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CCSS.Math:

let f of X equal negative x squared plus ax plus B over x squared plus CX Plus D where ABC and D are unknown constants which of the following is a possible graph of y is equal to f of X and it tells dashed lines indicate asymptotes so this is really interesting here and they give us four choices we see four from a three of them right now then if I scroll a little bit over you can see choice D and so I encourage you to pause the video and think about how we can figure it out because it is interesting because they haven't given us a lot of details they haven't given us what these coefficients or these constants are going to be all right now let's think about it so one thing we could think about is horizontal asymptotes so let's think about what happens as X approaches positive or negative infinity well as X so as as X approaches infinity or X approaches negative infinity f of X f of X is going to be approximately equal to well we're going to look at the highest degree terms because these are going to dominate as the magnitude of X the absolute value of x becomes very large so f of X is going to be approximately negative x squared over x squared which is equal to negative or we could another way to think about it this is the same thing as negative 1 so f of X is going to approach f of X is going to approach negative 1 in either direction as X approaches infinity or X approaches negative infinity so we have a horizontal asymptote at y equals negative 1 let's see choice a here it does look like they have a horizontal asymptote at y is equal to negative 1 right over there and we can verify that because each each hash mark is two we go from two to 0 to negative 2 to negative 4 so this does look like it's at negative 1 so just based only on the horizontal asymptote choice a looks good choice B we have a horizontal asymptote at y is equal to positive 2 so we can rule we can rule that out we know that our horizontal asymptote as X approaches positive or negative infinity is at negative 1 y equals negative 1 here our horizontal asymptote is at y is equal to 0 the graph approaches it approaches the x axis from either above or below so it's not the horizontal asymptote is not y equals negative 1 so we can rule that one out and then similarly over here our horizontal asymptote is not y equals negative 1 a horizontal asymptote is y is equal to 0 so we can rule that one out and that makes sense because really they only gave us enough information to figure out the horizontal asymptote they didn't give us enough information to figure out how many roots or what happens in the interval and all of those type of things how many zeros and all that because we don't know what the what the actual with the actual coefficients or constants of the quadratic are all we know is what happens as the x squared terms dominate this thing is going to approach negative 1 and so we picked choice choice a