Question

How do you graph `y=-4/(x+4)+3` using asymptotes, intercepts, end behavior?

Answer 1

Find the asymptotes first, at x=-4 and y=3. Because the first term is negative, the left side goes up and the right side down. The graph is below.

Explanation:

You cannot divide by zero. Ever!! Google "what happens when you divide by zero" to see the world imploding. Really, you can't divide anything into zero parts. So, looking at the first term, `-4/(x+4)`, we know that the function can't exist where `x=-4`, because that would put zero on the bottom of a fraction.
`x=-4` is the vertical asymptote.

The horizontal asymptote for the parent function, `y=1/x`, is at `y=0`, and this function has a vertical translation up 3 units. So the horizontal asymptote is at `y=3`.

End Behavior: The parent function also has the right part of the graph above the horizontal asymptote, above the left part. It's like a positive slope, where the graph goes up on the right and down on the left. The `x` in this function is negative, so it goes up on the left and down on the right.

The asymptotes are shown as dashed lines, and the function you asked about is in solid red: