The Problem
We are given an equation and a graph with a set of points. The points on the graph ( being the graphed function) can be defined as a set: . The given equation is: . The question is, what is the resultant points when applying the given transformations from the equation.
Approaching the problem
Before Thought
While it is possible to add an abstraction layer (and is used in my class), it is not necessary as there is reasoning behind these rules. Let’s solve this problem with pure algebra, and on the way we can see behind the curtains of this “abstraction layer.”
Problem Analysis
Before approaching the problem, we should first answer a few questions. For me, the most important being: What are we solving for?
We are solving for in the equation given. ( in particular)
It should also be noted that the given transformations are only horizontal transformations. This means that each point is only moved along the x-axis and it’s y component will not be changed.
Solving
Taking in one of the points and knowing that the function’s value will be the same, we can determine the following assumption to be true.
Why can we say this?
In this case we are looking at the first point in the function, . We are essentially saying that , the resultant transformed function, can be equal to . Since we know that the y value will not change ( in this case) then the value of when the two functions are equal will be the x value of the transformed point.
And now since we know that these two functions are equal, we can now state that
Further we can now solve for using basic algebra.
That means that the transformations applied to the first point is around
This can be done for all the rest of the points, and a tailor-made function can be created by replacing with something like . Where is the x value of the point that will be transformed.
Sorry if these notes seem rushed and not well explained, wrote this before going to bed.