that:
Multiplication by a any number, c > 0, rescales the number line by a factor of c; if c > 1, this is a stretch (moving points away from the origin), while if c < 1, this is a shrink (moving points towards the origin); the term scaling is used to refer to either a stretch or shrink.
So far we have considered addition by any number (which then includes subtraction), and multiplication by any positive number (which then includes division by positive numbers). Once we understand multiplication by -1, or "negation", which we have called h, we can then view multiplication by any negative number (e.g., -2) as multiplication by a positive number (e.g., 2), which we already understand, followed by multiplication by -1.
We can picture h as:
Notice how each point gets "flipped" to the other side of the origin, which effectively "flips" the entire number line. Note: You saw this effect in a previous Exercise, as well. We observed earlier that h is identical to its inverse, since to "undo" a flip, you simply flip again. Algebraically, we can observe that the opposite of "multiplication by -1" is "division by -1", but x/(-1) = -x, which gives another way to see that these are the same operations.
Multiplication by -1 flips the number line; the standard mathematical term for this is a reflection.
We will use these same three operations over and over again as a basis for all other graphing principles that we will encounter. We will use them throughout this text to analyze functions and quickly sketch graphs by hand. To summarize:
| English | Algebraic | Numerical | Geometric | ||
|---|---|---|---|---|---|
| Addition | x + c | Add c | shift/translation by c | c > 0 | shift by c in the positive direction |
| c < 0 | shift by -c in the negative direction | ||||
| Multiplication | c x | Multiply by c | scaling by a factor of c | c > 1 | stretch by a factor of c, away from the origin |
| c < 1 | shrink by a factor of 1/c, towards the origin | ||||
| Negation | -x | Multiply by -1 | flip/reflection | ||
Since we will use these patterns so often, you should commit them to memory. Notice how the English, algebraic, and numerical descriptions are all quite similar, so the key is simply to remember the five key geometric facts:
Addition by a positive number shifts in the positive direction.
Addition by a negative number (i.e., subtraction) shifts in the negative direction.
Multiplication by a positive number greater than 1 stretches away from the origin.
Multiplication by a positive number less than 1 shrinks towards the origin.
Multiplication by a negative sign flips over the number line.
These will often occur in combination. As long as we pay attention to the order of composition, we can accurately describe the geometric effect of any such function. For example, k(x) = -2x + 1, can be decomposed as:
multiply the input by 2, then
multiply this result by -1, and finally
add 1.
This means that we may describe the geometric effect on points of the number line as:
stretch by a factor of 2, then
flip over, and finally
shift 1 units to the right.
We can clearly see these effects numerically by considering the points x = -1, 0, 1 and computing the value of k at each point.
Putting all this analysis together in a single chart, we can see the correspondence between the algebra, arithmetic, and geometry:
| Algebraic | Numerical | Geometric | ||||
|---|---|---|---|---|---|---|
| English | Values | English | Number Line | |||
| x | Take the input | -1 | 0 | 1 | Take the line |  |
| 2x | Multiply by 2 | -2 | 0 | 2 | Stretch by 2 |  |
| -2x | Multiply by -1 | 2 | 0 | -2 | Flip |  |
| -2x + 1 | Add 1 | 3 | 1 | -1 | Shift 1 right |  |
Practice analyzing functions algebraically, numerically, and geometrically by completing the following
exercises.
Go to Post-Composition and Graphing
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