If (4,2) is a point on the graph of y=f(x), which of the following points must be on the graph of y=f(2x).
a) (4,1)
b) (8,2)
c) (2,2)
d) (4,4)
If (4,2) is on the graph, then f(4) = 2. Let's replace the 4 with 2x and see if we can get that 2x to equal 4, since we know the f(4). Well, when x = 2, then 2x = 4. Therefore, f(2x) is the same as f(4), which we know = 2. So, when x = 2, then f(2x) = 2, and we have the point (2,2).
Well, well, well, let's think about this equation makeover, shall we?
We know that the point (4,2) lies on the graph of y = f(x). So, if we want to find a point on the graph of y = f(2x), we need to do a little substitution dance. We just have to replace the 'x' in the original equation with '2x'. Are you ready for some math magic?
Let's substitute x = 2x into the original equation y = f(x). We get y = f(2x). Ta-da!
Now, let's test each option to see which points satisfy y = f(2x):
a) (4,1): If we substitute x = 2, we get y = f(4). Hm, it doesn't match the equation f(2x). So, a) is not a valid point.
b) (8,2): If we substitute x = 4, we get y = f(8). Boom! It matches the equation f(2x). So, b) is a winner!
c) (2,2): If we substitute x = 1, we get y = f(2). Hm, it doesn't match the equation f(2x). So, c) is out of the game.
d) (4,4): If we substitute x = 2, we get y = f(4). D'oh! It's the same as option a), which we already ruled out. So, d) is a no-go!
In conclusion, the only point that must be on the graph of y = f(2x) is b) (8,2). It's like hitting the mathematical jackpot!
To find the point(s) that must be on the graph of y = f(2x), we need to substitute the x-coordinate of each point into the new function, f(2x).
Let's go through each option:
a) (4,1)
Substituting x = 4 into f(2x), we get f(2 * 4) = f(8). This point is not given in the question and cannot be determined with the given information.
b) (8,2)
Substituting x = 8 into f(2x), we get f(2 * 8) = f(16). This point is not given in the question and cannot be determined with the given information.
c) (2,2)
Substituting x = 2 into f(2x), we get f(2 * 2) = f(4). This point is given in the question, so it must be on the graph of y = f(2x).
d) (4,4)
Substituting x = 4 into f(2x), we get f(2 * 4) = f(8). This point is not given in the question and cannot be determined with the given information.
Therefore, the point (2,2) must be on the graph of y = f(2x).
The correct answer is c) (2,2).
To find the point(s) that must be on the graph of y = f(2x), we need to substitute the x-coordinate of each given point into the expression 2x and then find the corresponding y-coordinate.
Let's go through each option:
a) (4,1):
Substituting the x-coordinate, 4, into 2x gives us 2 * 4 = 8. So the corresponding point on the graph of y = f(2x) would be (8, y).
Answer a) does not have a matching y-coordinate of 1, so it is not on the graph of y = f(2x).
b) (8,2):
Substituting the x-coordinate, 8, into 2x gives us 2 * 8 = 16. So the corresponding point on the graph of y = f(2x) would be (16, y).
Answer b) does not have a matching y-coordinate of 2, so it is not on the graph of y = f(2x).
c) (2,2):
Substituting the x-coordinate, 2, into 2x gives us 2 * 2 = 4. So the corresponding point on the graph of y = f(2x) would be (4, y).
Answer c) has a matching y-coordinate of 2, so it is on the graph of y = f(2x).
d) (4,4):
Substituting the x-coordinate, 4, into 2x gives us 2 * 4 = 8. So the corresponding point on the graph of y = f(2x) would be (8, y).
Answer d) does not have a matching y-coordinate of 4, so it is not on the graph of y = f(2x).
Therefore, the point (2,2) must be on the graph of y = f(2x).
The correct answer is option c).
we know that 4 = f(2) = f(2*1)
so, (a)
we know nothing about f(4) or f(8)