# Thread: Transformation of Exponential Functions Rule

1. ## Transformation of Exponential Functions Rule

Please explain how to transform exponential functions
To graph function y=3^(x-1) we shift function y=3^x one unit to the right ( the opposite of -1). I understand this far. Now to graph y=3^(1-x) which is the same as y=3^(-x+1), we need to shift the graph of y=3^-x one unit to the right. My question is why to the right when the opposite of 1 is -1, why not to the left. The texbook does not explain, when I plug in the numbers in y=3^(-x+1) x=0, y=3, the shift should be to the right, but what is the rule?

Thank you

2. Originally Posted by oceanmd
Please explain how to transform exponential functions
To graph function y=3^(x-1) we shift function y=3^x one unit to the right ( the opposite of -1). I understand this far. Now to graph y=3^(1-x) which is the same as y=3^(-x+1), we need to shift the graph of y=3^-x one unit to the right. My question is why to the right when the opposite of 1 is -1, why not to the left. The texbook does not explain, when I plug in the numbers in y=3^(-x+1) x=0, y=3, the shift should be to the right, but what is the rule?

Thank you
Say we want to graph 3^(1-x). Which is 1 unit to the left of 3^(-x). But 3^(-x) is the same as 3^x but instead of going up from left to right it goes up from right to left. Draw that and you get 3^(-x) if you add 1 to get 3^(1-x) the graph is the same graph moved 1 unit to the left.

3. Say we want to graph 3^(1-x). Which is 1 unit to the left of 3^(-x).No, it's 1 unit to the right But 3^(-x) is the same as 3^x but instead of going up from left to right it goes up from right to left.Correct, or you can just reflect about y-axis. Draw that and you get 3^(-x) if you add 1 to get 3^(1-x) the graph is the same graph moved 1 unit to the left.No, it's the same graph moved 1 unit to the right, not to the left. You can check it using the graphing calculator. My question is: what is the general rule for shifting exponential functions with negative exponent?

4. General rule for shifting functions horizontally.

To shift a function (any function) y = f(x) h units to the left, we graph y = f(x +h). To shift a function y = f(x) h units to the right we graph y = f(x - h).

So to graph
$\displaystyle y = 3^{1 - x} = 3^{-x + 1} = 3^{-(x - 1)}$ we need to look at the graph of $\displaystyle y = 3^{-x}$ and shift it 1 unit to the right. So you are correct. You just have to put the argument in the correct form.

-Dan

5. Topsquark,

Thank you for putting the theory behind it.

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### how to shift an exponential graph to the right

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