In order to understand the SlingShot Effect you have to make a distinction between a system in which 2 objects and 3 objects are involved
The SlingShot Effect does not happen when there are 2 objects involved.
The general path in this case for each object is an ellipse.
Each path has a point where the two objects are the closest.
The speed of each object the same distance away (before and after) from this
point is the same; there is no exchange of energy.
The path of each object is symmetrical.
The following sketch 1 shows this situation for two objects m and M (in a very simplified way)
m v1
x0 . . . x1 . . . x2>
v0 v2
t4 t4
....M....>
t0
The above sketch shows:
 The path followed by m (identified by the points x0,x1,x2 and 6 dots) and by M (identified by the point M and 8 dots).
 The shortest distance at t0 when m (at x1) and M .
 4 points to the right of x1 (including x2), representing the position of m at the moments t1, t2, t3 and t4.
 4 points to the right of M, representing the position of M at the moments t1, t2, t3 and t4.
 For the point to the left of x1 and M the same is true but then for the monents t1,t2,t3 and t4
 The speed v0 of m at x0 is the same as the speed v2 of m at x2.
 The speed of m at x1,t0 is v1, and the speed of M at t0 is V
For the SlingShot to happen at least 3 objects are involved.
The easiest system is one Sun and two planets. One planet should move in a circle (identified by *) and one in an ellipse (identified by +).
The following sketch shows this.
***
* * S = Object 1: Sun
* +++M+++ * = Object 2: large planet
* ++ * ++ + = Object 3: small planet or Satellite
* + S * + M = Meeting point
* ++ * ++
* +++M+++
* *
***
The important point of the above sketch is that the path of each object around the closest point M is not symmetrical.
During the whole path around the Sun the two planets will almost not influence each other, except if they become close.
Only then the path of object 3 will change in direction and speed, because a) the path of both around M is asymmetrical and b) the mass of object 3 is smal.
The SlingShot effect comes in two flavors: Speed up and Speed down.
Speed up means (This is the true SlingShot effect), that the speed of
object 3 has increased, the same distance away before and after a close encounter with object 2.
Speed down means, that the speed of object 3 has decreased.
For a good understanding it is important to know
 that in both cases (speed up and speed down) when object 3 approaches object 2 the speed of object 3 will increase first end then decrease. In the case of speed up the increase will be larger as the decrease, resulting in an overall increase. For speed down the reverse is true.
 that speed up for one object implies that the speed of the other object will decrease.
There has been an exchange of energy as described by Newton's Law.
The following Sketch 2 shows Speed up for m (dotted line):
x2 t4 >
. v2
.
t0 .
x1
t1
.
....M....>
.
.
x0
v0
t4
The above sketch shows the path followed by m and M at 9 different moments.
The shortest distance is at t0 when m is at x1 and M is at its drawn position.
The following Sketch 3 shows Speed down for m (dotted line):
<m
x2
v2 .
t4 .
. t0
x1
t1
.
....M....>
.
.
x0 m
v0
t4
The above sketch shows the path followed by m and M at 9 different moments. m goes from left to right
The shortest distance is at t0 when m is at x1 and M is at its drawn position.
In all the three sketches the initial speed v0 of m at t4 is identical.
The same is true for the speed V of M.
Comparing Sketch 2 with Sketch 3 shows that:
 going from t4 to t1 the path and speed increase of both will be almost identical.
 going from t1 to t0 in sketch 2 the path will be longer because also M moves away.
This will result in a longer and larger increase in speed.
 going from t1 to t0 in sketch 3 the path will be shorter because also M moves towards m. This will result in a shorter and smaller increase in speed.
 after t0 the the speed of both will decrease. However
 because for sketch 2 this will be shorter then the increase, the net result will be an increase.
 because for sketch 3 this will be longer then the increase, the net result will be a decrease.
For additional reading about SlingShot effect see the following articles:
 "Computer Recreations, How close encounters with star clusters are achieved with a computer telescope" by A.K. Dewdney. Scientific American January 1986, Page 1216. This article explains the theory of n body simulations.
 "Black Holes in Galactic Centers " by Martin J. Rees. Scientific American November 1990, Page 2633. This article explains an example of slingshot effect.
 "Mathematical Recreations, A Short Trek to Infinity " by Ian Stewart. Scientific American December 1991, Page 100102. This article explains slingshot effect.
Comment: The above sketches and description is not 100% sound and slightly more complex.

The speed of M in sketch 1 showing a true 2 object situation should be to the left.

Starting point of sketch 1, sketch 2 and sketch 3 is a straight line.
This is not correct.
The line should be bent and point x1 should be closer to M.