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Curved square root graph originating from the origin (0,0) increasing to the right.
y = x size 12{y= sqrt {x} } {}
Curved square root graph originating from the origin (0,4) increasing to the right.
y = x + 4 size 12{y= sqrt {x} +4} {}
Curved square root graph originating from the origin (0,-1.5) increasing to the right.
y = x 1 1 2 size 12{y= sqrt {x} - 1 { { size 8{1} } over { size 8{2} } } } {}

Other vertical permutations

Adding or subtracting a constant from 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:

The sum of two functions. Likely a parabola and line.
y = f ( x ) size 12{y=f \( x \) } {}

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

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

The sum of two functions. Likely a parabola and line.
y = f ( x ) size 12{y=f \( x \) } {}
The sum of two functions stretched out with the y-values doubled.
y = 2f ( x ) size 12{y=2f \( x \) } {} ; All y size 12{y} {} -values are doubled

Similarly, y = 1 2 f ( 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 ) 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.

The sum of two functions. Likely a parabola and line.
y = f ( x ) size 12{y=f \( x \) } {}
The sum of two functions. Same graph as previous but flipped vertically with v-values sign's changed.
y = -f ( x ) size 12{y=2f \( x \) } {} ; All y size 12{y} {} -values change sign

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 ) size 12{y=f \( x \) } {} .

The sum of two functions. Likely a parabola and line.
y = f ( x ) size 12{y=f \( x \) } {} ; Contains the following points (among others): ( 3,2 ) size 12{ \( - 3,2 \) } {} , ( 1, 3 ) size 12{ \( - 1, - 3 \) } {} , ( 1,2 ) size 12{ \( 1,2 \) } {} , ( 6,0 ) size 12{ \( 6,0 \) } {}

Suppose you were asked to graph y = f ( x + 2 ) size 12{y=f \( x+2 \) } {} . Note that this is not the same as f ( 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 ) size 12{y=f \( x+2 \) } {} is an instruction to add 2 to every x-value before plugging it into the function.

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

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

The original graph f ( x ) size 12{f \( x \) } {} contains the point ( 6,0 ) size 12{ \( 6,0 \) } {} ; therefore, f ( x + 2 ) size 12{f \( x+2 \) } {} contains the point ( 4,0 ) size 12{ \( 4,0 \) } {} . The point has moved two spaces to the left.
The sum of two functions. Likely a parabola and line.
y = f ( x ) size 12{y=f \( x \) } {}
The sum of two functions. Likely a parabola and line shifted left two units.
y = f ( x+2 ) size 12{y=2f \( x \) } {} ; Each point is shifted 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 ( x 2 ) 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 = 5 size 12{x=5} {} it does what f ( x ) size 12{f \( x \) } {} does at x = 3 size 12{x=3} {} . At x = 10 size 12{x="10"} {} , it copies f ( 8 ) size 12{f \( 8 \) } {} . And so on. Because it is always copying f ( x ) size 12{f \( x \) } {} to its left , this graph ends up being a copy of 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 ( x 6 ) 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 ) size 12{y=2f \( x \) } {} vertically stretches a graph; y = 1 2 f ( 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.

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

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

The sum of two functions. Likely a parabola and line.
y = f ( x ) size 12{y=f \( x \) } {}
The sum of two functions scaled by a factor of 2 and pushed twice as close.
y = f ( 2x ) size 12{y=2f \( x \) } {} ; Each point is twice as close to the y size 12{y} {} -axis

We can explain this the same way we explained f ( x 2 ) size 12{f \( x - 2 \) } {} . In this case, 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 = 5 size 12{x=5} {} it does what f ( x ) size 12{f \( x \) } {} does at x = 10 size 12{x="10"} {} . At x = 3 size 12{x= - 3} {} , it copies 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 ) size 12{f \( x \) } {} moved inward ; ie a compression. Similarly, f ( 1 2 x ) 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 ) size 12{y=f \( - x \) } {} does precisely what you would expect: it flips the graph around the y-axis. f ( 2 ) size 12{f \( - 2 \) } {} is the old f ( 2 ) size 12{f \( 2 \) } {} and vice-versa.

The sum of two functions. Likely a parabola and line.
y = f ( x ) size 12{y=f \( x \) } {}
The sum of two functions same as above horizontally flipped. The x-values signs are changed.
y = f ( -x ) size 12{y=2f \( x \) } {} ; Each point flips around the y size 12{y} {} -axis

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 + 5 size 12{y= - 2 \( x+3 \) rSup { size 8{2} } +5} {} can be easily graphed. You take the (known) graph of y = x 2 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 = x size 12{y= sqrt {x} } {} in a variety of flavors.

Graph showing the square root of x.
y = x size 12{y= sqrt {x} } {} ; Generated by plotting points; Contains ( 0,0 ) size 12{ \( 0,0 \) } {} , ( 1,1 ) size 12{ \( 1,1 \) } {} , ( 4,2 ) size 12{ \( 4,2 \) } {} ; Domain: x 0 size 12{x>= 0} {} ; Range: y 0 size 12{y>= 0} {} ; Range: y 0 size 12{y>= 0} {}
Graph showing the square root of x+5, similar to x squared but shifted over 5 units to the left.
y = x + 5 size 12{y= sqrt {x+5} } {} ; Shifted 5 units to the left; Contains ( 5,0 ) size 12{ \( - 5,0 \) } {} , ( 4,1 ) size 12{ \( - 4,1 \) } {} , ( 1,2 ) size 12{ \( - 1,2 \) } {} ; Domain: x 5 size 12{x>= - 5} {} ; Range: y 0 size 12{y>= 0} {}
Graph showing the square root of -x, then minus 2. Flipped horizontally, shifted down 2.
y = x 2 size 12{y= sqrt { - x} - 2} {} ; Flipped horizontally, shifted down 2;Contains ( 0, 2 ) size 12{ \( 0, - 2 \) } {} , ( 1, 1 ) size 12{ \( - 1, - 1 \) } {} , ( 4,0 ) size 12{ \( - 4,0 \) } {} ; Domain: x 0 size 12{x<= 0} {} ; Range: y 2 size 12{y>= - 2} {}
Graph showing the square root of x-1, plus 5. Flipped vertically, shifted 1 to the right and 5 up
y = x 1 + 5 size 12{y= - sqrt {x - 1} +5} {} ; Flipped vertically, shifted 1 to the right and 5 up;Contains ( 1,5 ) size 12{ \( 1,5 \) } {} , ( 2,4 ) size 12{ \( 2,4 \) } {} , ( 5,3 ) size 12{ \( 5,3 \) } {} ; Domain: x 1 size 12{x>= 1} {} ;Range: y 5 size 12{y<= 5} {}

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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
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