The equation that we deal with when we translate parabolas is y=a(x-h)^2+k. When we deal with translating parabolas we learn the meaning of the variables h and k and how it can be used to determine before graphing a general idea on how the parabola will look like.
When you compare the two dolphins in the picture you could really tell the differences. The equation of the first parabola that is formed by the first dolphin has an equation that is y= -x^2. Whereas the second parabola that is formed by the second dolphin looks lower than the first one, the equation for this parabola would be y= -x^2-k (k can be any number).the reason why its -k is because the second dulphin is under water. In this case K is the vertical translation of a parabola. If k>0, then you move the parabola vertically upwards by k units. If k<0, than you move vertically downwards by k units. The equation that could represent a parabola being translated vertically downward could be something like y=x^2-3, and equation that could represent a parabola being translated vertically upwards could be something like y=x^2+3.
In the equation y=a(x-h)^2 h represent the horizontal translation. As you can see this image show dolphins jumping one after another. The first parabola that is formed has an equation of y= -x^2, where as the second parabola shows how the parabola has shifted to the right the equation of this parabola is y= -(x-h) ^2. When we are dealing with these kinds of equations like for example Y=(x+5)^2 the h value is 5, but on a graph the actual co-ordinate for that x-value is -5. The number is reversed. Therefore, if it’s +3 than on a graph the x co-ordinate is -3 or if it is -4 than on the graph it will be +4.
