Chapter 1 | Functions and Graphs
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functions. In the previous example, for instance, we subtracted 2 from the argument of the function y = x 2 to get the function f ( x ) = ( x −2) 2 . This subtraction represents a shift of the function y = x 2 two units to the right. A shift, horizontally or vertically, is a type of transformation of a function . Other transformations include horizontal and vertical scalings, and reflections about the axes. A vertical shift of a function occurs if we add or subtract the same constant to each output y . For c >0, the graph of f ( x )+ c is a shift of the graph of f ( x ) up c units, whereas the graph of f ( x )− c is a shift of the graph of f ( x ) down c units. For example, the graph of the function f ( x ) = x 3 +4 is the graph of y = x 3 shiftedup 4 units; the graph of the function f ( x ) = x 3 −4 is the graph of y = x 3 shifted down 4 units ( Figure 1.23 ).
Figure 1.23 (a) For c >0, the graph of y = f ( x )+ c is a vertical shift up c units of the graph of y = f ( x ). (b) For c >0, the graph of y = f ( x )− c is a vertical shift down c units of the graph of y = f ( x ).
A horizontal shift of a function occurs if we add or subtract the same constant to each input x . For c >0, the graph of f ( x + c ) is a shift of the graph of f ( x ) to the left c units; the graph of f ( x − c ) is a shift of the graph of f ( x ) to the right c units. Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let’s look at an example. Consider the function f ( x ) = | x +3 | and evaluate this function at x −3. Since f ( x −3) = | x | and x −3< x , thegraph of f ( x ) = | x +3 | is the graph of y = | x | shifted left 3 units. Similarly, the graph of f ( x ) = | x −3 | is the graph of y = | x | shifted right 3 units ( Figure 1.24 ).
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