WEBVTT
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Consider the following graphs.
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Which of these is the graph đť‘“ of đť‘Ą equals đť‘Ą plus one squared divided by đť‘Ą minus one times đť‘Ą plus three?
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So here we have a rational function.
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Itâ€™s a fraction, and rational functions have asymptotes.
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The denominators have to be nonzero.
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Because if you take anything and divide by zero, it will be undefined, so we donâ€™t want that!
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So what would make our denominator zero would be one and negative three.
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So how did we get that?
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We should take the two factors that are on the denominator and set them equal to zero.
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So now we need to add one to our left equation and subtract three from our right equation.
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So if we would plug in one for đť‘Ą, just looking at the denominator, we would get zero times negative two and zero times anything is zero.
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So if we plugged in one, that left expression would be zero, and if we plug in negative three, that right expression will be zero.
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But since you multiply them together, either one of those would give us a final answer of zero on our denominator, and again we donâ€™t want that.
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So here we can see every single graph actually has the asymptotes at negative three and one.
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So that doesnâ€™t eliminate any of our options.
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So our next step would be to see what the graphs are doing when theyâ€™re really close to these asymptotes, and then weâ€™ll know, so letâ€™s think about it.
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If we have two asymptotes at negative three and one, that means we will never actually be able to land on those values.
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It doesnâ€™t work for our function.
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So we can plug in numbers really close to them and see what happens.
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So we can plug in values to the left of negative three, so just a little bit more negative than negative three, so negative 3.01, negative 3.0009, and so on.
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And then we are also wanna plug in numbers to the right of that asymptote, so more positive, so negative 2.999, negative 2.9999999, and so on.
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And then the same thing for one, numbers are really close to one that are a little bit less than that and a little bit greater than that.
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So letâ€™s first begin with numbers to the left of our furthest left asymptote, đť‘Ą equals negative three.
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So for example negative 3.01, letâ€™s plug that in.
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So taking our function instead of đť‘Ą, weâ€™re replacing it with negative 3.01.
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And we get 100.75, so something pretty high up.
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So now letâ€™s plug in a number thatâ€™s even closer to đť‘Ą equals negative three.
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So instead of negative 3.01, how about đť‘Ą equals negative 3.001?
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And when we plug that in, we get 1000.75, so way higher than that!
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So it must be going up in the đť‘¦-direction, becoming much greater.
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So we can go ahead and exclude option a because just to the left of our asymptote of đť‘Ą equals negative three, it should be going up, increasing.
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So now letâ€™s look to the right of that asymptote, so we can go ahead and exclude option a because it should be increasing; the number should be coming pretty pretty large for đť‘¦ as we get really close to that asymptote of đť‘Ą equals negative three.
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So now letâ€™s plug in numbers to the right of it such as đť‘Ą equals negative 2.9 and đť‘Ą equals negative 2.99.
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So at đť‘Ą equals negative 2.9, we get a negative number, negative 9.26.
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So now letâ€™s plug in the number thatâ€™s even closer to that dash line, negative 2.99, and we get a number even more negative, negative 99.25.
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So this is decreasing as we approach đť‘Ą equals negative three from the right side of it.
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And all three of our remaining options do that; however, a few of them are a little bit higher than the others, so why donâ€™t we plug in negative one for đť‘Ą and see where we should land?
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So plugging in negative one, we get zero squared divided by negative two times two.
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And zero divided by negative four is zero.
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So at đť‘Ą equals negative one, we should be at zero for đť‘¦, which would mean our answer would have to be graph c.
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However just to double check, letâ€™s keep checking the values around đť‘Ą equals one for the asymptote and make sure.
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So the numbers on the left side of the asymptote đť‘Ą equals one or a little less than one, so letâ€™s plug in đť‘Ą equals 0.9 and 0.99.
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When we plug in 0.9, we get negative 9.26.
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And now weâ€™re plugging in a number a little bit closer to that asymptote đť‘Ą equals one, and we get negative 99.25.
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So we are decreasing again, which is whatâ€™s happening in our graph.
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So thatâ€™s good!
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So finally we will plug in numbers to the right of that asymptote đť‘Ą equals one, so little bit bigger than one.
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So we will plug in 1.01 and 1.001.
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When we plug in 1.01, we get 100.75, so a pretty big number.
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And now a number thatâ€™s closer to that line, 1.001, we get an even larger number, so it will be increasing, which is exactly whatâ€™s happening in this option of graph c.
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So this graph is what represents our function.