Vertical Shift 630862

1. **State the problem:** We are given two sinusoidal functions $g(x)$ and $f(x)$ where $g(x)$ is the blue graph shifted 8 units up from $f(x)$, which is the red graph. We want to express $g(x)$ in terms of $f(x)$ in at least two different ways. 2. **Understand the vertical shift:** A vertical shift of a function $f(x)$ by $k$ units up is represented as $f(x) + k$. 3. **First way to write $g(x)$:** Since $g(x)$ is $f(x)$ shifted 8 units up, we have $$g(x) = f(x) + 8$$ 4. **Second way to write $g(x)$:** We can also write $f(x)$ in terms of $g(x)$ by subtracting 8 units down: $$f(x) = g(x) - 8$$ Rearranging this to express $g(x)$ gives: $$g(x) = f(x) + 8$$ which is the same as the first expression but confirms the relationship. 5. **Alternative expression using function transformation notation:** Another way to express the vertical shift is by defining a new function $h(x)$ such that $$h(x) = f(x) + 8$$ and then $g(x) = h(x)$. **Final answer:** $$g(x) = f(x) + 8$$ or equivalently, $$g(x) = h(x) \text{ where } h(x) = f(x) + 8$$