The function `f(x)=x^2` . The graph of `g(x)` is `f(x)` translated to the left 5 units and down 3 units. What is the formula for `g(x)?`

Expert Answers

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To translate any graph to the left 5 units, we replace `x` with `x+5,` giving `y=(x+5)^2.` Now, to shift this graph down 3 units, we subtract 3, getting

`g(x)=(x+5)^2-3.`

The way I thought about this when first learning was to shift the graph of `f(x)` to the left or right (in other words, horizontally), it makes sense that we should do something to the input, which is represented on the horizontal ` `axis. Thus, something like `f(x+5)` makes sense. To shift a graph up or down (vertically), we should do something to the output, which is just `f(x).` Thus, `f(x)-3` makes sense. Now if we combine the two we get

`g(x)=f(x+5)-3=(x+5)^2-3,` just as we got above. Here's the graph of  `y=(x+5)^2-3` in red and `y=x^2` in black.

 

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