Sean C. answered • 02/24/20
SAT, Math, Physics, ESL, Writing Tutor
Mallory,
What we have here is a shifted inverse function. Don't let that confuse you, that just means it's an inverse function - something of the form y = a/x (where a is some number) - that is shifted vertically or horizontally. In this case, it's both, as shown by the x + 7 in the denominator (horizontal shift) and the - 8 stuck on to the end (vertical shift).
When you get used to them, these will be simple to figure out. Let's start by figuring out the asymptotes of the most basic inverse function, y = 1/x. After this, you should just remember it.
To figure out the asymptotes, you want to ask yourself: Are there any values of x or y where this function cannot go? These values of x and y will be asymptotes or holes, and there are different ways it can happen. One common one is when the denominator equals 0. For this equation, this happens when x = 0. But is it an asymptote or a hole? Well, as x gets smaller and smaller - 0.1, 0.01, 0.001, and so on - y will become bigger and bigger- it will approach infinity. There is no limit to what it can reach. x isn't allowed to be 0 because it's not in the domain (because it would make the denominator equal 0), so this means that there is one vertical asymptote at x = 0. In quadrant 1, the graph will extend upwards forever, moving closer and closer to the y axis but never touching it. In quadrant 3, it will extend downwards forever.
Another helpful question to ask is, What happens when x goes to positive or negative infinity? Imagine x gets bigger and bigger. What happens to y? When we divide any number by a bigger and bigger number, we get a smaller and smaller answer- but it will never be 0. What that means for the equation y = 1/x is that y can never reach 0. The graph in quadrants 1 and 3 will extend outwards forever, moving closer and closer to the x axis but never touching it. (You can google the function to see a picture.) So, the asymptotes for y = 1/x are x = 0 (vertical) and y = 0 (horizontal), making a cross. It'll be helpful for you to remember this from now on.
So, what do the shifts of our function f(x) do to the asymptotes? They shift them! The horizontal shift (x + 7) will move the VERTICAL asymptote to the left by 7 so that it becomes x = -7, and the - 8 means the HORIZONTAL asymptote will move down 8 to become y = -8. If you need to, review shifting. The fact that f(x) has a 5 instead of a 1 in the numerator doesn't change anything about the asymptotes or general shape of the function.
Hope that helps!
