# Video: Finding the Asymptotes of a Hyperbola

The graph of equation 𝑦 = (𝑎𝑥 + 𝑏)/(𝑐𝑥 + 𝑑) is a hyperbola only if 𝑐 ≠ 0. In that case, what are the two asymptotes?

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### Video Transcript

The graph of equation 𝑦 is equal to 𝑎𝑥 plus 𝑏 over 𝑐𝑥 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 𝑥 equals 𝑘 when either the limit as 𝑥 approaches 𝑘 from below of 𝑦 is equal to double negative infinity or the limit as 𝑥 approaches 𝑘 from above of 𝑦 is equal to positive or negative infinity. Since 𝑦 is a rational function, this will happen at 𝑥 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 𝑐𝑥 plus 𝑑 is equal to zero for 𝑥. We obtained that there’s a vertical asymptote as 𝑥 is equal to negative 𝑑 over 𝑐.

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