Parabola with Point D FINAL
The Parabola and Point D
Using the equation for the parabola f(x)=a((x-p)^2)-q we see a parabola with a vertex of (p,q). This parabola has concavity and when 'a' is positive in the function the graph is concave up with the shape of the graph is upward. When 'a' is negative in the function it is concave down and the shape of it is downward.
When we consider point D at an x-coordinate of -1 and find the line tangnet to it, we are referring to an instantaneous rate of change that happened on f(x) specifically at point D. In addition no matter how much you vary the origins of the graph (altering p,q) or change the concavity of it (altering 'a') the line tangent to point D will always yield at x=-1 although it's slope will change as it adjusts to the dimensions of the parabola. This shows that altering the dimensions p,q, and a will in fact cause a change in the slope.
If you were to just adjust q however withouth changing p and a, the slope will stay the same because you're just making a change along the y-plane.It's also important to remark that our vertex will maintain despite in how the slope changes.