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Problem 17 Medium Difficulty

Starting with the graph of $ y = e^x $, write the equation of the graph that results from
(a) shifting 2 units downward.
(b) shifting 2 units to the right.
(c) reflecting about the x-axis.
(d) reflecting about the y-axis.
(e) reflecting about the x-axis and then about the y-axis.

Answer

(a) To find the equation of the graph that results from shifting the graph of $y=e^{x} 2$ units downward, we subtract 2 from the original function to get $y=e^{x}-2$
(b) To find the equation of the graph that results from shifting the graph of $y=e^{x} 2$ units to the right, we replace $x$ with $x-2$ in the original function to get $y=e^{(x-2)}$
(c) To find the equation of the graph that results from reflecting the graph of $y=e^{x}$ about the $x$ -axis, we multiply the original function by -1 to get $y=-e^{x}$
(d) To find the equation of the graph that results from reflecting the graph of $y=e^{x}$ about the $y$ -axis, we replace $x$ with $-x$ in the original function to get $y=e^{-x}$
(e) To find the equation of the graph that results from reflecting the graph of $y=e^{x}$ about the $x$ -axis and then about the $y$ -axis, we first multiply the original function by -1 (to get $y=-e^{x}$ ) and then replace $x$ with $-x$ in this equation to get $y=-e^{-x}$

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Video Transcript

okay, We're starting with y equals E to the X, and we're going to write the equation that results from each transformation. And these transformations work the same way. No matter what kind of function you're using, whether it's why equals e to the X or something triggered a metric or something polynomial the same things happen. So when you shift something down to you, subtract two from the equation. So we get why equals each of the X minus two for part B. When you shift something to the right to you end up subtracting two from X in the equation. So we get why equals e to the X minus two power part C. When you reflect something across the X axis, all the positive why values air made negative and all the negative y values air made positive. So we're taking the opposite of everything. So that looks like why equals the opposite of E to the x parte de When you reflect across the y axis, all your negative X values become positive and all your positive X values become negative. So you're taking the opposite of the exes, so you get y equals E to the opposite of X, And if you reflect across both axes, then both actions that we saw happen in part C and D will happen. So you'll have a negative in front of your E to the X, and you will also have a negative in front of your ex.