You can also show this with the pythagorean theorem. Assuming that the distance is the hypotenuse, the leg lengths of the triangle are the distance perpendicular from the path the source takes, and the distance between the point when the source is closest to the observer and the source's current position. a^2+b^2=c^2 is the formula for the length of the hypotenuse of a right triangle with leg lengths a and b. You can use the function f(x)=sqrt((x-a)^2+b^2) to model the displacement from the wave source from the perspective of a person standing at "a" with "b" being the offset from the point where the source is directly in front of the observer, as you said. You'll notice it makes a hyperbola. Because this is the displacement

You can also show this with the Pythagorean theorem. Assuming that the distance is the hypotenuse, the leg lengths of the triangle are the distance perpendicular from the path the source takes, and the distance between the point when the source is closest to the observer and the source's current position.

The formula for the length of the hypotenuse of a right triangle with leg lengths $a$ and $b$ is $a^{2} + b^{2} = c^{2}$.

You can use the function $f \left( x \right) = \sqrt{\left( x - a \right)^{2} + b^{2}}$ to model the displacement from the wave source from the perspective of a person standing at $a$ with $b$ being the offset from the point where the source is directly in front of the observer, as you said. You'll notice it makes a hyperbola. Because this is the displacement.