Transformations of Equations and Functions Section 1 3 Shifting 2 Consider the graph of the relation x y 2 1 the unit circle 2 2 If we shift the center to the point a b we have the relation x a y b 1 2 2 Example 1 The circle x 2 y 3 1 has center 2 3 The unit circle has been shifted 2 units to the right and 3 units down Rule To shift a graph to the right a units for a 0 replace x with x a in the relation To shift the graph to the left a units for a 0 replace x with x a in the relation Similarly To shift a graph b units up for b 0 replace y with y b To shift down b units for b 0 replace y with y b The same rules work in functions Example 2 y y 3 1 x 1 x 2 2 2 shifted to the right 2 units and down 3 units becomes which we write as y 2 1 x 2 3 Distortions Stretching Shrinking Let us transform the unit circle into an ellipse centered at 0 0 If the horizontal axis is the interval 2 2 and the vertical axis is 3 3 we are stretching horizontally by a factor of 2 and vertically by a 2 2 x y factor of 3 The equation of the ellipse is 1 2 3 Rule To stretch a graph horizontally by a factor of c for c 1 replace x by x in the relation or function c To stretch vertically by a factor of c for c 1 replace y by y c Shrinking is the same except c 1 in the relation or function Example 3 Stretch the function y x vertically by a factor of 3 and horizontally by a factor of 4 Write the new function as y f x Now shift this function to the right 5 units and down 6 units Write the new function as y g x Reflections To reflect the graph across the x axis replace y with y in the equation To reflect the graph across the y axis replace x with x in the equation Example 4 Sketch the graphs to see how reflections work 2 a y x Compositions y x 2 b y x 2 y 2 x f g x f g x means to substitute g x for the variable in the function f We perform the operation of g on x and then perform the operation f on the result Example 5 f u u g x 1 x 2 f g x 1 x 2 Example 6 Find g f x g f x for the functions of example 5 3 Example 7 Find f and g so that h x 2 sin x 4 f g x There are different possibilities
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