MHB Write down the equation of the function corresponding to the graph

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The equation of the function corresponding to the graph $y=(x-1)^2-6$ when translated 1 unit to the left is $y=x^2-6$. This is derived by substituting $x+1$ into the original function, resulting in $y((x+1)-1)^2-6$. The general rule for horizontal translations states that to shift a function left by $k$ units, one should use $f(x+k)$. The discussion clarifies the process of applying this translation and confirms the resulting equation. Understanding the derivation of this transformation is emphasized as crucial for grasping function translations.
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A graph in the form of $y=(x-1)^2-6$

Problem

Write down the equation of the function corresponding to the graph obtained when the above graph is translated 1 unit in the negative direction of the $x$ axis

Workings:

-

Where do I need help

In writing the equation of the funtion.
 
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If we are given a function $f(x)$, then to translate the graph of that function $k$ units to the left, we may use $f(x+k)$.

Can you proceed?
 
MarkFL said:
If we are given a function $f(x)$, then to translate the graph of that function $k$ units to the left, we may use $f(x+k)$.

Can you proceed?

$\displaystyle f(x)=a(x-h)^2+k$

The function is in this form.

Substituting the values,

$y=(x-1)^2-6$

k=-6

Is it $f(x-6)$? but I am not sure whether this is correct?
 
You are given:

$$y(x)=(x-1)^2-6$$

So, to translate this one unit to the left, use:

$$y(x+1)=((x+1)-1)^2-6=x^2-6$$
 
MarkFL said:
You are given:

$$y(x)=(x-1)^2-6$$

So, to translate this one unit to the left, use:

$$y(x+1)=((x+1)-1)^2-6=x^2-6$$

Thank you very much (Yes) , Now I see it in desmos

[graph]rqbf6pohnc[/graph]

MarkFL said:
If we are given a function $f(x)$, then to translate the graph of that function $k$ units to the left, we may use $f(x+k)$.

Can you proceed?

But what I don't still understand is it's derivation.

Now what has actually happened here $$y(x+1)=((x+1)-1)^2-6=x^2-6$$;

Using $f(x+k)$ Do you plug in the values for $f,x,k$ from the given function.

What does $f$ stand for & from which part of the function was a value taken.

A comment here would be highly appreciated :)
 
Last edited:
Suppose we are given a function $f(x)$ and we wish to shift the graph of that function horizontally. If we wish to shift the function to the left $k$ units, then we can think of accomplishing this by moving our coordinate axes horizontally to the right $k$ units, that is:

$$X=x+k$$

So now what we have is that:

$$f(X)=f(x+k)$$

is the graph of $f(x)$ shifted to the left by $k$ units. Now, in this problem, we are given the function:

$$y(x)=(x-1)^2-6$$

And we are told to shift it 1 unit to the left.

So, in terms of what I posted above, we have $y=f,\,k=1$ and so there results:

$$y(x+1)=((x+1)-1)^2-6=x^2-6$$
 
MarkFL said:
Suppose we are given a function $f(x)$ and we wish to shift the graph of that function horizontally. If we wish to shift the function to the left $k$ units, then we can think of accomplishing this by moving our coordinate axes horizontally to the right $k$ units, that is:

$$X=x+k$$

So now what we have is that:

$$f(X)=f(x+k)$$

is the graph of $f(x)$ shifted to the left by $k$ units. Now, in this problem, we are given the function:

$$y(x)=(x-1)^2-6$$

And we are told to shift it 1 unit to the left.

So, in terms of what I posted above, we have $y=f,\,k=1$ and so there results:

$$y(x+1)=((x+1)-1)^2-6=x^2-6$$

(Yes) Thank you very much again

To shift the function to the left y(x+1) and vice versa to shift the function right y(x-1)
 

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