Linear graph transformation

[AvgStudent]

Hello, I'm reviewing graph transformation and I stumbled on this question. I'm looking for why my method is incorrect.
Question: If $f(x) =3x+4$, then state the transformation for $f(x+3)$.
The answer is obvious it shifted to the left 3, however, I did it a bit differently.
If $f(x) =3x+4$, then $f(x+3)=3(x+3)+4= 3x+4+9=f(x)+9$. So I said shifted up 9.
Why is this wrong?

 

[Cubist]

Try plotting the original line. Then draw it shifted left by 3. Then draw it shifted up by 9. What do you notice?

 

[AvgStudent]

Try plotting the original line. Then draw it shifted left by 3. Then draw it shifted up by 9. What do you notice?

Oh, they're the same line which makes sense because they're algebraically equivalent. Still, why is shifting left 3 is same as shifting up 9?
Are both answers correct?

 

[The Highlander]

Hello, I'm reviewing graph transformation and I stumbled on this question. I'm looking for why my method is incorrect.
Question: If $f(x) =3x+4$, then state the transformation for $f(x+3)$.
The answer is obvious it shifted to the left 3, however, I did it a bit differently.
If $f(x) =3x+4$, then $f(x+3)=3(x+3)+4= 3x+4+9=f(x)+9$. So I said shifted up 9.
Why is this wrong?

It's not "wrong"; they are just the same lines (ie similar translations).
Go here and click on the green (wavy) button to turn the green line "on" & "off". ?

 

[Steven G]

Oh, they're the same line which makes sense because they're algebraically equivalent. Still, why is shifting left 3 is same as shifting up 9?
Are both answers correct?

If they are the same line, then both answers are the correct.
Some transformations can be done in different ways. In your case, shifting 3 to the right or shifting up 9 yields the same line.

why is shifting left 3 is same as shifting up 9?
Because you proved it!!!

 

[Steven G]

Suppose f(x) = mx+b
Then f(x+a) = m(x+a) + b = mx + b + ma = f(x) + ma.
This means shifting a line a units to the left is the same as shifting the line up ma units. I never knew this!!

 

[Steven G]

If two lines are parallel, it makes sense that moving one of the line to the left (or right) OR up (or down) will get you the other line.

 

[topsquark]

Hello, I'm reviewing graph transformation and I stumbled on this question. I'm looking for why my method is incorrect.
Question: If $f(x) =3x+4$, then state the transformation for $f(x+3)$.
The answer is obvious it shifted to the left 3, however, I did it a bit differently.
If $f(x) =3x+4$, then $f(x+3)=3(x+3)+4= 3x+4+9=f(x)+9$. So I said shifted up 9.
Why is this wrong?

Does your answer key say it's wrong? Either way f(x+3) = 3x + 13 no matter how you describe it.

-Dan

 

[Dr.Peterson]

Hello, I'm reviewing graph transformation and I stumbled on this question. I'm looking for why my method is incorrect.
Question: If $f(x) =3x+4$, then state the transformation for $f(x+3)$.
The answer is obvious it shifted to the left 3, however, I did it a bit differently.
If $f(x) =3x+4$, then $f(x+3)=3(x+3)+4= 3x+4+9=f(x)+9$. So I said shifted up 9.
Why is this wrong?

I'm not happy with the problem.

If they are asking what transformation of the given line is represented by f(x+3), then the only answer is, shifting left by 3 units. That is the transformation you are actually carrying out.

If they are asking for the transformed line (that is, the result of the transformation), then the answer is y = 3x+13.

That line can also be obtained by other transformations, such as shifting up by 9 units; but that transformation is a different transformation of the plane, and transforms a point on the original line to a different point on the new line. So I would not say that the transformation they are asking about is a shift up by 9. Rather, it is equivalent, as far as this particular line is concerned.

Did you quote the exact wording? And were you explicitly told that your answer was wrong?

 

[Steven G]

AvgStudent said the transformation was obviously a shift to the left 3 units.

Then after simplifying f(x+3), noticed that f(x+3) = f(x) + 9. At this point AvgStudent asked what he did wrong.

 

[AvgStudent]

Suppose f(x) = mx+b
Then f(x+a) = m(x+a) + b = mx + b + ma = f(x) + ma.
This means shifting a line a units to the left is the same as shifting the line up ma units. I never knew this!!

I was considering a point and its transformation. For example, if we have a point at (0,4), so shifting left 3 would yield (-3,4) and shifting up 9 would yield (0,13). Clearly, the 2 transformations gave 2 different coordinates. However, I realized now that I have to consider the whole line and not just a single point. The line connecting those points yields the exact same line as shifting the function shifting left 3 or shifting up 9.

I'm not happy with the problem.

If they are asking what transformation of the given line is represented by f(x+3), then the only answer is, shifting left by 3 units. That is the transformation you are actually carrying out.

If they are asking for the transformed line (that is, the result of the transformation), then the answer is y = 3x+13.

That line can also be obtained by other transformations, such as shifting up by 9 units; but that transformation is a different transformation of the plane, and transforms a point on the original line to a different point on the new line. So I would not say that the transformation they are asking about is a shift up by 9. Rather, it is equivalent, as far as this particular line is concerned.

Did you quote the exact wording? And were you explicitly told that your answer was wrong?

This answer was in the back of my book. The problem states "transformation" and the answer is "shifting left 3". Based on your answer, they're correct. I've never made the distinction between "transformation" vs. "transformed line" so that's why it troubled me. Your answer cleared that.

Thank you all.