Graph transformations

Objective. Describe translations and reflections of y=f(x).

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Learn: the key idea

Graph transformations change the position or shape of y=f(x) using simple rules applied to the function or its input. y=f(x)+a translates the graph vertically by a, y=f(x+a) translates it horizontally by −a, y=−f(x) reflects in the x-axis, and y=f(−x) reflects in the y-axis. For example, if f(x)=x², then y=f(x)−3 shifts the parabola down 3 units, and y=f(x−2) shifts it right 2 units. It helps to track one key point, such as the vertex or an intercept, through the transformation to check the new graph's position. Always state both the type of transformation and its direction and size clearly.

Worked example

The graph of y=f(x) has a minimum point at (3, −2). State the coordinates of the minimum point on y=f(x)+5.

1 $The transformation y=f(x)+5 is a vertical translation by 5 units up$
2 $The x-coordinate of the minimum is unchanged: x = 3$
3 $The y-coordinate increases by 5: -2 + 5 = 3$
4 $New minimum point is (3, 3)$

Answer: (3, 3)

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Common mistake. Pupils think y=f(x+2) shifts the graph right by 2, when adding inside the bracket actually shifts it left by 2. For f(x)=x², y=f(x+2)=(x+2)² has its vertex at x=−2, not x=2. Always remember the horizontal shift is opposite to the sign inside the bracket.

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