The discussion revolves around the meanings of the expressions 2f(x) and f(2x) in the context of function notation and graph transformations. Participants explore how these expressions relate to the values of functions and their graphical representations, particularly focusing on transformations such as stretching and shifting of graphs.
Participants do not reach a consensus on all points, particularly regarding the graphical interpretation of transformations and the effects on the vertex of functions. Some express confusion while others provide clarifications, leading to ongoing debate about the correct understanding of these concepts.
Participants mention specific figures from a textbook that may not be universally understood, indicating that the discussion is dependent on visual aids that are not present in the thread. There are also unresolved questions about the implications of transformations on the graphs of functions.
This discussion may be useful for students learning about function notation, graph transformations, and those seeking clarification on how to interpret and manipulate mathematical expressions involving functions.
2*f(x) means two multiplied by the function f.jaysquestions said:
Assuming that you have the graph of y = f(x), a new function y = 2*f(x) has y values that are exactly twice the value of those on the graph of y = f(x). This can be thought of as an expansion of the graph of f away from the x-axis to get y = 2*f(x). Don't think of this transformation as a shift, which is a transformation that rigidly moves all of the points on one graph by a set amount.jaysquestions said:
jaysquestions said:
You're misinterpreting the graph. Each y value on the graph of y = x2 is doubled to get the graph of y = 2f(x). Part of the black graph (y = f(x)) lies below the y-axis, so the points on the red graph (y = 2f(x)) are twice as far below the x-axis as those on the black graph. All points on the red graph are twice as far away from the x-axis as those on the black graph.jaysquestions said:
Again, 2f(x) would be 2x2, not x2. For y = x2, the vertex is at (0, 0), so doubling the y-value has no effect. The graph of y = 2x2 will also have (0, 0) as its vertex. All other points will be twice as far away from the x-axis as those on the graph of y = x2.jaysquestions said:
jaysquestions said:
OK I am getting this now. I was making two strange assumptions: 1) that the function represented in fig 1.36 as y=f(x) is f(x) = x^2. (it can't be because the vertex is well below the origin) and 2) I was thinking parabolas can't have vertexes below x-axis because of the x^2, but obviously they can be shifted, which is what fig 1.36 is and the whole point of all the examples. So in answer to your g(x) = x^2 -1 example, if we use 2f(x) then vertex would not stay in same place in your example, it would shift.DrewD said:
1) 2f(x) = 2(x2-3) is correct.jaysquestions said:
Neither. Let us take it step by step. First f(x)=x^{2}-3. This also means that f(u)=u^{2}-3 (we only changed the name of the variable, nothing else). So if we substitute 2x for u, we get f(2x)=(2x)^{2}-3.jaysquestions said: