Transformations of graphs

Objective. Apply translations, stretches and reflections to known graphs.

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Learn: the key idea

Graph transformations change the position or shape of a curve y=f(x) using simple rules: y=f(x)+a shifts vertically by a, y=f(x+a) shifts horizontally by −a, y=af(x) stretches vertically by factor a, and y=f(ax) stretches horizontally by factor 1/a. Reflections occur with y=−f(x) (flip in the x-axis) and y=f(−x) (flip in the y-axis). For f(x)=x², the graph of f(x−2)+3 is the standard parabola moved 2 units right and 3 units up, with vertex at (2,3) instead of (0,0). Apply transformations to key points to track the new curve accurately.

Worked example

Describe the single transformation that maps y=x² onto y=(x−4)², including direction.

1 $Compare the bracket term: x becomes x-4$
2 $Recall that f(x-a) shifts the graph right by a$
3 $Here a=4, so the shift is in the positive x-direction$
4 $State the transformation as a translation 4 units to the right$

Answer: Translation 4 units in the positive x-direction

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Common mistake. Students shift y=f(x+3) three units right instead of left, confusing the sign; adding inside the bracket always moves the graph in the negative x-direction, so f(x+3) shifts left by 3.

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