The domain, ( − ∞, ∞ ) ( − ∞, ∞ ) remains unchanged.Observe the results of shifting f ( x ) = 2 x f ( x ) = 2 x vertically: Both vertical shifts are shown in Figure 5. For example, if we begin by graphing a parent function, f ( x ) = 2 x, f ( x ) = 2 x, we can then graph two vertical shifts alongside it, using d = 3 : d = 3 : the upward shift, g ( x ) = 2 x + 3 g ( x ) = 2 x + 3 and the downward shift, h ( x ) = 2 x − 3. The first transformation occurs when we add a constant d d to the parent function f ( x ) = b x, f ( x ) = b x, giving us a vertical shift d d units in the same direction as the sign. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied. Just as with other parent functions, we can apply the four types of transformations-shifts, reflections, stretches, and compressions-to the parent function f ( x ) = b x f ( x ) = b x without loss of shape. Transformations of exponential graphs behave similarly to those of other functions. Graphing Transformations of Exponential Functions Observe how the output values in Table 1 change as the input increases by 1. Recall the table of values for a function of the form f ( x ) = b x f ( x ) = b x whose base is greater than one. Graphing Exponential Functionsīefore we begin graphing, it is helpful to review the behavior of exponential growth. It gives us another layer of insight for predicting future events. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. Most of the time, however, the equation itself is not enough. Working with an equation that describes a real-world situation gives us a method for making predictions.
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