Moving and Flipping Graphs

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Moving and Flipping Graphs

If you know what y = x² looks like, what about y = (x + 3)² or y = −x²? These are the same curve, just moved or flipped! Understanding transformations means you can sketch any related graph instantly.
Three graphs and a worked example. Up and down: the curve y = x² with y = x² + 1 moved up 1 and y = x² − 1 moved down 1. Left and right: f(x) with f(x + 2) moved left 2, marked −2, and f(x − 2) moved right 2, marked 2. Reflections of f(x) = 2 to the power x: −f(x) is the reflection in the x axis and f(−x) is the reflection in the y axis. Worked example: y = x² becomes y = (x − 3)² + 2, which is right 3 for the part inside the brackets and up 2 for the part outside.

f(x) + a moves a graph up by a and f(x) − a moves it down by a. f(x + a) moves it left by a and f(x − a) moves it right by a. −f(x) reflects in the x axis and f(−x) reflects in the y axis. Golden rule: changes outside f(x) change the y values and behave as you expect, in the same direction. Changes inside f(x) change the x values and behave the opposite way to what you expect. In the worked example, (x − 3)² is inside the brackets, so the graph moves right, opposite to the minus sign, and + 2 is outside, so the graph moves up.

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