Adding or subtracting a constant from
, 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
look like? We can make a table similar to the one we made before.
so
contains this point
–3
2
4
–1
–3
–6
1
2
4
6
0
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,
yields a graph that is vertically compressed, with each point moving toward the x-axis.
Finally, what does
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
.
Suppose you were asked to graph
. Note that this is
not the same as
! The latter is an instruction to run the function, and then add 2 to all results. But
is an instruction to add 2 to every x-value
before plugging it into the function.
changes
, and therefore shifts the graph vertically
changes
, 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.
so
contains this point
–5
–3
f(–3)=2
–3
–1
f(–1)=–3
–1
1
f(1)=2
4
6
f(6)=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 graphcontains the point
;
therefore,contains the point
. 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.
is an instruction that says to each point, “look two spaces to your left, and copy what the original function is doing
there .” At
it does what
does at
. At
, it copies
. And so on. Because it is always copying
to its
left , this graph ends up being a copy of
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:
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
vertically stretches a graph;
vertically compresses . Just as with addition and subtraction, we will find that the horizontal equivalents work backward.
so
contains this point
–1½
–3
2
–½
–1
–3
½
1
2
3
6
0
The original graph
contains the point
; therefore,
contains the point
. Similarly,
becomes
. Each point is closer to the y-axis; the graph has horizontally
compressed .
We can explain this the same way we explained
. In this case,
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
it does what
does at
. At
, it copies
. And so on. Because it is always copying f(x)
outside itself, this graph ends up being a copy of
moved
inward ; ie a compression. Similarly,
causes each point to look inward toward the y-axis, so it winds up being a
horizontally stretched version of the original.
Finally,
does precisely what you would expect: it flips the graph around the y-axis.
is the old
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
can be easily graphed. You take the (known) graph of
, 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
in a variety of flavors.
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?