By how much does y = 3(x-2) translate the line y=3x horizontally?

1 Answer
Jul 28, 2016

By 2 in the positive direction.

Explanation:

I'll first explain conceptually before giving the direct solution:
When a factor is added directly to the x of a function, that is, with parenthesis like you've shown above, it has the same effect as making every single input less by 2.

For example this means that when x = 0 for y = 3(x -2) it is the same as inputting x = -2 to y = 3x.

Naturally, this means that for the shifted function to have the same value as the unshifted one, x will need to be 2 more than the input of the unshifted function. This logic can be extended to any modification of x: it will always have the opposite effect on the shape of the function. A negative number results in a positive shift and visa-versa.

But to show this directly, consider the x-intercept of each function, the point where y = 0:

y = 3x
0 = 3x
x = 0

vs

y = 3(x-2) 0 = 3(x-2)
0 = 3x - 6
6 = 3x
x = 2

So before the shift, the y intercept was (0,0). Afterward it was (2,0). This shows us that our function had a shift of 2 in the positive direction!