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)?`

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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.

**Sources:**

A good way to think about it for whenever you have to deal with it is:

g(x)=f(x-d)+c

Where d is a translation to the right or left

- If d is positive it goes to the right (x-(d)) = (x-d)
- If d is negative it goes to the left (x-(-d)) = (x+d)

and c is a translation up or down

- If c is positive then it translates up
- If c is negative then it translated down

Hope this helps you with your future work.

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