Video: Finding the Asymptotes of a Hyperbola

The graph of equation 𝑦 = (π‘Žπ‘₯ + 𝑏)/(𝑐π‘₯ + 𝑑) is a hyperbola only if 𝑐 β‰  0. In that case, what are the two asymptotes?

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Video Transcript

The graph of equation 𝑦 is equal to π‘Žπ‘₯ plus 𝑏 over 𝑐π‘₯ plus 𝑑 is a hyperbola only if 𝑐 is not on zero. In that case, what are the two asymptotes?

Let’s start by finding the vertical asymptote of this function. Vertical asymptotes occur at π‘₯ equals π‘˜ when either the limit as π‘₯ approaches π‘˜ from below of 𝑦 is equal to double negative infinity or the limit as π‘₯ approaches π‘˜ from above of 𝑦 is equal to positive or negative infinity. Since 𝑦 is a rational function, this will happen at π‘₯ values, where the denominator of 𝑦 is equal to zero. Therefore, we can find our vertical asymptotes by setting the denominator of 𝑦 equal to zero. Now, we solve 𝑐π‘₯ plus 𝑑 is equal to zero for π‘₯. We obtained that there’s a vertical asymptote as π‘₯ is equal to negative 𝑑 over 𝑐.

In order to find the horizontal asymptote of 𝑦, we need to consider the limit as π‘₯ approaches positive or negative infinity of 𝑦. In order to find this limit, we can multiply the numerator and denominator by one over π‘₯. Then we use the fact that the limit as π‘₯ approaches infinity of one over π‘₯ is equal to zero, in order to say that when we take this limit, 𝑏 over π‘₯ and 𝑑 over π‘₯ will both tend to zero. And this leaves us with π‘Ž over 𝑐. Now, we have found our horizontal asymptote. So we have found that the two asymptotes of our hyperbola are π‘₯ is equal to negative 𝑑 over 𝑐 and 𝑦 is equal to π‘Ž over 𝑐.

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