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Function Transformations / Translations (page 3 of 4)

Sections: Basic rules, Additional rules, Moving the points, Working backwards from the graph

  • Given the following graph of f(x), graph the transformation f(x + 1) 3
    • graph of f(x)

    This transformation formula has just about everything: there's a left-shift of one (the "+1" inside), a move-down by three (the "3" outside), and a flip-upside-down (the "minus" sign out front). And, worse yet, I have no formula for f(x), so I can't cheat; I have to do the transformation.

    The way the original graph is drawn, there are a few points that I can use to keep track of things. If I move those points successfully, then I can draw the rest of the graph at the end.

    What I'm going to show you below is not what you would hand in. You would not show all this work in your homework, and certainly not on a test. But this displays the thinking that should be going through your head as you transform each point you've chosen.

    I'll do the points from left to right, starting with the point (3, 2).

point (–3, –2)

point moved to (–4, –2)

point moved to (–4, 2)

The first point I'll work with is the point (3, 2).

First, I shift the point left by one unit, to (4, 2).

Then I flip the point over the x-axis, up to (4, 2).


point moved to (–4, –1)

point (–2, –4)

point moved to (–3, –4)

Then I move the point down three units, to (4, 1).

Now I'll look at the second point marked on the graph, (2, 4).

I move the point back to the left by one unit, to (3, 4).


point moved to (–3, 4)

point moved to (–3, 1)

point (2, –3)

Flipping across the x-axis moves the point to (3, 4).

Moving the point three units down takes it to (3, 1).

The third point is at (2, 3).

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point moved to (1, –3)

point moved to (1, 3)

point moved to (1, 0)

Shifting left by one takes the point to (1, 3).

Flipping across the x-axis takes the point to (1, 3).

Moving down three units takes the point to (1, 0), on the axis.


point (4, 2)

point moved to (3, 2)

point moved to (3, –2)

The last point that I need to move is (4, 2).

Shifting the point one unit to the left takes it to (3, 2).

Flipping across the x-axis takes the point to (3, 2).


point moved to (3, –5)


graph of transformation of f(x)

Moving the point three units down takes it to (3, 5).


Now that I've moved all the points, I can graph the transformation.

If you're not sure of the order in which to do the transformation's steps, then stick to working from the inside out (from the argument, to the function, to anything done outside the function), like I did above. Pick a point to move, and trace out the movements with your pencil tip, drawing in the point once you reach the final location. Once you've moved all the points, draw in the transformation.

For the above transformation, I could have factored the minus sign out front of the expression and viewed the transformation as being [f(x + 1) + 3]. Then the point movements would have been "left one, up three, and then flip across the x-axis". The end result would have been the same.

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Cite this article as:

Stapel, Elizabeth. "Function Transformations / Translations: Moving the Points." Purplemath.
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