What are the two asymptotes of the hyperbola 𝑦 equals eight over four 𝑥 minus three plus five-thirds?
When we look at this function, we know that the two asymptotes will be places where there are no valid solutions — where the function doesn’t exist. From what we know about algebra, we know that we cannot divide by zero, which means this bottom term four 𝑥 minus three cannot be equal to zero.
At zero, there will be an asymptote. So we set four 𝑥 minus three equal to zero and we solve for 𝑥. We add three to both sides. Now, we have four 𝑥 equal to three. Divide both sides by four. The fours cancel out, leaving us with 𝑥 equal to three-fourths. At 𝑥 equal to three-fourths, there’s no solution to this equation. So there’s an asymptote there.
Something else we can know based on the principles of algebra is that this fraction will never be equal to zero. No matter what we input for 𝑥, eight divided by four 𝑥 minus three cannot equal zero. By setting that whole term equal to zero and then bringing down our five-thirds, we find our second asymptote. There is no solution to this equation at 𝑦 equals five-thirds.
Therefore, our asymptotes would fall at 𝑥 equals three-fourths and 𝑦 equals five-thirds.