If (-3,6) is a point on the graph of y= f(x), what point do you know is on the graph of y=f(x+3)?

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If `x=-6,` we have

`f(-6+3)=f(-3)=6,`

so`(-6,6)`is on the graph of`f(x+3).`

You might also know that the graph of `f(x+3)` is the same as the graph of `f(x)`shifted to theleft by 3 units. Thus if `(-3,6)` is on the graph of `f(x),``(-6,6)` must be...

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If `x=-6,` we have

`f(-6+3)=f(-3)=6,`

so `(-6,6)` is on the graph of  `f(x+3).`

You might also know that the graph of `f(x+3)` is the same as the graph of `f(x)` shifted to the left by 3 units. Thus if `(-3,6)` is on the graph of `f(x),` `(-6,6)` must be on the graph of `f(x+3).`

Here are the graphs of two functions to serve as an example. I picked `f(x)=-x+3`. The black graph is the graph of `f(x)` and the red graph is the graph of `f(x+3)=-(x+3)+3=-x.` Notice that `(-3,6)` is on the graph of `f(x)` and `(-6,6)` is on the graph of `f(x+3).`

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You should know that a point `(x_1,y_1) ` is on a graph of a function `y = f(x) if y_1 = f(x_1).`

Since the problem provides the information that (-3,6) is on the graph of the function `y = f(x)` , yields that `6 = f(-3)` .

Since the problem provides the information that `y = f(x+3)` and `f(x) = y` , yields:

`f(x + 3) = f(x) = y`

Substituting -3 for x and 6 for y, yields:

`f(-3 + 3) = f(-3) = 6 => f(0) = f(-3) = 6`

Hence, evaluating the coordinates of the point that belongs to the graph `y = f(x + 3)` , under the given conditions, yields `x = 0` and `y = 6` , thus, the point (0,6) is on the graph `y = f(x + 3).`

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