Absolute Values (1 Viewer)

[Leo.T]

Could any help me by explaing how to graph and plot/sketch absolute vaules?


y=f(x)=|x| + | x-1 | / <------ u dont have to explain both try that one first y= |x| +|x| + |x+2|


could anyone please try to explain this to me step by step because i am having heaps of trouble at this



note: please if u can also explain how do u actually know where to plot your dots and join them together cause that part i dont quite get


thanks heaps!

 

[undalay]

Q1

y=|x| + | x-1 |

okay absolutes are really easy once u get them :]

Firstly i look at this and i know that the whole function is made up of three different lines. Its easy to explain in person, harder over the forum.

|x| is made up of 2 different lines but they connect. (eg. |x| < 0 is y=-x, |x|>0 is y=x)

now for y=|x| + | x-1 |, on the x axis put the dots on 0, and 1.

Now to the left of 0, the equation of the line will be
y = -x + -(x-1)(anything less then 0 will cause both |x| and |x-1| to be negative)
simplifies to:
y = -2x +1
Subin the 0 into that and u get y=1, so thats the point at x = 0, thats 1 changing point, to the left of this point is just the equation y = -2x +1.

Now between 0 and 1 the equation of the line will be.
y = x + -(x-1) (anything between 0 and 1 will only cause |x-1| to be negative)
y = x-x+1
y= +1
So its a straight line between x=0 and x=1, where the equation is y=1

now to the right of 1 the equation of the line will be:
y=(x)+ ( x-1 )(anything greater then x=1, both |x| and |x-1| to be positive)
y=2x -1
So after x=1, theres a straight line with the equation y=2x-1

I hope this makes sense :/ Its much easier if i explain with diagrams, but cbb drawing them.

edit Q2:

y= |x| +|x| + |x+2|
This is exactly the same method as the first question. First group the like absolutes.

y= 2|x| + |x+2|

There will be 3 different lines.

edit2: attached picture

 

[Leo.T]

OMG THANKS

-----------------------
Now to the left of 0, the equation of the line will be
y = -x + -(x-1)(anything less then 0 will cause both |x| and |x-1| to be negative)
simplifies to:
y = -2x +1

k i get everything up to there

----------------------------------
Subin the 0 into that and u get y=1, so thats the point at x = 0, thats 1 changing point, to the left of this point is just the equation y = -2x +1.

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i dont get that part!!









=]



but how to actually plot it on a graph



my teacher graphed it for me it i got a |x| but with a flat bottom like v with no sharp part with a flat bottom

 

[undalay]

Hmmm did you look at the diagram i posted up?

Basically you got three lines right now right?

x<0 y=-2x+1
0<x<1 y=1
x>1 y=2x-1

And you just draw them like normal lines, but only for the certain domains.
I trust you know how to draw normal lines : D

"Subin the 0 into that and u get y=1, so thats the point at x = 0, thats 1 changing point, to the left of this point is just the equation y = -2x +1."

Don't worry about this.

 

[Leo.T]

EXPLAIN HOW TO DRAW NORMAL LINES! PLEASE !


thank you ! hAI!

 

[Leo.T]

how you get those line in that graph!

 

[undalay]

lol , you sub in the points.

x<0, y=-2x+1
Okay firstly subin x=0 into the equation.
Turns into y = -2(0) + 1)
y = 1.
So the first point is (0,1)
Next sub in any point that is less then 0.
I'll pick say -4
y = -2(-4)+1
y = 9
So the point is (-4,9)
Now join these points, and draw an arrow head the (-4,9) place to indicate to the reader that the line keeps going in the direction and doesnt just stop at (-4,9).



0<=x <1 , y=1
Subin 0 and 1 into y=1. Since theres no x in y=1, both the y values must be 1.
So the two points are (0,1) and (1,1) join these points.


Now subin final values into
x>=1 y=2x-1

subin 1 into y=2x-1
which is y=1
So you got (1,1)

now subin any number which is x>=1, lets take 4.

y =2(4)-1
y = 7

You got (4,7)
Now connect dots once again, make sure u put an arrow head at (4,7) to indicate that the line continues and does not stop at (4,7)

DOES THAT HELP ? i hope it makes sense.

 

[Leo.T]

A Tear Came Out!! Thank You!


W8 nO IM ConFUSED AGEN


Okay firstly subin x=0 into the equation.


why zero?1 shouldnt it be a number less than 0?

 

[undalay]

Well its a little bit tricky to explain.

You subin both 0 and a number less then 0.

because X<0, however -0.00000000000000000000000000001, still satisfies this equation. When you come across something like x<0, but not x<=0, you would normall subin that point, but not colour it in. Rather you would put an open circle on that point, to indicate that its x<0 rather then x<=0, however since we know that the whole line has no breaks in it and the whole thing eventually joins up together, we won't use open circles for simplicity.

I hope this helps ! Absolute values are very similar to piecemeal functions .

 

[Leo.T]

oh lol K still doesnt explain why u subin 0 >< should u subin -1 -2? in stead?

whyy 0 tell me telll meeeE!! also also do u have to subin zero?

 

[undalay]

Well, if you sub in x=-1, and x=-4, you'll still have a gap between x=-1 and x=0.
You need 2 points for any line. Since x<0, can be any number infinitely close to 0, just not exactly 0.

But i guess you could subin x=-1, as long as you continued the line up to x=0.

:O? make sense ahah

 

[Leo.T]

hmmm THANK YOU!! EVERYMUCH