What happens to the graph of a line when you add or subtract a constant to/from x

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embizze | High School Teacher | (Level 1) Educator Emeritus

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What happens to the graph of a line when you add/subtract a constant to/from x?

(1) If the line is vertical it is of the form x=a. If you add a constant h to x we get:

(x-h)=a ==> x=a+h. This results in the graph of the vertical line moving left/right. (Left if h<0, right if h>0.)

(2) If the line is not vertical we can write its equation as y=mx+b.

Then y=m(x-h)+b. We can use the distributive property to rewrite this:

y=m(x-h)+b

  =mx-mh+b

  =mx+(b-mh)

Thus the resulting line is parallel to the original line with a horizontal translation (shift left/right) of h units.

For example given the line y=2x+3; suppose we add 4 to x.

Then we get y=2(x+4)+3=2x+8+3=2x+11. This line is parallel to y=2x+3 but shifted 4 units to the left.

Note that this is equivalent to a vertical translation (shift up/down) of mh units -- in the example a shift of 4 units left is the same as a shift of 8 units up. This is true for lines.

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In general, given a function f(x), the transformed function f(x-h) will have a graph that is the same as f(x) but translated horizontally h units. For example if we replace `f(x)=x^3` with `f(x+3)=(x+3)^3` , the graphs will be the same but the transformed graph has been translated 3 units left.

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