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Transformations of a function worksheets

Change a function's rule and its graph moves. These free printable worksheets give the parent function and the new one and ask what happened — a shift, a stretch, a shrink or a reflection — with the whole effect readable from the rule and no graph paper needed.

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Transformations of a function
Transformations of a function worksheet — Shifts, stretches, reflections, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

Two questions settle every one of them. Is the change OUTSIDE the function, added to it or multiplying it? Then it acts on the output and the graph moves or stretches vertically, in the direction the number says. Is it INSIDE, done to x before the function sees it? Then it acts on the input and the graph moves sideways — against the sign.

Why this sheet works

That last part is the page. (x + 4)² moves the graph LEFT, and it is the fact students get wrong reliably, for years. The horizontal and vertical shifts are deliberately on the same sheet: x² + 4 goes up and (x + 4)² goes left, the same number and the same plus sign, so a reader cannot answer from the sign alone. The worked walk explains why rather than asserting it — the bracket is zero four steps earlier, and earlier along the x-axis is to the left.

How this one works

What happens to the graph?

g(x) = x² − 3—outside
Down3
  1. The subtraction is OUTSIDE the function: f is worked out first and then 3 is taken off. That changes the OUTPUT, which is the up-and-down direction.
  2. Every point comes out 3 lower than it did, so the whole graph moves down by 3.

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