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CCSS.Math: ,

we're told the graph of y equals 2 to the X is shown below so that's the graph it's an exponential function which of the following is the graph of y is equal to negative 1 times 2 to the x plus 3 plus 4 and when they give us 4 choices down here and before we even look closely at those choices let's just let's just think about what this would look like if it was transformed into that and you might notice that what we have here this Y that we want to find the graph of is a transformation of this original one how do we transform it well we've replaced X with X plus 3 then we multiplied that by negative 1 and then we add 4 so let's take it step by step so this is y equals two to the X what I want to do next is let's graph y is equal to 2 to the X plus 3 power well if you replace X with X plus 3 you're going to shift the graph to the left the left to the left by 3 and that might be a little bit counterintuitive but when we actually think about some points it'll hopefully make some sense here for example over in our original graph when X is equal to 0 Y is equal to 1 well how do we get y equal how do we get y equal 1 for our new graph for this thing right over here well to get y equals 1 here the exponent here still has to be 0 so that's going to happen at x equals negative 3 so that's going to happen at x equals negative 3 y is equal to 1 so notice we shifted to the left by 3 likewise in our original graph when X is 2 y is 4 well how do we get y equals 4 in our in this thing right over here well for y to be equal to 4 this exponent here needs to be equal to 2 and so for this exponent to be equal to 2 because 2 squared is 4 for this exponent to be equal to 2 X is going to be equal to negative 1 so when X is equal to negative 1 y is equal to 4 when X is equal to negative 1 Y is equal to 4 notice we shifted to the left by we shifted to the left by 3 and so this thing which wasn't isn't our final graph that we're looking for or it's gonna look something like it's going to look at something like something like like that which shifted its y2 equals to the X shifted to the left by three now let's figure out what the graph of now let's multiply this expression times negative one notice we're slowly building up to our goal so now let's figure out the graph of y is equal to negative one times two to the X plus three well here when y equals two to the X plus three if we multiply that times negative one whatever Y we had we're gonna have the negative of that so instead of when X is equal to negative three having positive one when x equals negative so you're gonna have negative one we multiplied by a negative one when X is equal to negative one instead of having four you're going to have negative four so our graph it's going to be flipped over its flipped over the x-axis and it's going to look something something something like this and this is not a perfect drawing but it'll give us a sense of things so then we can look at which of these graphs match up to that and then finally we want to add that four there so we want to figure out the graph of y equals negative one times two to the X plus three plus four so we want to take what we just had and shift it up by four so instead of this being a negative one right over here this is going to be a negative one plus four is three instead of this being a negative four negative four plus four is zero instead of our horizontal asymptote being at y equals zero our horizontal asymptotes going to be at y equals four so it's gonna look like let me draw I could do a better job than that our horizontal asymptote it's going to be right over there so our graph is going to look at something like you're gonna look at something like something like this we just shifted that red graph up by four shifted it up by shifted it up by four and we have a horizontal asymptote at y equals four so let's look at which of these choices match that so choice a right over here has a horizontal asymptote of y equals four but it is shifted on the horizontal direction in appropriately in fact it looks like it might have not been shifted to the left so we can rule this one out so let's rule that one out this one over here well this one just this one the this one approaches our asymptote as X increases so that's not right it should approach our asymptote as X decreases so you ruled that one out choice C looks like what we just graphed horizontal asymptote at x equals 4 when X is equals negative 3 y is equal to 3 that's what we got when X is equal to negative 1 y is equal to 0 so this looks right and you can even try those points out and so we like choice C and D is clearly off