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# Graphing

Module by: Kenny M. Felder. E-mail the author

Summary: This module describes how to graph basic functions.

Graphing, like algebraic generalizations, is a difficult topic because many students know how to do it but are not sure what it means.

For instance, consider the following graph:

If I asked you “Draw the graph of y=x2y=x2 size 12{y=x rSup { size 8{2} } } {}” you would probably remember how to plot points and draw the shape.

But suppose I asked you this instead: “Here’s a function, y=x2y=x2 size 12{y=x rSup { size 8{2} } } {}. And here’s a shape, that sort of looks like a U. What do they actually have to do with each other?” This is a harder question! What does it mean to graph a function?

The answer is simple, but it has important implications for a proper understanding of functions. Recall that every point on the plane is designated by a unique (x,y)(x,y) size 12{ $$x,y$$ } {} pair of coordinates: for instance, one point is (5,3)(5,3) size 12{ $$5,3$$ } {}. We say that its x x size 12{x} {} -value is 5 and its y y size 12{y} {} -value is 3.

A few of these points have the particular property that their y y size 12{y} {} -values are the square of their x x size 12{x} {} -values. For instance, the points (0,0)(0,0) size 12{ $$0,0$$ } {}, (3,9)(3,9) size 12{ $$3,9$$ } {}, and (5,25)(5,25) size 12{ $$- 5, 25$$ } {} all have that property. (5,3)(5,3) size 12{ $$5,3$$ } {} and (2,4)(2,4) size 12{ $$- 2, - 4$$ } {} do not.

The graph shown—the pseudo-U shape—is all the points in the plane that have this property. Any point whose y y size 12{x} {} -value is the square of its x x size 12{x} {} -value is on this shape; any point whose y y size 12{y} {} -value is not the square of its x x size 12{x} {} -value is not on this shape. Hence, glancing at this shape gives us a complete visual picture of the function y=x2y=x2 size 12{y=x rSup { size 8{2} } } {} if we know how to interpret it correctly.

## Graphing Functions

Remember that every function specifies a relationship between two variables. When we graph a function, we put the independent variable on the x x size 12{x} {} -axis, and the dependent variable on the y y size 12{y} {} -axis.

For instance, recall the function that describes Alice’s money as a function of her hours worked. Since Alice makes $12/hour, her financial function is m(t)=12tm(t)=12t size 12{m $$t$$ ="12"t} {}. We can graph it like this. This simple graph has a great deal to tell us about Alice’s job, if we read it correctly. • The graph contains the point (3,300)(3,300) size 12{ $$3,"300"$$ } {}.What does that tell us? That after Alice has worked for three hours, she has made$300.
• The graph goes through the origin (the point (0,0)(0,0) size 12{ $$0,0$$ } {}). What does that tell us? That when she works 0 hours, Alice makes no money.
• The graph exists only in the first quadrant. What does that tell us? On the mathematical level, it indicates the domain of the function ( t0t0 size 12{t >= 0} {}) and the range of the function ( m0m0 size 12{m >= 0} {}). In terms of the situation, it tells us that Alice cannot work negative hours or make negative money.
• The graph is a straight line. What does that tell us? That Alice makes the same amount of money every day: every day, her money goes up by $100. ($100/day is the slope of the line—more on this in the section on linear functions.)

Consider now the following, more complicated graph, which represents Alice’s hair length as a function of time (where time is now measured in weeks instead of hours).

What does this graph h(t)h(t) size 12{h $$t$$ } {} tell us? We can start with the same sort of simple analysis.

• The graph goes through the point (0,12)(0,12) size 12{ $$0,"12"$$ } {}.This tells us that at time (t=0)(t=0) size 12{ $$t=0$$ } {}, Alice’s hair is 12" long.
• The range of this graph appears to be 12h1812h18 size 12{"12" <= h <= "18"} {}. Alice never allows her hair to be shorter than 12" or longer than 18".

But what about the shape of the graph? The graph shows a gradual incline up to 18", and then a precipitous drop back down to 12"; and this pattern repeats throughout the shown time. The most likely explanation is that Alice’s hair grows slowly until it reaches 18", at which point she goes to the hair stylist and has it cut down, within a very short time (an hour or so), to 12". Then the gradual growth begins again.

## The rule of consistency, graphically

Consider the following graph.

This is our earlier “U” shaped graph ( y=x2y=x2 size 12{y=x rSup { size 8{2} } } {}) turned on its side. This might seem like a small change. But ask this question: what is yy size 12{y} {} when x=3x=3 size 12{x=3} {}? This question has two answers. This graph contains the points (3,9)(3,9) size 12{ $$3, - 9$$ } {} and (3,9)(3,9) size 12{ $$3,9$$ } {}. So when x=3x=3 size 12{x=3} {}, yy size 12{y} {} is both 9 and –9 on this graph.

This violates the only restriction on functions—the rule of consistency. Remember that the x x size 12{x} {} -axis is the independent variable, the y y size 12{y} {} -axis the dependent. In this case, one “input” value (3)(3) size 12{ $$3$$ } {} is leading to two different “output” values (9,9)(9,9) size 12{ $$- 9,9$$ } {} We can therefore conclude that this graph does not represent a function at all. No function, no matter how simple or complicated, could produce this graph.

This idea leads us to the “vertical line test,” the graphical analog of the rule of consistency.

Definition 1: The Vertical Line Test
If you can draw any vertical line that touches a graph in two places, then that graph violates the rule of consistency and therefore does not represent any function.

It is important to understand that the vertical line test is not a new rule! It is the graphical version of the rule of consistency. If any vertical line touches a graph in two places, then the graph has two different y y size 12{y} {} -values for the same x x size 12{y} {} -value, and this is the only thing that functions are not allowed to do.

## What happens to the graph, when you add 2 to a function?

Suppose the following is the graph of the function y=f(x)y=f(x) size 12{y=f $$x$$ } {}.

We can see from the graph that the domain of the graph is 3x63x6 size 12{ - 3 <= x <= 6} {} and the range is 3y23y2 size 12{ - 3 <= y <= 2} {}.

Question: What does the graph of y=f(x)+2y=f(x)+2 size 12{y=f $$x$$ +2} {} look like?

This might seem an impossible question, since we do not even know what the function f(x)f(x) size 12{f $$x$$ } {} is. But we don’t need to know that in order to plot a few points.

Table 1
x x size 12{x} {} f ( x ) f ( x ) size 12{f $$x$$ } {} f ( x + 2 ) f ( x + 2 ) size 12{f $$x+2$$ } {} so y=f(x)y=f(x) size 12{y=f $$x$$ } {} contains this point and y=f(x)+2y=f(x)+2 size 12{y=f $$x$$ +2} {} contains this point
–3 2 4 ( 3,2 ) ( 3,2 ) size 12{ $$- 3,2$$ } {} ( 3,4 ) ( 3,4 ) size 12{ $$- 3,4$$ } {}
–1 –3 –1 ( 1, 3 ) ( 1, 3 ) size 12{ $$- 1, - 3$$ } {} ( 1, 1 ) ( 1, 1 ) size 12{ $$- 1, - 1$$ } {}
1 2 4 ( 1,2 ) ( 1,2 ) size 12{ $$1,2$$ } {} ( 1,4 ) ( 1,4 ) size 12{ $$1,4$$ } {}
6 0 2 ( 6,0 ) ( 6,0 ) size 12{ $$6,0$$ } {} ( 6,2 ) ( 6,2 ) size 12{ $$6,2$$ } {}

If you plot these points on a graph, the pattern should become clear. Each point on the graph is moving up by two. This comes as no surprise: since you added 2 to each y-value, and adding 2 to a y-value moves any point up by 2. So the new graph will look identical to the old, only moved up by 2.

In a similar way, it should be obvious that if you subtract 10 from a function, the graph moves down by 10. Note that, in either case, the domain of the function is the same, but the range has changed.

These permutations work for any function. Hence, given the graph of the function y=xy=x size 12{y= sqrt {x} } {} below (which you could generate by plotting points), you can produce the other two graphs without plotting points, simply by moving the first graph up and down.

## Other vertical permutations

Adding or subtracting a constant from f(x)f(x) size 12{f $$x$$ } {}, as described above, is one example of a vertical permutation: it moves the graph up and down. There are other examples of vertical permutations.

For instance, what does doubling a function do to a graph? Let’s return to our original function:

What does the graph y=2f(x)y=2f(x) size 12{y=2f $$x$$ } {} look like? We can make a table similar to the one we made before.

Table 2
x x size 12{x} {} f ( x ) f ( x ) size 12{f $$x$$ } {} 2f ( x ) 2f ( x ) size 12{2f $$x$$ } {} so y=2f(x)y=2f(x) size 12{y=2f $$x$$ } {} contains this point
–3 2 4 ( 3,4 ) ( 3,4 ) size 12{ $$- 3,4$$ } {}
–1 –3 –6 ( 1, 6 ) ( 1, 6 ) size 12{ $$- 1, - 6$$ } {}
1 2 4 ( 1,4 ) ( 1,4 ) size 12{ $$1,4$$ } {}
6 0 0 ( 6,0 ) ( 6,0 ) size 12{ $$6,0$$ } {}

In general, the high points move higher; the low points move lower. The entire graph is vertically stretched, with each point moving farther away from the x-axis.

Similarly, y=12f(x)y=12f(x) size 12{y= { { size 8{1} } over { size 8{2} } } f $$x$$ } {} yields a graph that is vertically compressed, with each point moving toward the x-axis.

Finally, what does y=f(x)y=f(x) size 12{y= - f $$x$$ } {} look like? All the positive values become negative, and the negative values become positive. So, point by point, the entire graph flips over the x-axis.

## What happens to the graph, when you add 2 to the x value?

Vertical permutations affect the y-value; that is, the output, or the function itself. Horizontal permutations affect the x-value; that is, the numbers that come in. They often do the opposite of what it naturally seems they should.

Let’s return to our original function y=f(x)y=f(x) size 12{y=f $$x$$ } {}.

Suppose you were asked to graph y=f(x+2)y=f(x+2) size 12{y=f $$x+2$$ } {}. Note that this is not the same as f(x)+2f(x)+2 size 12{f $$x$$ +2} {}! The latter is an instruction to run the function, and then add 2 to all results. But y=f(x+2)y=f(x+2) size 12{y=f $$x+2$$ } {} is an instruction to add 2 to every x-value before plugging it into the function.

• f(x)+2f(x)+2 size 12{f $$x$$ +2} {} changes yy size 12{y} {}, and therefore shifts the graph vertically
• f(x+2)f(x+2) size 12{f $$x+2$$ } {} changes xx size 12{x} {}, and therefore shifts the graph horizontally.

But which way? In analogy to the vertical permutations, you might expect that adding two would shift the graph to the right. But let’s make a table of values again.

Table 3
x x size 12{x} {} x + 2 x + 2 size 12{x+2} {} f ( x + 2 ) f ( x + 2 ) size 12{f $$x+2$$ } {} so y=f(x+2)y=f(x+2) size 12{y=f $$x+2$$ } {} contains this point
–5 –3 f(–3)=2 ( 5,2 ) ( 5,2 ) size 12{ $$- 5,2$$ } {}
–3 –1 f(–1)=–3 ( 3, 3 ) ( 3, 3 ) size 12{ $$- 3, - 3$$ } {}
–1 1 f(1)=2 ( 1,2 ) ( 1,2 ) size 12{ $$- 1,2$$ } {}
4 6 f(6)=0 ( 4,0 ) ( 4,0 ) size 12{ $$4,0$$ } {}

This is a very subtle, very important point—please follow it closely and carefully! First of all, make sure you understand where all the numbers in that table came from. Then look what happened to the original graph.

### Note:

The original graph f(x)f(x) size 12{f $$x$$ } {} contains the point (6,0)(6,0) size 12{ $$6,0$$ } {}; therefore, f(x+2)f(x+2) size 12{f $$x+2$$ } {} contains the point (4,0)(4,0) size 12{ $$4,0$$ } {}. The point has moved two spaces to the left.

You see what I mean when I say horizontal permutations “often do the opposite of what it naturally seems they should”? Adding two moves the graph to the left.

Why does it work that way? Here is my favorite way of thinking about it. f(x2)f(x2) size 12{f $$x - 2$$ } {} is an instruction that says to each point, “look two spaces to your left, and copy what the original function is doing there.” At x=5x=5 size 12{x=5} {} it does what f(x)f(x) size 12{f $$x$$ } {} does at x=3x=3 size 12{x=3} {}. At x=10x=10 size 12{x="10"} {}, it copies f(8)f(8) size 12{f $$8$$ } {}. And so on. Because it is always copying f(x)f(x) size 12{f $$x$$ } {} to its left, this graph ends up being a copy of f(x)f(x) size 12{f $$x$$ } {} moved to the right. If you understand this way of looking at it, all the rest of the horizontal permutations will make sense.

Of course, as you might expect, subtraction has the opposite effect: f(x6)f(x6) size 12{f $$x - 6$$ } {} takes the original graph and moves it 6 units to the right. In either case, these horizontal permutations affect the domain of the original function, but not its range.

## Other horizontal permutations

Recall that y=2f(x)y=2f(x) size 12{y=2f $$x$$ } {} vertically stretches a graph; y=12f(x)y=12f(x) size 12{y= { { size 8{1} } over { size 8{2} } } f $$x$$ } {} vertically compresses. Just as with addition and subtraction, we will find that the horizontal equivalents work backward.

Table 4
x x size 12{x} {} 2x 2x size 12{2x} {} f ( 2x ) f ( 2x ) size 12{f $$2x$$ } {} so y=2f(x)y=2f(x) size 12{y=2f $$x$$ } {} contains this point
–1½ –3 2 ( 1 1 2 , 2 ) ( 1 1 2 , 2 ) size 12{ $$- 1 { { size 8{1} } over { size 8{2} } } ,2$$ } {}
–½ –1 –3 ( 1 2 ; 3 ) ( 1 2 ; 3 ) size 12{ $$- { { size 8{1} } over { size 8{2} } } ; - 3$$ } {}
½ 1 2 ( 1 2 ; 2 ) ( 1 2 ; 2 ) size 12{ $${ { size 8{1} } over { size 8{2} } } ;2$$ } {}
3 6 0 ( 3,0 ) ( 3,0 ) size 12{ $$3,0$$ } {}

The original graph f(x)f(x) size 12{f $$x$$ } {} contains the point (6,0)(6,0) size 12{ $$6,0$$ } {}; therefore, f(2x)f(2x) size 12{f $$2x$$ } {} contains the point (3,0)(3,0) size 12{ $$3,0$$ } {}. Similarly, (1;3)(1;3) size 12{ $$- 1; - 3$$ } {} becomes (12;3)(12;3) size 12{ $$- { { size 8{1} } over { size 8{2} } } ; - 3$$ } {}. Each point is closer to the y-axis; the graph has horizontally compressed.

We can explain this the same way we explained f(x2)f(x2) size 12{f $$x - 2$$ } {}. In this case, f(2x)f(2x) size 12{f $$2x$$ } {} is an instruction that says to each point, “Look outward, at the x-value that is double yours, and copy what the original function is doing there.” At x=5x=5 size 12{x=5} {} it does what f(x)f(x) size 12{f $$x$$ } {} does at x=10x=10 size 12{x="10"} {}. At x=3x=3 size 12{x= - 3} {}, it copies f(6)f(6) size 12{f $$- 6$$ } {}. And so on. Because it is always copying f(x) outside itself, this graph ends up being a copy of f(x)f(x) size 12{f $$x$$ } {} moved inward; ie a compression. Similarly, f(12x)f(12x) size 12{f $${ { size 8{1} } over { size 8{2} } } x$$ } {} causes each point to look inward toward the y-axis, so it winds up being a horizontally stretched version of the original.

Finally, y=f(x)y=f(x) size 12{y=f $$- x$$ } {} does precisely what you would expect: it flips the graph around the y-axis. f(2)f(2) size 12{f $$- 2$$ } {} is the old f(2)f(2) size 12{f $$2$$ } {} and vice-versa.

All of these permutations do not need to be memorized: only the general principles need to be understood. But once they are properly understood, even a complex graph such as y=2(x+3)2+5y=2(x+3)2+5 size 12{y= - 2 $$x+3$$ rSup { size 8{2} } +5} {} can be easily graphed. You take the (known) graph of y=x2y=x2 size 12{y=x rSup { size 8{2} } } {}, flip it over the x-axis (because of the negative sign), stretch it vertically (the 2), move it to the left by 3, and move it up 5.

With a good understanding of permutations, and a very simple list of known graphs, it becomes possible to graph a wide variety of important functions. To complete our look at permutations, let’s return to the graph of y=xy=x size 12{y= sqrt {x} } {} in a variety of flavors.

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