Parabolic Relationships
LetÕs begin with graphing the parabola:
f(x) = 2x2 +
3x – 4
Next, we want to observe what happens when we replace each x with (x-4).
The graph below shows the original function in purple and the new function in orange.
f(x) = 2x2+3x-4
f(x-4) = 2(x-4)2+3(x-4)-4
When x was replaced with (x-4) for every x then the graph remained congruent to the parent function is terms of the size and shape. This occurs because every single x was affected in the same manner. If every x was transformed in a different way then the new parabola would not have been similar to the original.
LetÕs manipulate the function to move the vertex of the graph into the second quadrant!
In order to make the parabolaÕs vertex in the second quadrant, I substituted every x for (x+4). And I also translated the function up, instead of down by making the constant positive.
y =2x2+3x-4
yÕ = 2(x+4)2+3(x+4)+4
LetÕs change the equation to produce a graph concave down BUT that shares the same vertex!
To make this function concave down but at the same vertex I began by reflecting the parabola. This ensures that the parabola would be concave down. Then I had to figure out how to make the two graphs share a vertex.
y = 2x2+3x-4
yÕÕ = -(2x2)-3x-6
LetÕs generalize!
When given the
equation y = a(x-h)2
+ b(x-k) +c , when a 
0, there are several general types of
transformations. Some of these include: horizontal translations, horizontal
scaling, vertical translations, and vertical scaling.
Horizontal translations will affect the functions domain. When a translation is acting as a horizontal shift then the number ÒhÓ will occur inside of the parentheses. This will affect the parent function as either a shift left or right. If you refer to the orange graph above, the transformation was (x-4), which shifted the function to the right four unit from the parent function. Whatever appears ÒinsideÓ the parentheses then the opposite shift occurs.
Next, letÕs consider vertical translations. This is given by the number Òc.Ó This number will move the function in a positive or negative direction along the y-axis, depending on the number.
Lastly, ÒaÓ is the vertical stretch. If this number is less than 0 then the parabola becomes concave down.