y = e − ( x − 3 ) . This graph has a reflection over the y-axis and is shifted right 3 units.Why right instead of left, though. Considering that this is equivalent to y = e ( − x + 3 ) I thought that the shift would be opposite of the sign being that it is in parentheses.Take y = e x + 3 for example. The graph of this function is shifted left 3 because of the parentheses.I want to supposed to figure out the shifts in the graphs. Am I missing a detail here?

Question

y  =      e          −      (      x      −      3      )      . This graph has a reflection over the y-axis and is shifted right 3 units.Why right instead of left, though. Considering that this is equivalent to   y  =      e          (      −      x      +      3      )       I thought that the shift would be opposite of the sign being that it is in parentheses.Take   y  =      e          x      +      3       for example. The graph of this function is shifted left 3 because of the parentheses.I want to supposed to figure out the shifts in the graphs. Am I missing a detail here?

Plutbantonavv · Accepted Answer

All you have written is correct. You only have to take care on the order of the transformations. For this, ask: &#039;What happens to x?&#039; and reverse the order and the operations.In the case of       e          −      (      x      −      3      )      , x is first decreased by 3, then multiplied by −1. If we reverse these operations, we see that first we have to reflect the graph of       e    x   along the y-axis and then shift it to the right by 3 (shift it to the left by −3).For the same       e          −      x      +      3      , we find that x is first multiplied by −1 then the gotten expression is increased by 3, so, reversing these, we first shift, indeed to the left, and then reflect.

Hunter Shah · Accepted Answer

Update:The transformation for       e          −      (      x      −      3      )       corresponds to the substitions: let u:=x−3. First, from   u  ↦      e    u   we go to   u  ↦      e          −      u       by reflecting the original graph on the y axis. Then making the substition   x  ↦  x  −  3 i.e.   x  ↦  u in the variable will give us the second step. You will be convinced if you plug in (enough) concrete values of x: e.g. if x=3 then u=0 and then       e          −      (      x      −      3      )        =      e          −      u        =  1. If x=4 then u=1, and so on..In general, the graph of g(x)=f(x−3) is shifted to the right (to the left by −3) compared to the graph of f(x), because  g  (  x  +  3  )  =  f  (  x  )    .

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