Question

Show why, for linear functions, a vertical translation is equivalent to a horizontal translation. For a linear function, what horizontal translation is equivalent to a vertical translation of 3 units up?

Answer 1

Explanation:

A) For the function ...

f(x) = mx + b

the vertical translation by k makes the function ...

g(x) = f(x) + k = mx + b + k

This can be rewritten as ...

g(x) = m(x +k/m) +b = f(x+k/m)

That is, the vertical translation by k is equivalent to a horizontal translation by -k/m, where m is the slope of the linear function.

___

B) For a vertical translation of 3, the equivalent horizontal translation is ...

-k/m for k=3

= -3/m . . . . . where m is the slope of the function

_____

Please note that there is no equivalent for m=0.

Answer 2

Answer:

Vertical is the same as horizontal for linear equations because when moving the function up, it moves the whole line left along the x-axis too. When moving it down, it moves the equation right on the x-axis.