What does 2f(x) mean, in words?

[jaysquestions]

Also, what about f(2x)? I am confused about how to enter 2 multiplied by the function in a ti-84 calc. I am wondering if its because I am mixing up function notation. I am learning about transforming graphs.
What does 2f(x) mean in words? Also what does f(2x) mean in words? thanks

 

[mathman]

f(2x), value of function at point 2x.
2f(x), twice value of function at point x.

 

[symbolipoint]

Also, what about f(2x)? I am confused about how to enter 2 multiplied by the function in a ti-84 calc. I am wondering if its because I am mixing up function notation. I am learning about transforming graphs.
What does 2f(x) mean in words? Also what does f(2x) mean in words? thanks

2*f(x) means two multiplied by the function f.
f(2x) means the function at 2x; or the value of the function evaluated at 2x.

Giving a name f to a function for the function using independent variable x will be named as f(x), to be read, "the function f of x". Shown alone, f and x are not factors, but are a complete name.

 

[jaysquestions]

thanks for the answers , so how would I enter 2f(x) into a calc then? for example, f(x) = x^2, .. how do I enter two times the function into calc?

 

[jaysquestions]

The attached graph from textbook is what is confusing me. I don't understand the reason for the vertical shift?

 

[Mark44]

The attached graph from textbook is what is confusing me. I don't understand the reason for the vertical shift?

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.

The graph of y = f(x) + 3 shifts (or translates) all of the points on the graph of f by 3 units upward. For example, if (2, 5) is on the graph of f, then (2, 8) will be on the shifted, or translated, graph.

 

[jaysquestions]

Sorry I am still not understanding and thanks for the reply. Its figure 1.36 that is giving me problem. For example, if y = f(x) = x^2, how would the new function 2f(x) = x^2 transform the function downward, as the graph is showing? I think I am assuming something wrong about the graph but I can't see what my error is ? thank you

 

[SteamKing]

Sorry I am still not understanding and thanks for the reply. Its figure 1.36 that is giving me problem. For example, if y = f(x) = x^2, how would the new function 2f(x) = x^2 transform the function downward, as the graph is showing? I think I am assuming something wrong about the graph but I can't see what my error is ? thank you

Let's say that the function f(x) = -1 when x = 0. What is y = 2f(x) when x = 0? How would that look plotted on the same axes as y = f(x)?

 

[symbolipoint]

A factor on the function either STRETCHES or SHRINKS it vertically. This is not a movement of the function from one place to another place. The figures 1.36, 1.37, 1.38, 1.39, are the book's attempt to show this using examples.

 

[jaysquestions]

When it shrinks or stretches does the vertex stay in the same place though? For example, does f(x) = x^2 and 2f(x) = x^2 have the same vertex coordinates? thanks

 

[Mark44]

Sorry I am still not understanding and thanks for the reply. Its figure 1.36 that is giving me problem. For example, if y = f(x) = x^2, how would the new function 2f(x) = x^2 transform the function downward, as the graph is showing? I think I am assuming something wrong about the graph but I can't see what my error is ? thank you

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.

Also, if f(x) = x², then 2f(x) = 2x², not x² as you wrote.

When it shrinks or stretches does the vertex stay in the same place though? For example, does f(x) = x^2 and 2f(x) = x^2 have the same vertex coordinates? thanks

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.

 

[DrewD]

When it shrinks or stretches does the vertex stay in the same place though? For example, does f(x) = x^2 and 2f(x) = x^2 have the same vertex coordinates? thanks

The example given does not use $f(x)=x^2$. Let's assume that it uses $g(x)=x^2-1$. Then the vertex is at $(0,-1)$. If $2f(x)$ means that the $y$ value of each point is multiplied by two, would the vertex stay in the same place?

 

[jaysquestions]

The example given does not use $f(x)=x^2$. Let's assume that it uses $g(x)=x^2-1$. Then the vertex is at $(0,-1)$. If $2f(x)$ means that the $y$ value of each point is multiplied by two, would the vertex stay in the same place?

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.
thank you

 

[jaysquestions]

I want to make sure I am understanding the diff between 2f(x) and f(2x) using an example. So if, f(x) = x² - 3, am I correct in saying that:

1) 2f(x) = 2(x²-3)?
and
2) f(2x) = 2x²-3? or should this be = (2x²)-3?

thanks

 

[SteamKing]

I want to make sure I am understanding the diff between 2f(x) and f(2x) using an example. So if, f(x) = x² - 3, am I correct in saying that:

1) 2f(x) = 2(x²-3)?
and
2) f(2x) = 2x²-3? or should this be = (2x²)-3?

thanks

1) 2f(x) = 2(x²-3) is correct.

2) f(2x) = 2x²-3? or should this be = (2x²)-3?
Neither.

f(x) = x² - 3, therefore f(2x) = (2x)² - 3 = 4x² - 3

Look at it like this. Take a function f(t) = t² - 3 and set t = 2x.

 

[Svein]

2) f(2x) = 2x2-3? or should this be = (2x2)-3?

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]

Thanks for the replies , ..believe it or not I actually meant to say f(2x) = (2x)²-3, I just overlooked where the bracket started.
thanks everyone for the help