Math Problem?

Question

If the graph of y = (ax + b) / (x + c) has a horizontal asymptote y = 2 and a vertical asymptote x = -3, then a + c = ?

Be precise with steps, please.

Answer 1

First of all, it helps to sketch out the axes and the asymptotes, even if you're not sure what the graph is going to look like.

Now, let's consider y = (ax+b) / (x+c)

Let's take the easier-to-visualise asymptote (for me at least) first, the vertical asymptote at x = -3. This means that as x approaches -3, y approaches infinity, or negative infinity. In this sort of graph, this will always happen as the denominator of y approaches 0 (since when a fraction's denominator is 0, y is technically infinity). So, given that knowledge, we know that at x = -3, the denominator (x+c) is 0. So that's easy, c = 3.

Now for the horizontal asymptote, it helps to break the fraction up. So instead of
y = (ax+b) / (x+c) , we can write
y = (ax) / (x+c) + b / (x+c)

Now, since y varies based on x, and not the other way around, we can't go about identifying the horizontal asymptote by asking "When does x approach infinity?". So instead, we ask "What happens when x approaches infinity?" Looking at our equation above, as x approaches infinity, what happens?

Take it term by term. Let's start with b / (x+c). As x approaches infinity, what happens? That's easy, b / (x+c) approaches 0 due to the huge denominator.

Now, what happens for (ax) / (x+c)? Think of it as a[ x / (x+c)]. [x / (x+c)] will approach 1 as x approaches infinity, as the c in the denominator becomes more and more insignificant (if x is 1, [1 / (1+3)] is nowhere near 1. But as x gets bigger, e.g. [99999 / (99999+3)], it gets pretty darn close to 1). So as x approaches infinity, a[x / (x+c)] approaches a(1) = a.

So now we combine the terms together - as x approaches infinity, y approaches a(1) + 0 = a. Therefore, a is 2, which is what y approaches as x tends to infinity!

Alright, so a = 2, and c = 3. Go figure.

I hope my answer wasn't too convoluted. Anyway, in general, for math questions involving graphs, intercepts, asymptotes, tangents etc. I find Graphmatica to be an absolutely awesome tool. Just type in the equation and poof! you get a graph. Playing around with the equations in Graphmatica will help your understanding of graphs tremendously.

Answer 2

The numerator and denominator are both of the same degree so a horizontal asymptote will occur. It will occur y= coefficient of x in numerator / the coefficient of x denominator.
Since y=2 , then a/1 = 2 so a =2.

The vertical asymptote occurs when the denominator equals 0. Thus since x=-3 c must = 3 to make denominator 0.

So a+c = 2+3=5

Answer 3

todays 420