Does Shifting a Linear Equation Change its Points on the Graph?

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In summary: It's a "fake shift". That's why there isn't a shift. In summary, the function g(x), which is identical to f(x+2)+5, does not actually shift the curve of f(x) as it may seem. The constants a and b in g(x) are engineered in such a way that the linear shift specified in g(x) does not produce any different outputs from f(x). Therefore, while there is technically a shift, it does not actually shift anything.
  • #1
primarygun
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I am confused of shifting a linear equation.
Let f(x)=ax+b
And g(x) is identical to f(x+2)+5
For example, we create a specific condition, g(x)=f(x) and (1,2) is a point on f(x) [Does this implies that (1,2) is also a point on g(x)?]
Next step is to find f(x): By using the given conditions, f(x)= -5x/2+9/2


The contradiction appears: g(x)=f(x+2)+5
That's mean shifting the whole curve of f(x) to left parallel to x-axis by 2 units, then by shifting it upwards by 5 units, we get g(x).
My answer to the previous question ( typed in bold ) is yes but I am not certain with my answer. If I am correct, then the point hasn't moved away.
However, it's clear to know that the shifting must move the point upward DUE TO A VECTOR NATURE.
My contradiction is here, anyone helps me solve it?
 
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  • #2
You haven't bolded anything, but I'll try to help you out.

If g(x) is f(x+a)+b, then it isn't necessary for them to intersect at all. It may happen though. I think this is what your question was:

If

[tex] f(x) = \frac{-5}{2}x +\frac{9}{2} [/tex] then

[tex] g(x) = f(x+2)+5 = \left(\frac{-5}{2}(x+2) + \frac{9}{2}\right) + 5 [/tex]

The large brackets surround f(x+2).
 
  • #3
You say:
"Let f(x)=ax+b
And g(x) is identical to f(x+2)+5"

Given that, what does "For example, we create a specific condition,
g(x)=f(x)" mean?
 
  • #4
whozum said:
You haven't bolded anything, but I'll try to help you out.

If g(x) is f(x+a)+b, then it isn't necessary for them to intersect at all. It may happen though. I think this is what your question was:

If

[tex] f(x) = \frac{-5}{2}x +\frac{9}{2} [/tex] then

[tex] g(x) = f(x+2)+5 = \left(\frac{-5}{2}(x+2) + \frac{9}{2}\right) + 5 [/tex]

The large brackets surround f(x+2).
[Does this implies that (1,2) is also a point on g(x)?]
This is the bolded question. Sorry, that's not the solution.
Once the function f(x) is changed to g(x)=f(x+2)+5,
that means it shifts left by 2 and upwards by 5.
Obviously, the new point is not the original point as it has moved upward along the original line.
This is a general case for all linear equations.
However, as I mentioned, if I let f(x)=g(x), moreover, (1,2) lies on the equation,...then f(1)=2 ---> g(1)=2
That means the point didn't move.
How to explain the italic sentence?
My solution: The line is shifted 2 units leftwards and then 2 units rightwards for that specific equation.
 
  • #5
primarygun said:
[Does this implies that (1,2) is also a point on g(x)?]
This is the bolded question. Sorry, that's not the solution.
Once the function f(x) is changed to g(x)=f(x+2)+5,
that means it shifts left by 2 and upwards by 5.
Obviously, the new point is not the original point as it has moved upward along the original line.
(1,2) is a point on f(x), and
[tex]g(x) = \left(\frac{-5}{2}(x+2) + \frac{9}{2}\right) + 5 [/tex]

[tex] g(1) = 2 [/tex]

We're all dandy up til here.

However, as I mentioned, if I let f(x)=g(x), moreover, (1,2) lies on the equation,...then f(1)=2 ---> g(1)=2
That means the point didn't move.
How to explain the italic sentence?
My solution: The line is shifted 2 units leftwards and then 2 units rightwards for that specific equation.

I really don't know what you mean by this part, but if you notice, f(x) = g(x) as it is. Simplify g(x), you'll find that it already = f(x), the two lines are incident and equivalent. Thus any point on f(x) is a point on g(x).


If you give me a specific question, I can address it for you, other than that, I really don't know what you are lookin for.
 
  • #6
Is the graph shifted? If it is shifted, can you tell me how the shift is?
 
  • #7
The graph is not shifted. It is just written in a form where it appears so.
 
  • #8
But g(x)=f(x+2)+5? If the condition [g(x)=f(x)] is not told, everyone will shift the graph.
I'm really confused here, please explain it to me.
 
  • #9
primarygun said:
But g(x)=f(x+2)+5? If the condition [g(x)=f(x)] is not told, everyone will shift the graph.
I'm really confused here, please explain it to me.

But the condition was told.

I have to admit, I really don't understand the original problem statement so I could be totally off base but this is what I think is happening.

I think the original problem statement is this:

Given:
f(x)=ax+b
g(x)=f(x+2)+5

Find constants a and b such that f(x) = g(x) and f(1)=2.

And you did this.

So, without the condition [f(x)=g(x)] you could not solve the problem and because of the condition the linear shift does not matter. You can think of it as engineering the constants a and b in such a way that the particular linear shift specified by g(x) doesn't produce outputs different from f(x).

Does this seem right to anybody else?
 
  • #10
egsmith, that's exactly what is going on here. I just don't know how to explain the general "g(x) f(x-a)+b then shift f(x) to the right a and up b" in this context. The general rule applies and the shift occurs, but the final result is equivalent to the original result.
 
  • #11
what on Earth is the problem?


if f(x)=ax+b, and g(x)=f(x+2)+5 then g(x)=a(x+2)+b+5 = ax+2a+b+5, now just get on with what ever it is you want to do with it and stop mucking around.
 
  • #12
He doesn't understand why there isn't a shift.
 
  • #13
He doesn't understand why there isn't a shift.
Only whozum knows what my problem is.
Why isn't there a shift? I've proved it mathematically with evidence. I really need to be "solved".
 
  • #14
Look at the constant 'a' that you solved for and look at g(x). You sould see some similarities. Basically g(x) shifts f(x) on top of itself so that you can't tell the difference between f(x) and the function after the shift by design.

There is a shift but it doesn't matter.
 
  • #15
So is the original point (1,2) arrives (1,2) after the shift?
 
  • #16
Yes, f(1)=2 g(1)=2 by design and as a consequence of the given condition f(x)=g(x).
Maybe this would help you.
Graph f(x) with a=-5/2 and b=9/2
On the same graph, perhaps in a different color, graph f(x+2)
In a third color, graph f(x+2)+5.
Hopefully this will show you that the shift does exist. It is just that you end up back to where you started because this is what the problem asked you to do ( when it said let f(x)=g(x) ).
 
  • #17
But, if the point arrive at the original point, then, the shift is not just horizontal or vertical.
 
  • #18
primarygun said:
So is the original point (1,2) arrives (1,2) after the shift?

If I'm interpreting your question properly, then no. If you shift (1,2) left 2 units and up 5 then it ends up at (-1,7), which you'll see is still on the line.

This shouldn't be suprising, for any line y=mx+b where m is non zero, a horizontal shift left by k units yields the line y=m(x+k)+b=y=mx+(mk+b). Notice this ls just the original line shifted up mk units. In particular, if m is non zero then any horizontal shift is equivalent to some vertical shift.

In the case at hand, knowing that going 2 left and 5 up gets the line back to where we started tells us that m=-5/2, since it means a shift left by 2 gives the same line as a shift down by 5 (so k=2, mk=-5 in the last paragraph). The extra condition f(1)=2 is what gave the b constant of 9/2.
 
  • #19
If I'm interpreting your question properly, then no. If you shift (1,2) left 2 units and up 5 then it ends up at (-1,7), which you'll see is still on the line.
But, the prove is :
f(x+2)+5=f(x)
Just let f(u+2)=f(v) first
Note: if u=v and f(u+2)=f(v), then the points are at the same position.-----1
But, there was actually two shifts of the original function graph-----------2
The problem is here, if there was really a shift and shift 2 units left and 5 units upwards. With given the equation as I mentioned before, the point obviously
arrives at a higher position.
so,1 and 2, one of them must be wrong. If both are correct, then the shift is not just horizontally and vertically.
Someone please tell me where my mistake is.
 
  • #20
primarygun said:
Only whozum knows what my problem is.
Why isn't there a shift? I've proved it mathematically with evidence. I really need to be "solved".


You haven't proved there is a problem. You only think there is a problem.

We have functions f and g such that f(x+2)+5=g, and such that f=g. The problem is that you think that in order to find, say, g(1) you need to shift f of 1 along 2 and up 5. That isn't true, you need to shove f(3) up 5, that is you shift x along two THEN move f(x) up 5. Which isn't the same thing as what you're claiming.
 
  • #21
matt grime said:
You haven't proved there is a problem. You only think there is a problem.

We have functions f and g such that f(x+2)+5=g, and such that f=g. The problem is that you think that in order to find, say, g(1) you need to shift f of 1 along 2 and up 5. That isn't true, you need to shove f(3) up 5, that is you shift x along two THEN move f(x) up 5. Which isn't the same thing as what you're claiming.

f(3)+5=g(1) Yes this is correct.
But, y=f(x)
If f(u)=g(u), then the final point reaches the original point exactly.
Now f(3), 3 is not the x-coordinate, it is 1 as f(1+2)+5,
i.e. f(1)=g(1).
 
  • #22
What the...? Should I give up?

f(1)=g(1), yes, and f(3)=g(3) and f(1)=g(1)=f(3)+5=g(3)+5


So what?

The problem is you seem to think that to find g(1) I plot f, then find f(1), shift 2 along the x-axis and then 5 up the y axis, which is what you're describing, ie you think g(1)= f(1)+5, when we know g(1)=f(3)+5


This shifitng busniess is all well and good for plotting the graphs on the same axis, however the y-coordinate when x=1 on the graph of g is equal to the (y coordinate of f at x=3) plus 5.

There is no issue there.
 
Last edited:
  • #23
Ok Thanks dear.
f(x+a)=g(x),
The points should be (x+a,y) , (x,y).
sorry for wasting you scientists' time, sorry.
 

What is graphical linear shifting?

Graphical linear shifting is a mathematical technique used to transform a graph by shifting it horizontally or vertically. It involves adding or subtracting a fixed value to each point on the original graph in order to create a new graph.

What is the purpose of graphical linear shifting?

The purpose of graphical linear shifting is to manipulate the position of a graph in order to make it easier to analyze or compare with other graphs. It can also be used to represent real-world data more accurately.

What are the steps involved in graphical linear shifting?

The steps involved in graphical linear shifting are:

  • Identifying the original graph and the desired transformation (horizontal or vertical shifting).
  • Determining the amount of shift (represented by a value, k).
  • Applying the formula y = f(x) +/- k to each point on the original graph to create a new graph.

Can graphical linear shifting be used for non-linear functions?

No, graphical linear shifting can only be used for linear functions. Non-linear functions cannot be transformed using a single fixed value like linear functions can.

How can graphical linear shifting be applied in real-world situations?

Graphical linear shifting can be used to adjust graphs that represent real-world data, such as temperature over time or population growth. It can also be used to analyze trends or compare data sets more accurately.

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